M Rodrigues
email: rodrigues@lagep.univ-lyon1.fr

H Hammouri

C Mechmeche

N Benhadj Braiek

A High Gain Observer based LMI Approach

Keywords: Nonlinear systems, uniform observability, nonlinear observer, LMI

This paper deals with the observability analysis and the observer synthesis of a class of nonlinear systems. The observer that we consider has a constant gain and concerns a class of uniformly observable systems. A sufficient condition permitting to design such a constant gain observer is given. The calculation of this gain is based on LMI technics.

INTRODUCTION

In this paper, we propose sufficient conditions under which a controlled nonlinear system admits an exponential observer. The need to study the observer design problem for nonlinear dynamical systems is, from a control point of view, well understood by now.

The existence of an observer relies on observability properties which are a bit more involved than in the linear case. Many techniques have been developed for designing an observer for nonlinear systems. Among these techniques, a rather natural approach consists in considering systems which can be steered by a change of coordinates into state affine systems up to output injection. From observability point of view, these systems possess similar properties as linear systems and an extended Luenberger observer can be designed. Several authors [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF], [START_REF] Krener | Nonlinear observers with linealizable error dynamics[END_REF] [START_REF] Xia | Nonlinear observer design by observer canonical forms[END_REF], have characterized such nonlinear systems. An extension of these results consists to ask how one can design a constant gain observer of nonlinear systems (so-called an extended Luenberger observer). The idea consists to develop normal forms characterizing the class of nonlinear systems which are observable independently on the inputs (called uniformly observable systems). Based on these normal forms and the high gain technics, the authors in [START_REF] Hammouri | Sur la théorie de l'observation et des observateurs des systèmes non linéaires[END_REF], [START_REF] Bornard | A high gain observer for a class of uniformly observable systems[END_REF], [START_REF] Gauthier | A simple observer for nonlinear systems -application to bioreactors[END_REF], [START_REF] Gauthier | Observability and observers for nonlinear systems[END_REF] gave an extended Luenberger. Many extensions of these results to multi-output uniformly observable systems have been proposed (see for instance H. [START_REF] Armanet | High gain observer based on a triangular structure[END_REF], [START_REF] Hammouri | Nonlinear observer for locally uniformly observable systems[END_REF]).

Under a geometric condition (called uniform observability structure), the authors in [START_REF] Hammouri | Nonlinear observer for locally uniformly observable systems[END_REF] gave a normal form which extend those proposed in [START_REF] Hammouri | Sur la théorie de l'observation et des observateurs des systèmes non linéaires[END_REF], [START_REF] Gauthier | A simple observer for nonlinear systems -application to bioreactors[END_REF], [START_REF] Gauthier | Observability and observers for nonlinear systems[END_REF].

Based on this normal form, the authors in [START_REF] Hammouri | Nonlinear observer for locally uniformly observable systems[END_REF] propose a constant gain observer. The algorithm permitting to calculate the gain of the observer is based on the existence of some cone of matrices. The computation of this cone is very difficult to check in practical cases. Based on the LMI technics, our aim here consists to replace the above condition by a more simpler. Indeed, using LMI approach in this new observer gain design, gives a sufficient condition which is more easy to be verified and implemented.

The paper is organized as follows. The next section resumes some previous necessary results and gives the problem statement. In Section 2, a high gain observer is designed through LMI technics under a new condition. We end this paper by an illustrative example before concluding.

OBSERVER SYNTHESIS

The system that we consider [START_REF] Hammouri | Nonlinear observer for locally uniformly observable systems[END_REF]) has the following normal form:

ż = F (u, z) y = Cz (1)
where

F (u, z) =    F 1 (u, z) . . . F q (u, z)   , z =    z 1 . . . z q    ∈ R n ; u ∈ U a compact subset of R m ; z i ∈ R ni ; n 1 ≥ n 2 ≥ . . . ≥ n q ; n 1 + . . . + n q = n.
Each function F i (u, z), i = 1, . . . , q -1 satisfies the following triangular structure:

F i (u, z) = F i (u, z 1 , . . . , z i+1 ), z i ∈ R ni (2)
with the following rank condition:

Rank( ∂F i ∂z i+1 (u, z)) = n i+1 ∀z ∈ R n ; ∀u ∈ U (3)
Nonlinear systems that can be steered by a change of coordinates to the form (1), ( 2), ( 3) are those satisfying some geometrical condition (called a U -uniform observability structure [START_REF] Hammouri | Nonlinear observer for locally uniformly observable systems[END_REF]).

Definition 2.1. A constant gain exponential observer for system (1) is a dynamical system of the form:

ż = F (u, ẑ) + G(C ẑ -y) (4)
where G is a constant matrix such that: ẑ(t)z(t) ≤ λe -µt ẑ(0)z(0) where λ > 0 and µ > 0 are constants which do not depend on the input u ∈ L ∞ (R + , U ) nor on ẑ(0), z(0).

The existing results

In [START_REF] Hammouri | Nonlinear observer for locally uniformly observable systems[END_REF], the authors gave some sufficient conditions under which a constant gain observer was designed for system (1). These conditions may be formulated as follows:

H1) Global Lipschitz condition:

∃c > 0; ∀u ∈ U ; ∀z, z ′ ∈ R n , F (u, z) -F (u, z ′ ) ≤ c z -z ′ .
H2) The Cone condition:

∀k, 1 ≤ k ≤ q -1, there exists n k × n k+1 constant matrix S k,k+1 such that ∂F k ∂z k+1 (u, z) ∈ C(n k , n k+1 ; -1; S k,k+1 ); ∀(u, z) ∈ U × R n where C(n k , n k+1 ; -1) is the cone defined by {M ∈ M(n k , n k+1 ; R); s.t. M T S + S T M < αI k+1 }. M(n k , n k+1 ; R)
is the space of n k ×n k+1 real matrices and I k+1 is the (k + 1)× (k + 1) identity matrix.

Set Z = (Z 1 , . . . , Z q-1 ) ∈ R n × . . . × R n . Under the above hypotheses, the authors in [START_REF] Hammouri | Nonlinear observer for locally uniformly observable systems[END_REF],

show that there exists a symmetric positive definite matrix P and constants ρ > 0 and η > 0 such that for every (u, Z 1 , . . . , Z q-1 ) ∈ U × R n × . . . × R n , we have:

P A(u, Z) + A T (u, Z)P -ρC T C ≤ -ηI, where A(u, Z)=             0 A 1 (u, Z 1 ) 0 . . . 0 . . . 0 A 2 (u,
. . . A q-1 (u, Z q-1 ) 0 . . . . . . . . . . . . 0             with A k (u, Z k ) = ∂F k ∂Z k+1 k (u, Z k ).
Using this construction, the authors show that an exponential observer for system (1) takes the form:

ż = F (u, ẑ) + ∆ θ P -1 C T (C ẑ -y) (5)
where

∆ θ =      θI n1 0 . . . 0 0 θ 2 I n2 0 . . . . . . . . . 0 0 . . . 0 θ q I nq      , I n k is the n k × n k
identity matrix, k = 1, . . . , q and θ .

The drawback of the above observer's gain construction lies into the fact that the observer's gain depends on the matrices S k,k+1 and that their construction is a very difficult task.

In what follows, we will give a more simpler construction based on the LMI technics. As in the control problems, this new construction by LMI opens a new field of investigation where the observer gain may take into account of the parameter uncertainty for example. We present this new observer gain design in the following section.

HIGH GAIN OBSERVER BASED LMI TECHNICS

As in [START_REF] Hammouri | Nonlinear observer for locally uniformly observable systems[END_REF], the assumption H1) is maintained. However, assumption H2) will be reformulated as follows:

A(u, Z) is the matrix defined above, from H1), the set of matrices

E = {A(u, Z); (u, z) ∈ U × R n } is a bounded sub- set of M(n, n; R).

H3) Polytopic condition:

E is contained in a polytopic convex set P=Co{M 1 , . . . , M l } of M(n, n; R), where the M i 's are the vertices of P for which there exist a n×n symmetric positive definite matrix P and a n × p matrix W such that:

for 1 ≤ i ≤ l, (M i + KC) T P + P (M i + KC) < 0 (6) This inequality (6) can be linearized as follows for 1 ≤ i ≤ l, M T i P + P M i + C T W T + W C < 0 (7) where W = P K.

Remark 3.1. Since the polytopic set P is a compact convex set, it follows that there exists α > 0 such that:

for 1 ≤ i ≤ l, M T i P + P M i + C T W T + W C ≤ -αI (8) for every M ∈ P.

Theorem 3.1. Under hypotheses H1) and H3), an exponential observer for system (1) takes the form: ż = F (u, ẑ) + ∆ θ P -1 W (C ẑy) (9) where ∆ θ is the diagonal matrix defined above. The gain of the observer is K = P -1 W .

Proof 3.1. We will show that there exists θ 0 , such that ∀θ ≥ θ 0 , the error e(t) = ẑ(t)z(t) exponentially converges to 0,where z(t) is the unknown state and ẑ(t) is the its estimate given by ( 9).

Set F i (u, z, ẑ) = F i (u, ẑ1 , . . . , ẑi , z i+1 ) for 1 ≤ i ≤ q -1; F q (u, z, ẑ) = F q (u, ẑ) and, F (u, z, ẑ) =    F 1 (u, z, ẑ) . . . F q (u, z, ẑ)   ,
we obtain:

ė = (F (u, ẑ) -F (u, z, ẑ)) + ( F (u, z, ẑ) -F (u, z)) +∆ θ P -1 W (C ẑ -y) (10)
To show that e(t) exponentially converges to 0, it suffices to show that ǫ(t) = ∆ -1 θ e(t) exponentially converges to 0.

From (10), we deduce:

ǫ = ∆ -1 θ (F (u, ẑ) -F (u, z, ẑ)) + ∆ -1 θ ( F (u, z, ẑ) -F (u, z)) +P -1 W (C ẑ -y) (11)
Using the expression of F (u, z, ẑ) and the fact that F (u, z) has a triangular structure, the main value theorem yields to:

F (u, ẑ) -F (u, z, ẑ) = A(u, Z)e,
where Z = (Z 1 , . . . , Z q-1 ) and A(u, Z) is the upper diagonal matrix given above. Moreover the k th bloc of A(u, Z)

takes of the form A k (u, Z k ) = ∂F k ∂z k+1 (u, Z 1 k , . . . , Z k+1 k ),
where Z i k (t) = z i (t)+Λ i (t)e i (t) and Λ is a diagonal matrix hose coefficients are in [0, 1].

Using the structure of A(u, Z) and the fact that C = (I n1 , 0, . . . , 0), a simple calculation yields to:

∆ -1 θ (F (u, ẑ)-F (u, z, ẑ)) = ∆ -1 θ A(u, Z)e = θA(u, Z)ǫ and W C∆ θ = θW C. Hence (11) becomes: ǫ = θ(A(u, Z) + P -1 W C)ǫ + ∆ -1 θ ( F (u, z, ẑ) -F (u, z)) (12)
To end the proof of the theorem, it suffices to show that V (t) = ǫ T (t)P ǫ(t) exponentially converges to 0. A simple calculation yields to:

V (t) = θǫ T (t)((A(u, Z) + P -1 W C) T P + P (A(u, Z) +P -1 W C))ǫ(t) +2ǫ T (t)P ∆ -1 θ ( F (u, z, ẑ) -F (u, z)) (13)
From hypothesis H3) and remark 3.1, we deduce:

V (t) ≤ -αθ ǫ(t) 2 + 2ǫ T (t)P ∆ -1 θ ( F (u, z, ẑ) -F (u, z)) (14)
Using the triangular structure of F (u, z) and ∆ theta , we get:

∆ -1 θ ( F (u, z, ẑ) -F (u, z)) =         θ -1 (F 1 (u, z 1 , z 2 ) -F 1 (u, z 1 , z 2 )) . . . θ -k (F k (u, z 1 , . . . , z k , z k+1 ) -F 1 (u, z 1 , . . . , z k , z k+1 )) . . . θ -q (F q (u, z 1 , . . . , z q-1 , z q ) -F 1 (u, z 1 , . . . , z q-1 , z q ))        
From assumption H1):

F k (u, z 1 , . . . , z k , z k+1 ) -F 1 (u, z 1 , . . . , z k , z k+1 ) ≤ c
e 1 2 + . . . + e k 2 . Thus, for θ ≥ 1, we have θ -k F k (u, z 1 , . . . , z k , z k+1 ) -F 1 (u, z 1 , . . . , z k , z k+1 ) ≤λ e 1 2 + . . . + e k 2 , for some constant λ which doesn't depend on θ.

Combining this last fact with (15), we obtain:

V (t) ≤ -αθ ǫ(t) 2 + β ǫ(t) 2 (
15) where β is a constant which doesn't depend on θ.

To end the proof, it suffices to take θ > β α .

COMMENTS AND EXAMPLES

In this section, we will give some remarks and examples concerning the existence of a constant gain observer.

Remark 4.1. Assumption H1) and conditions (2), (3) are not generally sufficient for the existence of a constant gain observer for system (1).

Indeed, consider the following example:

           ż1 = u 1 z 3 ż2 = u 2 z 3 ż3 = 0 y = Cz = z 1 z 2 (16)
where u = (u 1 , u 2 ) belongs to the unit circle U = {u s.t. u = 1}.

Clearly system (16) takes the form (1) and satisfying (2), (3) and assumption H1). Let us show that system (16) doesn't admit a constant gain observer.

Assuming that there exists a constant gain observer of the form:

ż = A(u)ẑ + K(C ẑ -y) (17)
where,

A(u) = 0 0 u 1 0 0 u 2 0 0 0 , K = k 11 k 12 k 21 k 22 k 31 k 32
is a constant matrix and C = 1 0 0 0 1 0 .

Thus, for every u ∈ L ∞ (R + , U ), the error equation:

ė = (A(u) + KC)e (18)
is exponentially stable at the origin.

In particular, the error equations associated to inputs u(t) = (1, 0) and u(t) = (-1, 0) are exponentially stable. This implies that: In what follows, we will give an example which illustrate our LMI method. Let consider the following system with u(t) ∈ [0, 1]:

k 11 k
       ż1 = A 1 (u)z 2 ż2 = A 2 (u)z 3 ż3 = 0 y = Cz = z 1 (19)
with the following matrices

ż = A(u)z y = Cz = z 1 (20)
where,

A(u) =      0 0 1 u 0 0 0 -u 1 0 0 0 0 0 1 -u 0 0 0 0 u 0 0 0 0 0      , C = 1 0 0 0 0 0 1 0 0 0 (21)
The vertices of the polytope are A(0) and A(1).

We solve LMI from equation ( 7), with K = P -1 W , and we obtain: 33.54 2.71 -12.74 -4.78 -19.24 2.71 28.51 4.40 -13.68 -8.13 -12.74 4.40 27.78 6.01 -14.95 -4.78 -13.68 6.01 35.32 -16.14 -19.24 -8.13 -14.95 -16.14

P =     

CONCLUSION

In this paper, an observer synthesis of a class of nonlinear systems is presented. The paper focusses on the design of a high gain observer by LMI technics which allow more easier solutions to be verified. A short example illustrates the developed technic by the design of a constant gain.

In future works and as in the control problems, this new observer gain design may take into account of the parameter uncertainty which should be more easy by the use of LMI.

  Z 2 ) 0 . . .

						. . .
	. . .	. . .	. . .	. . .	. . .	. . .
	. . .	. . .	. . .	. . .	. . .	0
	. . .	. . .	. . .	. . .

  21 k 32 -k 31 k 22 < 0 and k 21 k 32 -k 31 k 22 > 0.

	12 1		k 11 k 12 -1
	k 21 k 22 0	and	k 21 k 22 0	are both Hurwitz ma-
	k 31 k 32 0		k 31 k 32 0
	trices.
	A simple calculation shows that this yields to the following
	contradiction: k


Christopher Gray

Meghdad Khazaee

Chris Francois

Hans Manner

Hans Manner Fully

Fully Automated, Physics-based Analysis of SCADA Temperature Signals for Drivetrain Monitoring

1. A Practical Approach to the Use of SCADA Data for Optimized Wind Turbine Condition Based Maintenance, Gray et al 2. Blundell, Stephen J., and Katherine M. Blundell. Concepts in thermal physics. Oxford University Press on Demand, 2010. 3. Schlechtingen, Meik, and Ilmar Ferreira Santos. "Comparative analysis of neural network and regression based condition monitoring approaches for wind turbine fault detection." Mechanical systems and signal processing 25, no. 5 (2011): 1849-1875. 4. Zhang, Zhen-You, and Ke-Sheng Wang. "Wind turbine fault detection based on SCADA data analysis using ANN." Advances in Manufacturing 2, no. 1 (2014): 70-78. 5. Zaher, A. S. A. E., S. D. J. McArthur, D. G. Infield, and Y. Patel. " ▪ Develop simple analytical models to describe the relationship between wind turbine operating conditions, environmental conditions and component temperature. ▪ Demonstrate how such models can be used to accurately detect incipient faults in critical components, triggering proactive inspection and repair. ▪ Achieve a system that can be used to analysis large fleets of wind turbines with full automation.

It is well understood that SCADA data can be used as an input to condition monitoring algorithms, and in particular the analysis of trends in temperature signals has been shown as an effective means of early fault detection, allowing proactive inspection and maintenance to be planned.

Due to the dynamic operating characteristics of a wind turbine, including variations in ambient conditions as well as the turbine control, the thermal response of the individual components is relatively complex. Accurate fault detection therefore requires simulation of the turbine in its healthy operational state, and then comparison with measured temperatures in order to identify outliers and deviations.

Previous work has shown that machine learning techniques such as neural networks can be used to simulate the relationship between input variables and a response variable. However, such techniques suffer from some limitations including lack of transparency, the requirement for training sets covering prolonged periods and over-fitting. This work aims to demonstrate that physics-based models can be used to model the thermal response of components inside the wind turbine with full transparency, low complexity and high accuracy. Using this approach, a number of faults have been detected across a wind turbine portfolio, including gearbox, main bearing and generator damage.

The method was embedded in a software application developed by i4SEE TECH and connected to a centralised SCADA system managing data continuously for a portfolio of over 500MW of wind turbines operated by CGN Europe Energy. By automating the entire process of data acquisition, model training, analysis, outlier detection and fault diagnosis, it was possible to detect a range of different faults.

In some cases, these faults related to temperature sensor errors or problems with the turbine cooling system. Other cases related to failure in large mechanical components such as the main bearing or generator bearings. Depending on the specific details of the failure mechanism, it was shown that faults can be detected up to 12 months in advance.

Results were fed directly into a digital tracking tool in order to record resulting inspection and service activities. Physics-based models were used to simulate the thermal behaviour of various components within the wind turbine, with specific focus on drivetrain components including the main bearing and generator bearings. The models describe the relationship between heat input and temperature response of the system components using coefficients to describe heat transfer rates, heat coefficients and thermal inertia. Referring to the sketch, SCADA variables P 1 and P 2 were used to calculate the heat input to the system. The heat loss was calculated as a function of the temperature difference between the body and its environment. Finally, the rate of change of body temperature was calculated using a thermal inertia coefficient, calculated for each individual turbine. All coefficients were trained using reference datasets from relatively short time periods, and then used to predict future temperature response of the same components. Prediction and measurement were compared to produce residual values which in turn were used to detect the presence of component faults.

Abstract

Results

Objectives

Conclusions Methods

Due to the high level of automation achieved and the use of available SCADA logs accessed via existing data infrastructure, a scalable and costeffective solution has been demonstrated. This has been used to regularly monitor a large global fleet of wind turbines. The capability for detecting a range of faults from critical mechanical systems in the wind turbine can contribute significantly to improvements in overall fleet reliability and availability, reduction in costs for service and related logistics.

Fully

  i4SEE TECH GmbH, Austria (2) CGN Europe Energy, France (3) University of Graz, Austria PO.218 Visit our homepage at i4see.com Replace with QR code Replace with QR code


Christopher Gray

Marko Vulikic

Phillip Elflein

Hans Manner

Component Life Assessment Using Incomplete Failure Datasets

1. A Review of Accelerated Test Models", L. A. Escobar, W. Q.Meeker, Statist. Sci. Volume 21, 552-577, Number 4 (2006) 2. Reliability performance and maintenance-a survey of failures in wind power systems. Ribrant J. , KTH, Stockholm, Sweden, 2006 i4SEE TECH GmbH would like to express thanks to the Österreichische Forschungsfördergesellschaft (FFG) and Austrian Wirtschaftsservice (aws) for supporting this work.

▪ Demonstrate how relatively small sets of failure data can be used to predict long term failure rates with defined levels of uncertainty. ▪ Integrate the results into a logistics management system to allow an operator to project spare parts requirements and financial risk Premature failure of the main wind turbine sub-systems such as generators, gearboxes or transformers can result in prolonged downtime, production losses and expensive repairs. The implications of such failures for fleet owners and operators can become extremely significant, as the overall financial liability is potentially high. Decisions concerning the choice of service contract, preferred service provider, level of insurance, required spare parts and tooling, or whether to extend the operational life of the turbine are all strongly influenced by expectations of the future reliability of the turbines. Therefore, the capability to project failure rates over relatively long time periods ranging from one to ten years becomes extremely important. Classical reliability theory is readily available for addressing such problems, including well proven techniques such as MTTF (Mean-Time-To-Failure) assessment and Weibull modelling. However, the practical application of such techniques is often limited due to the high effort required to prepare well classified historical failure datasets, or even a complete absence of such failure data.

In this poster we demonstrate a method that has been developed to deals with such limitations in a pragmatic manner, allowing component failure risk to be calculated with quantified level of uncertainty. The method has been successfully implemented for an aging fleet of around 100 turbines and the results are used to provide decision support for centralised maintenance management.

By relaxing typically strict requirements concerning taxonomy and failure mode classification, a solution was developed that could be deployed productively within a relatively short timescale. Limitations in the quality and quantity of available input data were accepted and considered in failure rate predictions, by stating the confidence intervals associated with all results.

As input data quality improves over time, these confidence intervals improve incrementally. Furthermore, as understanding of specific failure mechanisms improve, the data classification may be progressively updated for increasing results accuracy.

The resulting methodology has been embedded in an operational software system and is used to provide fleet level reliability projections that are updated on a monthly basis. This provides the fleet operator with valuable intelligence that is used for strategical decision support.

Abstract Results

Objectives

Conclusions Methods

Component Life Assessment Using Incomplete Failure Datasets

Christopher Gray 1 , Marko Vulikic 1 , Phillip Elflein 2 , Prof. Hans Manner 3

(1) i4SEE TECH GmbH, Austria (2) EnergieKontor, Germany (3) University of Graz, Austria

PO.248

Visit our homepage at i4see.com

Replace with QR code Replace with QR code Master data was gathered for a portfolio of around 100 turbines, all from the same manufacturer and type. Priority systems were selected including gearbox, generator, transformer, blades and blade bearings. Service records for all turbines were examined and a database was compiled including all major failure events leading to complete repair or replacement of the priority systems. A statistical model was then developed capable of calculating the overall failure probability for each priority system, for a time period of up to ten years into the future. The method allows taxonomy to be defined with varying levels of detail, is tolerant to small input datasets and is adaptive, increasing in accuracy with the addition of any future failure events.

Wind turbine failure rate analysis and reliability projection has been the subject of research for many years in the wind industry, with several well know large scale funded projects aiming to advance the capability in this field. However, a fundamental barrier has always been a lack of high quality input data. Nevertheless, the strategical and financial importance of this subject area is extremely high, and a pragmatic application of fundamental reliability theory can lead to huge improvements in O&M strategy and significant cost savings at many levels of the value chain. The developed approach is tolerant to imperfections in data quality, can be rolled out quickly and with low cost, and has been proven to provide immediate value in a practical use case.


Aurélia Pernin

Florence Dubois-Brissonnet
email: florence.dubois@agroparistech.fr

Stéphanie Roux

Marine Masson

Véronique Bosc

Marie-Noëlle Maillard

Phenolic compounds can delay the oxidation of polyunsaturated fatty acids and the growth of Listeria monocytogenes: structure-activity relationships

Keywords: phenolic compounds, antioxidant, AAPH, antibacterial, Listeria monocytogenes, structure-activity

BACKGROUND: Phenolic compounds present a potential solution to ensuring food quality and safety. Indeed, they can limit oxidation reactions and bacterial growth in food products. Although their antioxidant mechanisms of action are well known, their antibacterial ones are less well understood, especially in light of their chemical structures.

The aim of this study was to first quantify both aspects of a series of natural phenolic compounds and then link these activities to their chemical structure.

RESULTS: We evaluated antioxidant activity by measuring the capacity of phenolic compounds to delay free linoleic acid oxidation caused by the action of a hydrophilic azoradical initiator (AAPH). We evaluated antibacterial activity by measuring the growth inhibition of Listeria monocytogenes and determining the non-inhibitory and minimum inhibitory concentrations for each compound. Compounds with ortho-diphenolic structures were the best antioxidants, whereas those belonging to the simple phenol category were the best antibacterial compounds.

CONCLUSION: The physico-chemical properties of the compounds influenced both activities, but not in the same way. The chemical environment of the phenolic group and the presence of delocalization structures are the most important parameters for antioxidant activity, whereas the partition coefficient logP is one of the most important factors involved in antibacterial activity.

INTRODUCTION

Over the last decade, the fatty acids' profile of food products has been modified to respect nutritional recommendations. The proportion of saturated fatty acids has considerably decreased in favor of monounsaturated (MUFA) and polyunsaturated (PUFA) fatty acids, in particular those of the omega-3 family. [START_REF] Inra | Etude d'impacts des chartes d'engagements volontaires de progrès nutritionnel sur les volumes du nutriments mis sur le marché: étude actualisée[END_REF] However, these changes can limit food quality and safety. Indeed, the increase in the number of double bonds in fatty acids in food products leads to their higher oxidability during processing and storage. Oxidation reactions produce off-flavors due to the generation of volatile compounds, decrease nutritional quality by the partial loss of essential fatty acids and co-oxidation of vitamins, and increase the chemical risk by the generation of potentially toxic compounds during heating (cyclic monomers, polymers, furans, 4-hydroxynonenal). [START_REF] Jacobsen | Antioxidant strategies for preventing oxidative flavour deterioration of foods enriched with n-3 polyunsaturated lipids: a comparative evaluation[END_REF][START_REF] Cuvelier | Stabilité des huiles alimentaires au cours de leur stockage[END_REF] Moreover, recent studies have shown that the growth of pathogenic Gram-positive microorganisms, such as Staphylococcus aureus or Bacillus cereus, may be favored in the presence of exogenous unsaturated fatty acids (UFA). The mechanisms are not fully understood, but it has been reported that these bacteria can incorporate exogenous UFA in their membranes to regulate their fluidity when fatty acid synthesis is blocked by an FSAII inhibitor [START_REF] Brinster | Essentiality of FASII pathway for Staphylococcus aureus[END_REF] or impaired by low temperatures. [START_REF] De Sarrau | Unsaturated fatty acids from food and in the growth medium improve growth of Bacillus cereus under cold and anaerobic conditions[END_REF] Under these conditions, pathogenic microorganisms can grow and reach infectious doses faster than expected, causing potential food safety issues in food products.

One solution to limit these phenomena is to use molecules with both antioxidant and antibacterial properties, such as phenolic compounds, which are widely found in plants or present in by-products from agro-industry. Several studies have reported the antioxidant and antibacterial properties of phenolic compounds. The antioxidant mechanisms of phenolic compounds are quite well understood and are based on the ability of phenolic groups to release an electron or a hydrogen to a lipid radical. The phenolic radical formed is more stable than the lipid radical, due to the presence of delocalization structures. [START_REF] Berset | Antioxydants phénoliques -Structure, propriétés, sources végétales[END_REF] Fewer studies have focused on the antibacterial activity of phenolic compounds. Solely the mechanism of action of simple phenols is well described. They act by partitioning of the molecule into the membrane phospholipid layers, inducing the leakage of cell components and bacterial death. [START_REF] Burt | Essential oils: their antibacterial properties and potential applications in foods-a review[END_REF] The antibacterial activity of several phenolic acids and flavonoids has already been reported, but their mechanism of action is not fully understood. [START_REF] Daglia | Polyphenols as antimicrobial agents[END_REF] We aimed to quantify both antioxidant and antibacterial activities of a series of natural phenolic compounds and investigate structure-activity relationships to determine their potential to ensure food quality and safety. Molecules were selected based on their antioxidant and/or antibacterial activity reported in published studies [START_REF] Berset | Antioxydants phénoliques -Structure, propriétés, sources végétales[END_REF][START_REF] Burt | Essential oils: their antibacterial properties and potential applications in foods-a review[END_REF][START_REF] Kancheva | Lipid Oxidation in Homogeneous and Micro-Heterogeneous Media in Presence of Prooxidants, Antioxidants and Surfactants[END_REF] and their structures, which covered a broad range of chemical families, making it possible to study the role of specific chemical groups in their effectiveness. Listeria monocytogenes was used as the bacterial model because it is a ubiquitous Gram-positive pathogen which is widespread in the environment and a potential contaminant at all stages of the food chain. [START_REF] Buchanan | A review of Listeria monocytogenes: An update on outbreaks, virulence, dose-response, ecology, and risk assessments[END_REF] We tested the ability to delay oxidation with linoleic acid, because it is highly reactive and representative of PUFAs found in most food products.

EXPERIMENTAL CHEMICAL REAGENTS

2,2'-azobis(2-amidinopropane) dichloride (AAPH), linoleic acid (purity > 98%), sodium dihydrogen phosphate dihydrate, sodium phosphate dibasic dodecahydrate, and sodium borate were purchased from Sigma-Aldrich (St Quentin Fallavier, France). Absolute ethanol was purchased from VWR (Fontenay-sous-Bois, France), boric acid from Prolabo (Fontenay-sous-Bois, France), 1 mol.L -1 sodium hydroxide and acetone from Carlo Erba (Fontenay-aux-Roses, France), and Tween 20 from Sigma-Aldrich (St Quentin Fallavier, France).

BACTERIAL STRAIN AND GROWTH CONDITIONS

The strain used in this study was Listeria monocytogenes CNL 895805, serotype ½, isolated from sheep's brain. It was graciously provided by P. Velge (INRA, Nouzilly). [START_REF] Van Langendonck | Tissue culture assays using Caco-2 cell line differentiate virulent from non-virulent Listeria monocytogenes strains[END_REF] The strain, stored in cryovials at -80°C, was resuscitated in two successive subcultures in tryptic soy broth (TSB, Biomérieux, France) at 30°C before each experiment.

PHENOLIC COMPOUNDS

p-hydroxybenzoic acid, protocatechuic acid, vanillic acid, gallic acid, p-coumaric acid, caffeic acid, ferulic acid, chlorogenic acid, thymol, α-tocopherol, (-)-epicatechin, naringenin, kaempferol, and quercetin were purchased from Sigma-Aldrich (St Quentin Fallavier, France). Eugenol was purchased from Janssen (Beerse, Belgium), carvacrol from Fluka (Buchs, Switzerland), and eriodictyol, apigenin, luteolin, and rutin from Extrasynthèse (Genay, France). Chemical structures are shown Figure 1. The partition coefficients (logPs) of the compounds were found on the ChemSpider website. The predicted logPo/w from ACD/Labs Percepta Platform PhysChem Module were used.

For the evaluation of antioxidant activity, stock solutions of phenolic compounds were prepared by dissolution in absolute ethanol. For the evaluation of antibacterial activity, phenolic compounds were prepared according to various protocols we elaborated, depending on their solubility in the culture medium. p-hydroxybenzoic, protocatechuic, gallic, and chlorogenic acid powders were directly dissolved in the culture medium.

Vanillic, p-coumaric and ferulic acids, carvacrol, thymol, and eugenol were first dissolved in acetone, which was evaporated under nitrogen flow after addition to the culture medium. Caffeic acid, (-)-epicatechin, naringenin, and eriodictyol were first dissolved in acetone/distilled water (80/20 (v/v)) and the acetone evaporated under nitrogen flow. Bacterial growth controls were carried out to ensure the absence of inhibitory activity of trace amounts of acetone after evaporation.

EVALUATION OF ANTIOXIDANT ACTIVITY

Antioxidant activity was evaluated according to a method previously described by Peyrat-Maillard et al. [START_REF] Peyrat-Maillard | Antioxidant Activity of Phenolic Compounds in 2,2′-Azobis (2-amidinopropane) Dihydrochloride (AAPH)-Induced Oxidation : Synergistic and Antagonistic Effects[END_REF] with minor modifications. The test involves a multiphasic system and consists of measuring the capacity of phenolic compounds to delay the oxidation of linoleic acid. Conjugated diene hydroperoxides were generated by the action of a hydrophilic diazo initiator (AAPH, 2,2'-azobis (2-amidinopropane) dihydrochloride) on free linoleic acid and were followed by spectrophotometry at 234 nm [START_REF] Corongiu | Detection of conjugated dienes by second derivative ultraviolet spectrophotometry[END_REF] with a doublebeam spectrophotometer (Spectrophotometer Specord 210 Plus, Analytik Jena, Germany).

A mixture containing 2.81 mL phosphate buffer (0.05 mol.L -1 , pH 7.4), 30 µL substrate solution (0.016 mol.L -1 98% linoleic acid, 4.89 mmol.L -1 Tween 20, 20 mmol.L -1 sodium hydroxide in borate buffer (0.05 mol.L -1 , pH 9.0)), and 10 µL freshly prepared ethanolic solution of the phenolic compound were added in a spectrophotometric cuvette and incubated at 37°C for 5 min. At least five concentrations of antioxidant were tested from 0 to 9 µmol.L -1 for (-)-epicatechin, to 10 µmol.L -1 for eriodictyol, apigenin, luteolin, kaempferol, quercetin, and rutin, 12 µmol.L -1 for caffeic acid and chlorogenic acid, 20 µmol.L -1 for protocatechuic acid, ferulic acid, and α-tocopherol, 30 µmol.L -1 for eugenol and naringenin, 35 µmol.L -1 for gallic acid, 40 µmol.L -1 for vanillic acid, 50 µmol.L -1 for p-hydroxybenzoic acid and p-coumaric acid, and 70 µmol.L -1 for carvacrol and thymol.

The maximum concentrations tested were chosen to reach equivalent antioxidant effectiveness between compounds or their solubility threshold. At least three independent experiments were conducted for each phenolic compound. Then, 150 µL AAPH solution (0.04 mol.L -1 in phosphate buffer 0.05 mol.L -1 , pH 7.4) was added. The absorbance was measured at 234 nm for at least 5 h vs a control containing the same solution, except the substrate, which was replaced by 30 µL borate buffer (0.05 mol.L -1 , pH 9.0).

The antioxidant potential of a phenolic compound is defined by its effective concentration to delay oxidation for one hour (EC1h), relative to a control without antioxidant, and is expressed in µmol.L -1 .h -1 : the lower this value, the stronger the antioxidant. The EC1h was determined by first plotting the concentration of conjugated dienes (C=A234nm/(l.ε), with l = 1 cm and ε =27 000 L.mol -1 .cm -1 ) [START_REF] Corongiu | Detection of conjugated dienes by second derivative ultraviolet spectrophotometry[END_REF] as a function of time for several concentrations of a given phenolic compound. Then, the time to reach a given concentration of conjugated dienes (arbitrarily chosen to be 18.5 µmol.L -1 ) was determined (tCi) for each concentration of antioxidant. The difference between this time (tCi) and the time necessary to reach the same concentration with the control (tC0) was calculated (ΔtCi = tCi -tC0). ΔtCi was then plotted as a linear function of the concentration of the antioxidant and the slope calculated.

The effective concentration to delay oxidation for one hour (EC1h) corresponds to the inverse of this slope. For statistical purposes, the delayed oxidation times (ΔtCi = tCi -tC0) were directly used. An analysis of covariance (ANCOVA) between the concentrationdependent effect of the phenolic compound and a phenolic compound*concentration interaction was applied using XLSTAT 18.06 (Addinsoft, Paris, France). If significant effects were revealed (p < 0.05), an estimated mean for the delayed oxidation time was calculated and compared using the Newman Keuls multiple comparison test (confidence interval 95%) to define groups of phenolic compounds with similar antioxidant behavior.

EVALUATION OF ANTIBACTERIAL ACTIVITY soy broth (TSB) medium using a method previously described with minor modifications. [START_REF] Guillier | Growth response of Salmonella Typhimurium in the presence of natural and synthetic antimicrobials: estimation of MICs from three different models[END_REF] Bacterial growth was followed in an automatic spectrophotometer (Bioscreen C, Labsystems, Helsinki, Finland) by measuring Optical Density (OD) at 600 nm for 72 h in two 100-well microplates. Two hundred microliters of TSB pH 7.2, with or without phenolic compounds, was added to each well. The final concentrations of the phenolic compounds in the wells varied from 0 to 2.5 mmol.L -1 for carvacrol and thymol, to 2.6 mmol.L -1 for eriodictyol, 2.7 mmol.L -1 for naringenin, 8 mmol.L -1 for eugenol, 9.4 mmol.L -1 for (-)-epicatechin, 20 mmol.L -1 for p-coumaric acid, 30 mmol.L -1 for phydroxybenzoic, protocatechuic, vanillic, caffeic, and ferulic acids, 35 mmol.L -1 for chlorogenic acid, and 36 mmol.L -1 for gallic acid. The maximum concentrations tested were chosen to obtain total growth inhibition or the solubility threshold. Five to 40 concentrations, prepared from at least two different solutions, were tested per compound.

Each well was inoculated with a standardized inoculum at 1% (v/v from the second subculture: approximately 10 6 CFU.mL -1 ) and the microplates incubated at 30°C with slow and continuous shaking. At least two growth curves were acquired for each concentration. The pH values of the TSB after the addition of various concentrations of phenolic compounds were measured with a pH-meter (SI Analytics lab 870, Mainz, Germany) in an independent experiment. The inhibitory activity is described by the minimum inhibitory concentration (MIC) and non-inhibitory concentration (NIC). The MIC is the concentration at which no bacterial growth is recorded and the NIC that below which the compound has no inhibitory activity. These values are expressed in mmol.L -1 : the lower the MIC, the stronger the antibacterial effect. These values were obtained after two modeling steps. First, the maximum specific growth rates (µmax) were estimated from the growth kinetics by fitting the modified Gompertz model. [START_REF] Guillier | Growth response of Salmonella Typhimurium in the presence of natural and synthetic antimicrobials: estimation of MICs from three different models[END_REF] Second, the MIC and NIC were determined for each phenolic compound by plotting growth rates transformed by square-root as a function of the concentration and modeling µmax with the Lambert Pearson model. [START_REF] Lambert | Susceptibility testing: accurate and reproducible minimum inhibitory concentration (MIC) and non-inhibitory concentration (NIC) values[END_REF] The solver of Microsoft Excel ® (2013) was used to minimize the sum of squares and allowed estimation of the model parameters. The standard deviations (SD) of model parameters and sums of squares were calculated with SolverAid, a complementary macro. [START_REF] De Levie | Advanced Excel for scientific data analysis[END_REF]

RESULTS AND DISCUSSION

CHARACTERIZATION OF ANTIOXIDANT BEHAVIOR

We evaluated the antioxidant activity of the selected phenolic compounds by measuring their capacity to delay free linoleic acid oxidation (Table 1). Many of the tested molecules had similar antioxidant behavior to that of the well-known food antioxidant α-tocopherol, which also has a phenolic group. Phenolic compounds belonging to flavan-3-ols, flavonols, and hydroxycinnamic derivatives, with an ortho-diphenolic structure, had the highest antioxidant potential. The results highlight the strong relationship between chemical structure and antioxidant activity. The model obtained by ANCOVA was significant (F39,225 = 331.3, p < 0.0001). The concentration effect was significant (p < 0.0001), with the delayed oxidation time (ΔtCi = tCi -tC0) increasing proportionally with the concentration of the compound, whereas the phenol effect was not significant (p = 0.310), the intercept being equal for all phenolic compounds. The interaction effect of the phenolic compound*concentration was also significant: the slope of the regression curve was associated with the nature of the phenolic compound.

We observed three different kinetic behaviors when the production of conjugated dienes was measured during linoleic acid oxidation induced by the azo-initiator AAPH. Kaempferol, vanillic acid, and protocatechuic acid production illustrate the three kinetic behaviors (Figures 2a, 2b and2c), each including various phenolic families. Several studies have reported such kinetic behaviors, but the corresponding mechanisms have not been completely elucidated. [START_REF] Foti | Antioxidant properties of phenols[END_REF][START_REF] Laguerre | Evaluation of the ability of antioxidants to counteract lipid oxidation: Existing methods, new trends and challenges[END_REF][START_REF] Roche | Antioxidant activity of olive phenols: mechanistic investigation and characterization of oxidation products by mass spectrometry[END_REF] The first group of antioxidants belonged to chain breakers [START_REF] Roche | Antioxidant activity of olive phenols: mechanistic investigation and characterization of oxidation products by mass spectrometry[END_REF] and included kaempferol (Figure 2a), caffeic acid, chlorogenic acid, gallic acid, α-tocopherol, luteolin, quercetin, and rutin. The second group of antioxidants, considered to be retarders, [START_REF] Roche | Antioxidant activity of olive phenols: mechanistic investigation and characterization of oxidation products by mass spectrometry[END_REF] was composed of vanillic acid (Figure 2b), p-coumaric acid, p-hydroxybenzoic acid, apigenin, carvacrol, naringenin, and thymol. The third group of antioxidants, including protocatechuic acid (Figure 2c), ferulic acid, (-)-epicatechin, eriodictyol, and eugenol, showed intermediate behavior. Chain-breakers induce a lagtime phase, during which the lipidic substrate is not oxidized until complete consumption of the antioxidant, but is then oxidized with a peroxidation rate reaching the same value as the control without antioxidant. [START_REF] Foti | Antioxidant properties of phenols[END_REF] Conversely, retarders reduce the peroxidation rate [START_REF] Laguerre | Evaluation of the ability of antioxidants to counteract lipid oxidation: Existing methods, new trends and challenges[END_REF] without inducing a distinct lag-phase (Figure 2b) and, according to our study, need to be used at higher concentrations to preserve linoleic acid from oxidation. Indeed, phenolic compounds that act as retarders all showed a lower antioxidant potential, with EC1h > 8 µmol.L -1 .

Kinetic models have been presented in the literature [START_REF] Foti | Antioxidant properties of phenols[END_REF][START_REF] Roche | Antioxidant activity of olive phenols: mechanistic investigation and characterization of oxidation products by mass spectrometry[END_REF] to help explain these two types of behavior. According to Foti [START_REF] Foti | Antioxidant properties of phenols[END_REF] , an antioxidant should show effective chain-breaker behavior if lipid peroxyl radicals LOO • react faster with antioxidants AH (LOO • + AH → LOOH + A • ) than unsaturated fatty acids L'H (LOO • + L'H → LOOH + L' • ), suggesting a higher kinetic constant for the first reaction and thus implying a lag period. [START_REF] Foti | Antioxidant properties of phenols[END_REF] For retarders, kinetic constants should be similar, leading to the absence of a lag phase, since the antioxidant is slowly consumed. In reaction systems involving AAPH, Roche et al. [START_REF] Roche | Antioxidant activity of olive phenols: mechanistic investigation and characterization of oxidation products by mass spectrometry[END_REF] suggested that, in addition to inhibiting propagation by scavenging lipid radicals LOO • , antioxidants should also be able to scavenge the azo-initiator radicals ROO • (ROO • + AH → ROOH + A • ). Thus, antioxidants should inhibit the initiation step of lipid oxidation (ROO • + LH + O2 → ROOH + LOO • ). [START_REF] Roche | Antioxidant activity of olive phenols: mechanistic investigation and characterization of oxidation products by mass spectrometry[END_REF] Phenolic compounds which efficiently inhibit both initiation and propagation of oxidation are described as chain-breakers; their kinetic curves show a lag phase until complete consumption of the antioxidant. [START_REF] Roche | Antioxidant activity of olive phenols: mechanistic investigation and characterization of oxidation products by mass spectrometry[END_REF] In contrast, compounds which efficiently inhibit initiation, but not propagation, are described as retarders. [START_REF] Roche | Antioxidant activity of olive phenols: mechanistic investigation and characterization of oxidation products by mass spectrometry[END_REF] Their kinetics do not show a lag phase and they essentially scavenge azoinitiator radicals. [START_REF] Roche | Antioxidant activity of olive phenols: mechanistic investigation and characterization of oxidation products by mass spectrometry[END_REF] According to Foti [START_REF] Foti | Antioxidant properties of phenols[END_REF] , the values of the kinetic constant of the reaction LOO • + AH → LOOH + A • should be linked to specific chemical characteristics of the antioxidant molecules, such as: the bond dissociation energy (BDE) of the phenolic group; the steric hindrance effect of chemical structures near the phenolic group of phenolic antioxidants; and the presence of specific chemical structures, including catechol groups. Indeed, Liu and Wu 20 demonstrated that the spin-densities of the O • atom in antioxidant radicals could provide an explanation for the difference of the antioxidant effects among various compounds with small structural differences. Thus, chemical structure plays an important role in antioxidant activity. The kinetic constant values can also be linked to the location of the phenolic compound in a multiphasic system. In the AAPH system, as in the present study, lipid radicals are located in micelles of linoleic acid, whereas azo-initiator radicals are solubilized in the aqueous phase, thus adding an additional parameter to be taken into account. [START_REF] Kancheva | Lipid Oxidation in Homogeneous and Micro-Heterogeneous Media in Presence of Prooxidants, Antioxidants and Surfactants[END_REF][START_REF] Roche | Antioxidant activity of olive phenols: mechanistic investigation and characterization of oxidation products by mass spectrometry[END_REF] These studies highlight the lack of a well-defined classification between chain-breakers and retarders. The differences in the shapes of the curves may not only be due to differences in the mechanism of action of the antioxidant, but also the kinetic constants of a complex set of reactions occurring in multiphasic systems in which partitioning of the compounds must be taken into account.

CHARACTERIZATION OF THE ANTIBACTERIAL BEHAVIOR

We evaluated the inhibitory activity of the phenolic compounds against the growth of L. monocytogenes through the modeling of growth curves and determination of two characteristic parameters for each compound, the MIC and NIC (Table 1). The method used here offers two advantages relative to semi-quantitative determination of the MIC: a standard deviation is associated to the MIC value and a complementary parameter (the NIC) is provided to better characterize the activity.

Simple phenols had the highest inhibitory effect, as previously described. [START_REF] Burt | Essential oils: their antibacterial properties and potential applications in foods-a review[END_REF] Flavonoids also showed high inhibitory activity (naringenin) when their level of solubility allowed performance of the assay. MICs of flavonoids for L. monocytogenes and other Grampositive bacteria have been estimated in the literature, but the results are inconsistent. 21- 23 Although less effective than phenols, phenolic acids also showed good activity (Table 1).

The most effective were p-coumaric and ferulic acids. The least effective were chlorogenic and gallic acids. MICs obtained in the present study for several phenolic acids were of the same order of magnitude as those obtained against L. monocytogenes in two other studies. [START_REF] Vaquero | Influence of phenolic compounds from wines on the growth of Listeria monocytogenes[END_REF][START_REF] Wen | Antilisterial activity of selected phenolic acids[END_REF] The NIC provides additional useful information concerning the activity of the phenolic compound (Table 1 and Figure 3). Simple phenols and flavonoids were effective at very low concentrations, as shown for eugenol and naringenin (Figure 3). Other compounds were effective from a NIC that was slightly higher, as demonstrated for p-coumaric, ferulic (Figure 3), or vanillic acid. Finally, the NIC was much higher for many phenolic acids, including p-hydroxybenzoic acid and protocatechuic acid (Figure 3), as well as gallic, caffeic, and chlorogenic acids. These behaviors show that some compounds were active from the lowest tested concentration, whereas the concentration of others needed to reach a certain threshold before reducing the bacterial growth rate.

The mechanism of action of simple phenols has been very-well described. [START_REF] Burt | Essential oils: their antibacterial properties and potential applications in foods-a review[END_REF] The most studied simple phenols are eugenol, thymol, and carvacrol, which are found in aromatic plants and recovered in high concentrations in essential oils and hydrosols. [START_REF] Burt | Essential oils: their antibacterial properties and potential applications in foods-a review[END_REF][START_REF] Nazer | Combinations of food antimicrobials at low levels to inhibit the growth of Salmonella sv. Typhimurium: A synergistic effect?[END_REF][START_REF] Verma | Chemical Composition and Antimicrobial Potential of Aqueous Distillate Volatiles of Indian Peppermint (Mentha piperita) and Spearmint (Mentha spicata)[END_REF] These compounds intercalate into the phospholipid layers of the bacterial membrane and disturb the van der Walls interactions between the lipid acyl chains, leading to the disruption of phospholipid packing and membrane integrity. [START_REF] Burt | Essential oils: their antibacterial properties and potential applications in foods-a review[END_REF] Consequently, ion gradients are disrupted and vital constituents, such as ions and macromolecules, are released, leading to bacterial death. [START_REF] Helander | Characterization of the Action of Selected Essential Oil Components on Gram-Negative Bacteria[END_REF][START_REF] Ultee | Mechanisms of action of carvacrol on the foodborne pathogen[END_REF][START_REF] Ultee | The Phenolic Hydroxyl Group of Carvacrol Is Essential for Action against the Food-Borne Pathogen Bacillus cereus[END_REF] Some specific values for lipophilicity parameters have been suggested in the literature to be associated with antibacterial activity [START_REF] Ultee | The Phenolic Hydroxyl Group of Carvacrol Is Essential for Action against the Food-Borne Pathogen Bacillus cereus[END_REF] : a compound with a partition coefficient in octanol-water (logPo/w) above 3 can deeply partition into membranes, whereas a logPo/w above 4 does not always result in higher antibacterial effectivity. Indeed, the ranking of antibacterial efficacy may be related to logP (the log P of carvacrol, thymol, and eugenol are 3.28, 3.28, and 2.20, respectively). The NIC threshold also appeared to be quite low for these three compounds (Figure 3), reflecting intercalation into the membrane at low concentrations.

The mechanisms of action for the antibacterial properties of flavonoids are not yet fully understood. However, as naringenin has a logP similar to that of carvacrol and thymol (3.19), it is possible that naringenin has an antibacterial action similar to that of simple phenols, in accordance with the results of Tsuchiya et al. [START_REF] Tsuchiya | Comparative study on the antibacterial activity of phytochemical flavanones against methicillin-resistant Staphylococcus aureus[END_REF] Additionally, other mechanisms of action for flavonoids, including inhibition of nucleic acid synthesis and energy metabolism, have also been reviewed. [START_REF] Cushnie | Antimicrobial activity of flavonoids[END_REF] For these compounds, small NICs reflect high activity at the lowest concentration (Figure 3). Antibacterial mechanisms of phenolic acids are not fully understood due to their complex multifactorial activity. First, the decrease in pH due to the introduction of the phenolic compound into the medium can have a simple and direct impact on growth. Moreover, organic acids can act by a very well-known antibacterial mechanism through the penetration of their non-dissociated form into the cell and acidification of the cytoplasm, leading to K + efflux and cell death. The ionization rate of the molecule is fundamental and depends on both pH and the pKa: for the same pH, the higher the pKa, the higher the proportion of the non-dissociated form. The partition coefficients of phenolic acids may also contribute to their antibacterial activity, as some logPs can be as high as 2.43 for pcoumaric acid. Gallic and ferulic acids have been reported to irreversibly change the properties of bacterial membranes. [START_REF] Borges | Antibacterial activity and mode of action of ferulic and gallic acids against pathogenic bacteria[END_REF] Regarding the five phenolic acids with a high NIC (as shown for p-hydroxybenzoic and protocatechuic acids in Figure 3), the effective concentrations appeared to correlate with a pH below 6 when dissociation of acids had probably overcome the buffering capacity of the medium. In contrast, the three phenolic acids with a very low NIC were effective at pH > 6 (as shown for p-coumaric and ferulic acids in Figure 3), probably reflecting a mechanism of action that is less dependent on acid dissociation.

The possibility that phenolic acids may act as antibacterial compounds through combined mechanisms is supported by several studies. [START_REF] Campos | Cell membrane damage induced by phenolic acids on wine lactic acid bacteria[END_REF][START_REF] Ramos-Nino | Quantitative structure activity relationship for the effect of benzoic acids, cinnamic acids and benzaldehydes on Listeria monocytogenes[END_REF] Ramos-Nino et al. [START_REF] Ramos-Nino | Quantitative structure activity relationship for the effect of benzoic acids, cinnamic acids and benzaldehydes on Listeria monocytogenes[END_REF] could predict the antibacterial behavior of the phenolic acid as a function of the lipophilicity parameter (logK), as well as the effect of the acid dissociation constant (pKa). However, the logP cannot explain all the differences we observed for phenolic acids; p-coumaric and ferulic acids have different partition coefficients (2.43 and 1.64 respectively) and a similar pKa, but had similar antibacterial activity.

Further studies are needed to exhaustively explore the impact of extracellular pH, undissociated acid activity, and logP on the complex antibacterial mechanism of action of phenolic acids.

STRUCTURE-ACTIVITY RELATIONSHIPS

We assessed the antioxidant and antibacterial activities of phenolic compounds with various structures (Figure 1, Table 1), allowing us to describe some structure-activity relationships for both types of activities.

The formation of catechol groups by adding a second hydroxyl group in the ortho-position on the aromatic ring of phenolic acids (caffeic acid vs p-coumaric acid, protocatechuic acid vs p-hydroxybenzoic acid) or the B ring of flavonoids (eriodictyol vs naringenin, luteolin vs apigenin, quercetin vs kaempferol) significantly improved the antioxidant potential. Such higher antioxidant activity is due to stabilization by the catechol groups of their corresponding radicals by intramolecular hydrogen bonding or the regeneration of a catechol group, in parallel with quinone formation, from two catechol radicals. 12,35- 41 In contrast, adding a third hydroxyl group (gallic acid vs protocatechuic acid) slightly decreased the antioxidant potential, which may have been due to higher steric hindrance.

Concerning antibacterial activity, adding a second or third hydroxyl group to the phenolic ring of phenolic acids increased the MIC. This has also been reported for several bacteria, including Bacillus cereus [START_REF] Wen | Antilisterial activity of selected phenolic acids[END_REF][START_REF] Gutiérrez-Larraínzar | Evaluation of antimicrobial and antioxidant activities of natural phenolic compounds against foodborne pathogens and spoilage bacteria[END_REF][START_REF] Merkl | Antimicrobial and antioxidant properties of phenolic acids alkyl esters[END_REF][START_REF] Sánchez-Maldonado | Structure-function relationships of the antibacterial activity of phenolic acids and their metabolism by lactic acid bacteria[END_REF] , although it cannot be generalized for all strains or species. Such addition of phenolic functions tended to decrease the partition coefficient (e.g. 1.42 for caffeic acid vs 2.43 for p-coumaric acid), suggesting that the compound may be less easily partitioned into the lipophilic bacterial membrane, thus having lower antibacterial activity. [START_REF] Ramos-Nino | Quantitative structure activity relationship for the effect of benzoic acids, cinnamic acids and benzaldehydes on Listeria monocytogenes[END_REF] We obtained similar results in our study for flavonoids, for which eriodictyol (flavanone with a catechol group on the B ring), appeared to be less effective against L. monocytogenes than naringenin, its corresponding molecule with only one OH group (Table 1). This result supports the assumption that the decrease in lipophilicity induced by the addition of a hydroxyl group hinders the correct intercalation of the compound in the membrane (the logP of eriodictyol is 2.59 vs 3.19 for naringenin), as for phenolic acids. However, further studies are necessary to evaluate the impact of hydroxylation of the B ring on the antibacterial activity of flavonoids, [START_REF] Cushnie | Antimicrobial activity of flavonoids[END_REF][START_REF] Cowan | Plant products as antimicrobial agents[END_REF] as contradictory results have been reported in the literature. [START_REF] Mandalari | Antimicrobial activity of flavonoids extracted from bergamot (Citrus bergamia Risso) peel, a byproduct of the essential oil industry[END_REF] The presence of a methoxyl group in the ortho position of a phenolic group (vanillic acid vs p-hydroxybenzoic acid and ferulic acid vs p-coumaric acid) significantly increased antioxidant activity, as ortho substitution with the electron donor methoxyl group increases the stability of the produced radical. [START_REF] Liu | How many peroxyl radicals can be scavenged by hydroxylsubstituted Schiff bases in the oxidation of linoleic acid ?[END_REF][START_REF] Cuvelier | Comparison of the Antioxidative Activity of Some Acid-phenols: Structure-Activity Relationship[END_REF][START_REF] Maillard | Antioxidant Activity of Barley and Malt: Relationship with Phenolic Content[END_REF] This had a small and variable effect on antibacterial activity, as already shown for cinnamic acids in the literature. [START_REF] Wen | Antilisterial activity of selected phenolic acids[END_REF][START_REF] Sánchez-Maldonado | Structure-function relationships of the antibacterial activity of phenolic acids and their metabolism by lactic acid bacteria[END_REF] Esterification of caffeic acid by quinic acid, leading to chlorogenic acid, did not have any significant influence on the antioxidant potential, because the catechol group is still available. However, it resulted in an increase in the MIC, as reported in the literature. [START_REF] Peyrat-Maillard | Antioxidant Activity of Phenolic Compounds in 2,2′-Azobis (2-amidinopropane) Dihydrochloride (AAPH)-Induced Oxidation : Synergistic and Antagonistic Effects[END_REF][START_REF] Wen | Antilisterial activity of selected phenolic acids[END_REF][START_REF] Saavedra | Antimicrobial activity of phenolics and glucosinolate hydrolysis products and their synergy with streptomycin against pathogenic bacteria[END_REF] The addition of quinic acid decreased the partition coefficient (from 1.42 to -0.36) and increased steric hindrance, probably conferring lower penetration of the compound into the membrane.

The presence of alkyl groups in the ortho or para position of the phenolic group (carvacrol and thymol) did not have any significant influence on the antioxidant potential, confirming published data. [START_REF] Gutiérrez-Larraínzar | Evaluation of antimicrobial and antioxidant activities of natural phenolic compounds against foodborne pathogens and spoilage bacteria[END_REF] These two isomers have generally similar activities on many species, but some strains or species may be more sensitive to carvacrol, as was L. monocytogenes in the present study. [START_REF] Gutiérrez-Larraínzar | Evaluation of antimicrobial and antioxidant activities of natural phenolic compounds against foodborne pathogens and spoilage bacteria[END_REF] The addition of a cinnamic group (caffeic acid vs protocatechuic acid and ferulic acid vs vanillic acid) to the aromatic ring significantly increased both antioxidant and antibacterial activities. The capacity to stabilize the phenoxyl radical by resonance had a large effect on the antioxidant activity. [START_REF] Peyrat-Maillard | Antioxidant Activity of Phenolic Compounds in 2,2′-Azobis (2-amidinopropane) Dihydrochloride (AAPH)-Induced Oxidation : Synergistic and Antagonistic Effects[END_REF][START_REF] Brand-Williams | Use of free radical method to evaluate antioxidant activity[END_REF][START_REF] Chen | Structure-thermodynamics-antioxidant activity relationships of selected natural phenolic acids and derivatives: An experimental and theoretical evaluation[END_REF][START_REF] Cuvelier | Comparison of the Antioxidative Activity of Some Acid-phenols: Structure-Activity Relationship[END_REF][START_REF] Rice-Evans | Structure-antioxidant activity relationships of flavonoids and phenolic acids[END_REF] The increase in antibacterial activity may be related to the increased partition coefficient, due to the addition of a cinnamic group. [START_REF] Vaquero | Influence of phenolic compounds from wines on the growth of Listeria monocytogenes[END_REF][START_REF] Sánchez-Maldonado | Structure-function relationships of the antibacterial activity of phenolic acids and their metabolism by lactic acid bacteria[END_REF] For flavonoids, any chemical modifications that lead to higher resonance of the molecules, thus improving the stability of the phenoxyl radicals, increased the antioxidant activity, similar to the addition of a cinnamic group on phenolic acids [START_REF] Peyrat-Maillard | Antioxidant Activity of Phenolic Compounds in 2,2′-Azobis (2-amidinopropane) Dihydrochloride (AAPH)-Induced Oxidation : Synergistic and Antagonistic Effects[END_REF][START_REF] Maillard | Antioxidant Activity of Barley and Malt: Relationship with Phenolic Content[END_REF][START_REF] Rice-Evans | Structure-antioxidant activity relationships of flavonoids and phenolic acids[END_REF][START_REF] Heim | Flavonoid antioxidants: chemistry, metabolism and structure-activity relationships[END_REF][START_REF] Pietta | Flavonoids as antioxidants[END_REF] ; the presence of a double bond in the 2,3-position of the C ring (luteolin vs eriodictyol), the addition and/or availability of a hydroxyl group in the 3-position, in combination with the double bond in the 2,3 position of the C ring (kaempferol vs apigenin, quercetin vs luteolin or rutin), or the presence of a ketone in the 4-position conjugated with the double bond in the 2,3 position of the C ring (quercetin vs epicatechin) (Table 1). The impact of this type of chemical structure in flavonoids on antibacterial activity could not be tested due to solubility issues.

CONCLUSIONS

The chemical structure of phenolic compounds may have an opposite impact on antioxidant and antibacterial activities. These differences are attributable to completely different mechanisms of action between the two activities. The two main structural parameters that improve antioxidant activity are the chemical environment of the phenolic group and the presence of delocalization structures.

The partition coefficient may be one of the main parameters that influence antibacterial activity; in many cases, the higher the logP, the higher the bacterial growth inhibition. This is in accordance with the antibacterial activity being driven by a high lipophilicity parameter for optimized intercalation into the bacterial membrane. Nevertheless, our structure comparisons show that logP is not the only parameter to explain the compound's multifactorial antibacterial effectiveness. Finally, a series of phenolic compounds with both activities could be selected, based on these results, to conduct further studies in more complex food systems, including emulsions. The double activity of phenolic compounds can be applied in the food industry by using a mix of phenolic compounds, such as plant extracts or agro-industrial by-products, to take advantage of their complementary effects.

Figure 1 :

 1 FIGURES CAPTIONS

Figure 2 :

 2 Figure 2: Kinetics of linoleic acid oxidation in an AAPH system (followed by

Figure 3 :

 3 Figure 3: Effect of increasing concentrations of phenolic compounds on the growth of

TABLE CAPTION Table 1 :

 CAPTION1 Antioxidant and antibacterial activities of phenolic compounds: effective concentration to delay oxidation and non-inhibitory and minimum inhibitory concentrations for L. monocytogenes.For antioxidant activity, compounds are divided into groups (with letters) for which there is no significant difference in the estimated delay for oxidation, according to the Newman Keuls multiple comparison test (confidence interval 95%). For antibacterial activity, standard deviations are associated with NIC and MIC values. nd: not determined.

		Antioxidant activity	Antibacterial activity
	Phenolic		Groups
	compounds	EC1h	identified with	NIC ± SD	MIC ± SD
		(µmol.L -1 .h -1 )	Newman Keuls	(mmol.L -1 )	(mmol.L -1 )
			comparison
	Phenolic acids
	caffeic acid	1.2	CD	13.76 ± 0.50	18.03 ± 1.21
	chlorogenic acid	1.2	D	18.11 ± 0.81	33.44 ± 0.54
	protocatechuic acid	2.0	F	19.50 ± 0.74	24.63 ± 0.32
	ferulic acid	2.2	FG	5.04 ± 0.35	13.60 ± 0.31
	gallic acid	3.8	H	22.55 ± 1.17	31.70 ± 0.60
	vanillic acid	8.5	J	8.93 ± 0.50	21.30 ± 0.38
	p-coumaric acid	28.5	K	4.61 ± 0.49	15.44 ± 0.54
	p-hydroxybenzoic acid	239.6	K	10.27 ± 0.56	17.35 ± 0.29
	Simple phenols
	eugenol	2.2	G	2.88 ± 0.20	5.62 ± 0.13
	thymol	8.0	IJ	0.78 ± 0.04	1.31 ± 0.05
	carvacrol	8.1	IJ	0.33 ± 0.10	0.95 ± 0.07
	Flavonoids
	quercetin	0.8	A	nd	nd
	(-)-epicatechin	1.0	B	2.34 ± 0.30	> 9.44
	rutin	1.1	BC	nd	nd
	luteolin	1.2	D	nd	nd
	eriodictyol	1.5	E	0.23 ± 0.30	> 2.58
	kaempferol	4.8	HI	nd	nd
	apigenin	43.6	K	nd	nd
	naringenin	267.7	K	0.12 ± 0.11	1.85 ± 0.38
	α-tocopherol (referent)	2.4	G	nd	nd

ACKNOWLEDGEMENT

Aurélia Pernin received financial support for her doctoral studies from the French Government. The authors declare no competing financial interest.