Shanshan Chang

Françoise Quignard

Francesco Di Renzo

Bruno Clair
email: bruno.clair@univ-montp2.fr

SOLVENT POLARITY AND INTERNAL STRESSES CONTROL THE SWELLING BEHAVIOUR OF GREEN WOOD DURING DEHYDRATION IN ORGANIC SOLUTION

Keywords: Tension wood, Organic solvent, Longitudinal strain, Tangential strain, Stress release

The dimensional variations of green wood samples induced by organic solvents have been studied. The solvents used (ethanol, isopropanol, acetone, and acetonitrile) covered a wide range of polarity and were studied pure and in aqueous solutions over a wide range of concentrations. Samples of normal and tension wood of poplar were used in order to minimize the effect of hydrophobic extractives on the wood-solvent interactions. The evolution of wood volume and of tangential strain with the concentration of the organic solvents shows a behavior similar to gels, with a significant swelling for solutions of intermediate polarity. The similarity of volume obtained in water and less polar pure organic solvents strikingly contrasted the different effects of water and organic solvents on dry wood. Low-polarity solvents were extremely effective in the stress release of tension wood, as indicated by the pattern of longitudinal shrinkage. Solvent exchange does not affect the mesoporous structure of the cell walls of tension wood and is a promising way to reduce internal stress in wood products.

INTRODUCTION

The new context of the carbon disequilibrium produced by the use of petroproducts in the last decades makes it important to explore all ways of carbon storage. Among them, the more sustainable issue is the massive use of wood and wood-plant based products, since the resource is renewable, cheap, and abundant. The replacement of 5% of concrete building by wood would prevent the emission of about 100 Mt/y of CO 2 , corresponding to the emission of 50 million cars [START_REF] Metz | Climate Change 2007 -Mitigation of Climate Change, Contribution of Working Group III to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change[END_REF]. The extended use of wood as a material faces new challenges at each step of the production process. A fundamental challenge is the control of the dimension changes between the watersaturated state of the living plant and the air-dried equilibrium of final product. The consequent strains, involved in several phases of the drying process, affect the quality of the products and hence their usability by producing distortions and cracks. Wood structure is mainly responsible for the pattern of shrinkage experienced by living wood when its original water is removed by drying. The variations of wood moisture content below fibre saturation point (a parameter defined as the moisture content at which free water should be completely gone, while the cell walls are saturated with bound water) are responsible for dimensional variation in wood. Normally, the most stable dimension is in the longitudinal direction, and the least stable in the transverse direction. The departure of bound water results in anisotropic shrinkage, usually in the order of 0.1% to 0.2% along the longitudinal, 3% to 6% in radial, and 6% to 12% in tangential directions [START_REF] Skaar | Wood-Water Relations[END_REF].

The shrinkage pattern also depends on the large mechanical stresses, called maturation stresses, which originate in living trees during the cell maturation process [START_REF] Clair | Maturation stress generation in poplar tension wood studied by synchrotron radiation micro-diffraction[END_REF]). In the case of normal wood, newly formed fibers tend to contract longitudinally and to expand transversely, but this strain is restrained by their adherence to the older and stiffer wood cells. The resulting tensile stresses provide a better resistance against buckling. Adjusting this mechanism, a tree is able to create asymmetric growth stresses in inclined trees and to create a bending moment, allowing a trunk to recover verticality or a branch to maintain a given inclination. This unusually stressed wood is called reaction wood. Two kinds of reaction woods exist, depending on the kind of tree. While gymnosperms produce compression wood in the lower face of stems, angiosperms produce tension wood, generating high tensile stresses on their upper face [START_REF] Wardrop | The reaction anatomy of arborescent angiosperms[END_REF][START_REF] Fisher | Occurrence of reaction wood in branches of dicotyledons and its role in tree architecture[END_REF]. In tension wood, longitudinal tensile stresses can reach up to 70 MPa [START_REF] Okuyama | Growth stresses in tension wood: Role of microfibrils and lignification[END_REF]. Maturation stresses are present in the living stem as internal or residual stress even without the action of external forces. During the transformation process (falling of tree and sawing), a part of the stress is released, but internal stresses still remain in the log. To reduce stress in wood product, heating treatments, such as steaming can be applied (Nogi et al. 2003a[START_REF] Nogi | Reduction of longitudinal and tangential growth stress in logs by heat treatment[END_REF]. The phenomenon associated with this technique is known as 'hygrothermal recovery' (HTR, Kubler 1987;[START_REF] Gril | Tree mechanics and wood mechanics: Relating hygrothermal recovery of green wood to the maturation process[END_REF]. Recently, Chang et al. (2009a) showed that the use of alcohol has a similar effect and releases stress More generally, the treatment of wood with organic solvent is an important technological process and forms the basis of several methods of preservation and functionalisation of wood materials. In most cases, the interactions between wood and organic solvents have been studied on oven-dried wood. The degree of swelling of ovendried wood in solvents is a direct function of the polarity of the solvent [START_REF] Mantanis | Swelling of wood. Part II. Swelling in organic liquids[END_REF]. Most organic solvents are less polar than water, and oven-dried wood soaked in water usually swells much more than when immersed in organic solvents. The only exceptions are observed in the case of very polar solvents, for which swelling is comparable to the swelling in water. The sensitivity to polar solvents has suggested that swelling of dry wood depends on a mechanism similar to the swelling of other hydrophilic materials, such as natural polyelectrolyte gels [START_REF] Wilson | Colloidal phenomena and the adsorption formula[END_REF][START_REF] Anseth | Mechanical properties of hydrogels and their experimental determination[END_REF]. The extent of swelling of dry wood is more limited than the swelling of polysaccharide or polypeptide gels [START_REF] Draget | Swelling and partial solubilization of alginic acid gel beads in acidic buffer[END_REF][START_REF] Pourjavadi | New protein-based hydrogel with superabsorbing properties: Effect of monomer ratio on swelling behavior and kinetics[END_REF], due to the composite nature of wood, which includes a less hydrophilic lignin fraction and presents a complex cell structure.

In this paper, we present the first study on shrinkage/swelling behavior of green (never-dried) poplar wood samples in the aqueous solutions of several organic solvents at different concentration levels. This study contributes to a more fundamental understanding of the surface properties of wood cells. Swelling and shrinkage of gels are the result of the competition of the gel surface with the solvent of the surrounding solution. The osmotic pressure generated by the different affinity of a component of the solution for the solid adsorbent and the solvent accounts for the displacement of fluids between the gel and the solution. In the case of the interaction of some polyelectrolytes with organic solvents, it has been shown that swelling occurs only in the presence of solvents with a polarity closely matching the polarity of the polymer surface [START_REF] Phenix | The swelling or artists' paint in organic solvents. Part 2. Comparative swelling powers of selected organic solvents and solvent mixtures[END_REF][START_REF] Ono | Lipophilic polyelectrolyte gels as super-absorbent polymers for nonpolar organic solvents[END_REF]). The study of the competition between water and organics for the surface of green wood can provide information on the surface properties of wood before they are affected by drying treatments. Poplar wood was chosen for its low content of extractives, which are low-polar molecules that are able to modify the hydrophiliclipophilic balance of the surface.

EXPERIMENTAL

Wood Samples

One tilted and bent young poplar tree with a diameter of 8 cm was sampled for various testing conditions. This species produces tension wood with a typical gelatinous layer and a longitudinal drying shrinkage up to 10 times higher than that of normal wood [START_REF] Fang | Relationships between growth stress and wood properties in poplar I-69 (Populus deltoides Bartr. cv. "Lux" ex I-69/55)[END_REF]. Normal wood and tension wood samples were taken from the lower and upper side of the tilted trunk, respectively. The presence of tension wood was confirmed by anatomical observations, showing a large amount of fibers with a gelatinous layer and a thin S 2 layer. Care was taken to maintain samples in wet condition during storage from the time of sampling to experimentation. As a result, the moisture content of samples was always kept much higher than the fibre saturation point.

Five sets of specimens were sampled along the trunk axis; 9 specimens from pith to bark for normal wood and tension wood were obtained for each set. Specimen size was: 60 mm (longitudinal direction) × 2 mm (radial direction) × 7 mm (tangential direction). In order to compare them, the specimens in each set were located nearly at the same position.

Dehydration by Organic Solvents and Dimension Changes Measurements

Dehydrations were performed in ethanol, acetone, acetonitrile, and isopropanol. The choice of solvents was dictated by their miscibility with both water and the liquid CO 2 used for the final supercritical drying step. Some properties of the solvents are reported in Table 1. The octanol/water partition coefficient [START_REF] Ghose | Atomic physicochemical parameters for three dimensional structure directed quantitative structure-activity relationships III: Modeling hydrophobic interactions[END_REF], viz. the ratio of the concentrations of the solvent in octanol and water equilibrated phases, increases for decreased polarity of the solvent. The Hildebrandt solubility parameter is representative of the cohesive strength of the solvent, while the Hansen hydrogen bond parameter is the component of the cohesive force corresponding to hydrogen bonding, which tends to be higher for more polar protic solvents.

The first to fourth wood sets were dehydrated by immersion in serial solutions of increasing solvent concentration (10, 30, 50, 70, 90%, and anhydrous) during three days for each step. The fifth set was dehydrated by repeated three-day treatments with anhydrous ethanol, in order to check the effect of serial dehydration on final dimensions. This experiment was stopped when the weight difference between two subsequent treatments became less than 0.3%.

Wood specimens dimensions at different dehydration stages were measured along longitudinal and tangential directions using a digital micrometer (precision 0.001 mm) on a dedicated device allowing perfect repositioning of the specimen at each measurement. Bending or warping was weak enough to be neglected. Relative dimensional changes due to organic substitution in longitudinal and tangential directions were represented as strains ε L and ε T , respectively, and expressed in percent. The dimensional change was calculated as the dimensional variation between each dehydrated condition and green condition, divided by the dimension in the green state. A positive value of ε L or ε T indicates that the sample expands, and a negative one indicates shrinkage. The volume variations of the specimens were calculated as the product of longitudinal, tangential, and radial strain, and the radial strain being assumed as half of tangential strain [START_REF] Nakano | Analysis of cell wall swelling on the basis of the cylindrical model[END_REF].

N 2 Adsorption-Desorption Measurement

To evaluate the effect of solvent dehydration on gelatinous layer texture, nitrogen adsorption measurements were performed on the supercritical CO 2 dried samples. Such techniques, typically used for the characterization of adsorbents or inorganic catalysts, have been applied to the study of biological materials such as wood. Before nitrogen adsorption measurements, the organic solvent in dehydrated samples was replaced by liquid CO 2 , which was supercritically extracted in a Polaron 3100 apparatus. Then the dried samples were tested on a micromeritics ASAP 2010 volumetric apparatus at 77 K for adsorption measurement. This experimental technique allows the specific surface area (S BET ) [START_REF] Brunauer | Adsorption of gases in multimolecular layers[END_REF]) and the pore size distribution [START_REF] Broekhoff | Studies on pore systems in catalysts. 11. Pore distribution calculations from adsorption branch of a nitrogen adsorption isotherm in case of ink-bottle type pores[END_REF] of the samples to be evaluated. The volume of pores smaller than 50 nm was measured as the volume of liquid nitrogen adsorbed at p/p° 0.98. Only tension wood samples have been measured since only the specific gelatinous layer of tension wood contains significant mesoporosity [START_REF] Clair | Characterization of a gel in the cell wall to elucidate the paradoxical shrinkage of tension wood[END_REF]Chang et al. 2009b).

RESULTS

Dimensional Deformation during Organic Solvent Exchange

Figure 1 shows the variations of longitudinal strain (ε L ) during serial organic solvents substitution. As observed previously with alcohol (Chang et al. 2009a), normal wood slightly swelled during serial organic solvent substitution. For each solvent, a stable value of the strain (-0.02% to +0.1%) was rapidly attained, which was the same order as what is released with hygrothermal treatments [START_REF] Clair | Maturation stress generation in poplar tension wood studied by synchrotron radiation micro-diffraction[END_REF]. Tension wood shrunk greatly (up to 0.5 %) and seemed to continue to shrink at the end of the substitution without reaching an asymptote.

The range of longitudinal strain was different from solvent to solvent during the substitution process. Acetonitrile gave the largest longitudinal swelling or shrinkage (around -0.5%, at the end of the substitution), and was followed by acetone. Both isopropanol and ethanol showed similar smaller strains. It also can be seen that during acetone and acetonitrile substitution, more than 50% of the final strain appeared between 0 to 30% concentrations in tension wood specimens. During ethanol substitutions, strains appeared to be proportional to the solvent concentration. Along the tangential direction (Fig. 2a), the value of tangential strain (ε T ) was positive both in normal wood and tension wood, which means that the specimens expanded in tangential direction in all the organic solutions in the same order as what was measured after hygrothermal treatment [START_REF] Clair | Maturation stress generation in poplar tension wood studied by synchrotron radiation micro-diffraction[END_REF]. As the solvent concentration increased from 0 to 70%, the specimens swelled and reached a maximum swelling. These maximum strains of tension wood at 70% concentration largely depended on the nature of the solvent and varied from around 0.5% in ethanol to 1.4% in acetonitrile. At higher concentration, from 70 to 100%, the specimens shrank back and the swelling ratio gradually decreased to reach an average value of 0.35% in pure organic solvents. Independently of the nature of the solvent, normal wood always presents a smaller swelling than the corresponding tension wood. When samples were dehydrated in anhydrous ethanol directly, the deformation values (both in longitudinal and tangential directions) were similar to the final state reached after increasing ethanol concentration by steps (data not shown).

The volume variation as a function of the concentration of the organic solvent is given in Fig. 2b. It was found that at 70% acetonitrile, tension wood increased in volume by 1.6% and normal wood by 1.3%. At 100% acetonitrile, both types of wood were back to the original volume plus or minus 0.03%. In presence of the other solvents, volume variations followed a similar pattern, with a lower swelling maximum at intermediate concentrations and a small retention of swelling in pure organic solvent.

Mesoporosity of Tension Wood Samples

Isotherms for the adsorption and desorption of nitrogen at 77K on solventsubstituted poplar tension wood aerogels are given in Fig. 3. The shape of the isotherms was very similar for samples exchanged with different solvents. According to the IUPAC classification [START_REF] Sing | Reporting physisorption data for gas-solid systems[END_REF], the isotherm of all samples are type IV with a H3 type hysteresis loop, indicating the presence of slit-shaped pores with a non-uniform size. The characteristic termination of hysteresis at p/p° 0.45 corresponds to the threshold of stability of N 2 menisci [START_REF] Trens | A macrothermodynamic approach to the limit of reversible capillary condensation[END_REF]. The pore surface areas as measured by the B.E.T. method were 18.1, 14.8, 11.7, and 8.8 m 2 g -1 for tension wood samples under acetone, ethanol, acetonitrile, and isopropanol substitution, respectively. The surface area data were strongly correlated with the mesopore volume. The volume of pores smaller than 50 nm was 50, 46, 38, and 30 mm 3 g -1 for samples treated with acetone, ethanol, acetonitrile, and isopropanol, respectively. These volumes of pores are not significantly different considering the natural variability of amount of gelatinous layer from sample to sample. [START_REF] Meyer | Swelling green wood in solvents[END_REF]. "Swelling green wood in solvents," BioResources 7( 2 The peculiar shape and typical hysteresis loop of these curves are indicative of a highly mesoporous adsorbent. The pore size distributions determined from the nitrogen adsorption data by the method of [START_REF] Broekhoff | Studies on pore systems in catalysts. 11. Pore distribution calculations from adsorption branch of a nitrogen adsorption isotherm in case of ink-bottle type pores[END_REF] are plotted in Fig. 4. The majority of the accessible pores exhibited diameters between 2 and 50 nm, with the maximum in the distribution curves occurring at a diameter of 7 nm for all samples. The pore size distribution was not at all affected by the changes of the nature of the solvent (Fig. 4).

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DISCUSSION

Tangential and Volume Strain in Organic Solutions

The curves of volume swelling shown in Fig. 2b are quite similar to the curves of tangential swelling in Fig. 2a and are relatively unrelated to the longitudinal swelling data shown in Fig. 1. This is in agreement with the relative extent of the two kinds of strain, much larger in the case of tangential strain, as observed in virtually all kinds of woods. The swelling of dry wood in organic solvents was extensively studied, and it was observed that most non-polar or weakly polar solvents induced a swelling of oven-dried wood significantly smaller than the swelling induced by water or strongly polar solvents [START_REF] Mantanis | Swelling of wood. Part II. Swelling in organic liquids[END_REF][START_REF] Morisato | Adsorption of liquids and swelling of wood[END_REF]. If extrapolated from dry-oven wood to green wood, these data would suggest that replacement of water by non-polar solvents should bring about a tangential shrinkage of wood. Such behaviour does not correspond to the observed swelling of green poplar wood in mixed water-organic solvents (Fig. 2). This indicates that permanent modifications induced by drying affected the accessibility to solvents of some parts of the wood structure.

The tangential swelling of the green poplar studied here was very similar in normal and tension wood (Fig. 2a); hence it is not related with the microfibrils angle or the presence of G-layer. This is confirmed by the lack of significant modifications of the mesoporosity of the tension wood samples (Fig. 3), a typical property of the gelatinous layer but is in good agreement with tangential strain measured during stress release [START_REF] Clair | Evidence that release of internal stress contributes to drying strains of wood[END_REF]. The observed tangential swelling is solvent-related and it is more important for less polar solvents, as easily deduced from the comparison of the swelling data of Fig. 2 and the solvent properties reported in Table 1. This trend is completely opposite to the trend observed for the swelling of dry wood in organics [START_REF] Mantanis | Swelling of wood. Part II. Swelling in organic liquids[END_REF][START_REF] Morisato | Adsorption of liquids and swelling of wood[END_REF] and confirms the difference in surface properties of green and dry wood. The bell-shaped dependence of swelling with the concentration of organics (Fig. 2) is especially relevant, as it indicates a non-linear dependence from the composition of the solvent. [START_REF] Mantanis | Swelling of wood. Part II. Swelling in organic liquids[END_REF] discussed data of [START_REF] Hasselblatt | Der wasserdampfdruck und die elektrische Leiftähigkeit des Holzes in Abhängigkeit von seinem Wassergehalt[END_REF] and observed that the swelling of several woods in organic solvents was larger if the solvents contained some water. [START_REF] Hiroki | Volume phase transitions of poly (acryloyl-l-proline methyl ester) gels in response to water-alcohol composition[END_REF] for polypeptide gels and [START_REF] Nishida | Lignin gel with unique swelling properties[END_REF] in the case of polymeric lignin gels observed a maximum swelling at intermediate alcohol concentrations and a return to the unswelled volume for higher alcohol concentrations [START_REF] Hiroki | Volume phase transitions of poly (acryloyl-l-proline methyl ester) gels in response to water-alcohol composition[END_REF]. This non-linear behavior, typical of the swelling of many polymeric systems with solvents, suggests that an optimized solvent treatment can further open the wood structure and increase its accessibility to preservation treatments. Such an effect can also present some interest in the treatments needed for the separation of wood components as a first step for the production of second-generation bio-based chemicals in biorefineries. When different solvents were tested, swelling was only observed for solvents with values of the solubility parameter corresponding to the nature of the polymer. Lower or no swelling was observed for lower or higher values of the solvent solubility parameter for materials as different as coal [START_REF] Jones | On the solvent-induced swelling of low-rank coals[END_REF], paints [START_REF] Phenix | The swelling or artists' paint in organic solvents. Part 2. Comparative swelling powers of selected organic solvents and solvent mixtures[END_REF], polyelectrolytes [START_REF] Ono | Lipophilic polyelectrolyte gels as super-absorbent polymers for nonpolar organic solvents[END_REF], and partially-dried larch wood [START_REF] Loskutov | Swelling of larch wood in organic liquids[END_REF].

The order of the amount of swelling for the aqueous solutions of different solvents is acetonitrile > acetone > isopropanol ~ ethanol. This order is unrelated to the molecular volume or the density of the solvents and it is not strictly related to the values of the solubility parameter. However, it is in excellent agreement with the order of the octanol/water partition coefficient, representative of the lipophilic/hydrophilic balance of the solvent, and with the Hansen hydrogen-bonding parameter, representative of the hydrogen-bond component of the cohesive force of the solvent. It seems clear that the extent of swelling of green wood increases when the wood surface interacts with more lipophilic solvent and it is related to the decrease of the force of hydrogen bonds which connect wood sub-structures. However, the dependence of swelling on the lipophilicity of the solvent is not monotonous.

The swelling curves of Fig. 2 indicate that green wood exchanged with pure solvents presented a volume similar to its volume in pure water. This suggests that water and organic solvents of solubility parameter between 20 and 26 are at the equilibrium with a similar conformation of poplar green wood. The bell shape of the swelling curves corresponded to a significant swelling for mixed solutions of water and organics. This suggests that the most interacting solvent for undried poplar wood would present a solubility parameter intermediate between 26 (the value for ethanol, the most polar organic used) and 47 (the value for water).

Longitudinal Strain and Stress Release in Organic Solutions

Longitudinal strain follows a pattern of evolution with the concentration of organic solvents very different from the pattern observed by tangential strain, as evidenced by comparison of Figs. 1 and 2a. Previous results (Chang et al. 2009a) have suggested that the evolution of longitudinal strain with alcohol concentration is related to the structure of the cell walls. The cell wall is composed of a middle lamella, which is the highly lignified joint between two cells, then a primary wall and a secondary wall divided in three layers named S 1 , S 2 , and S 3 from the border to the centre of the cell. Whereas primary wall, S 1 , and S 3 are reinforced by crystalline cellulose microfibrils nearly perpendicular to the fiber axis, the S 2 layer is the thickest and the angles of the microfibrils are around 10° to 30° relative to the cell axis. The longitudinal shrinkage is known to be related to the microfibrils angle in the S 2 layer. Microfibrils angle is large in juvenile wood or compression wood, leading to higher longitudinal shrinkage than in normal wood where microfibrils angle is lower. In tension wood of some species, the peculiar gelatinous layer replaces the S 3 layer and part of the S 2 layer. In gelatinous layer, the microfibrils angle is close to 0, but a paradoxical high longitudinal shrinkage (up to 1.2%, [START_REF] Chow | A comparative study of the structure and composition of tension wood in beech (Fagus sylvatica L.)[END_REF][START_REF] Clair | Shrinkage of the gelatinous layer of poplar and beech tension wood[END_REF][START_REF] Clair | Relationship between growth stresses, mechanical -physical properties and proportion of fibre with gelatinous layer in chestnut (Castanea Sativa Mill.)[END_REF][START_REF] Yamamoto | Role of the gelatinous layer (G-layer) on the origin of the physical properties of the tension wood of Acer sieboldianum[END_REF] occurs due to the collapse of its gel-like structure [START_REF] Clair | Characterization of a gel in the cell wall to elucidate the paradoxical shrinkage of tension wood[END_REF]. This excessive shrinkage causes problems in wood utilization, including longitudinal crooking and bowing, and thus reduces the potential economic value of the material. Previous results (Chang et al. 2009a) have shown that the longitudinal swelling of normal wood under organic solutions mainly depends on the microfibrils angle of the S 2 layer in the cell wall. A swelling was observed in the normal wood of poplar, whereas a large shrinkage was observed in poplar tension wood, in which the gelatinous layer presents a microfibrils angle close to zero. The behavior of tension wood, which shrunk in all organic solutions, is consistent with what was observed for tension wood shrinking in ethanol (Chang et al. 2009a). Since the variations of shrinkage of tension wood are very different from the changes in porosity of the gelatinous layer measured by nitrogen adsorption-desorption, the longitudinal strain in tension wood is not induced by a collapse of the gelatinous layer as during the evaporative drying, but would be the expression of the release of residual tensile stress, a basic feature shared by all types of tension wood. The formation of tension wood is characterized by the development of considerable maturation stresses [START_REF] Wardrop | The reaction anatomy of arborescent angiosperms[END_REF][START_REF] Okuyama | Growth stresses in tension wood: Role of microfibrils and lignification[END_REF]. Even after the release of macroscopic stress resulting from wood sample cutting, some residual stresses remain in the microstructure. At the cell wall level, the gelatinous layer is under tension [START_REF] Clair | Precautions for the structural analysis of the gelatinous layer in tension wood[END_REF][START_REF] Clair | Maturation stress generation in poplar tension wood studied by synchrotron radiation micro-diffraction[END_REF] and this tensile stress is supported by the other layers, which are subjected to compression stress. During wood-solvent interaction, the water-solvent molecule exchange may provoke the increase of molecular mobility within the amorphous substance of the cell wall, i.e., the matrix. Thus, molecular mobility affects essentially the other layers which are compressive stressed, and then a negative strain occurs.

The results shown in Figs. 1 and 2 reveal that the less polar solvents, acetonitrile and acetone, induced greater strains both in longitudinal and tangential directions. The larger volume swelling observed for less polar solvents contributes to more extended molecular fluxes inside the samples.

CONCLUSIONS

The mechanical behavior of normal and tension poplar green wood immersed in organic solvents and their aqueous solutions were investigated, and the solvent-induced strains were measured. The following main conclusions can be drawn:

1. In the longitudinal direction, normal wood moderately swells during solvent substitution; however, tension wood shrinks by an amount depending on the polarity of the solvent. The longitudinal strains of tension wood are not due to the presence of a given solvent but to the release of pre-existing internal stress during the exchange process. The less polar the solvent, the greater the strain observed, suggesting that disruption of hydrogen bonds within the amorphous substance of the cell wall is an important mechanism of stress release.

2. Organic solvents, as well as water can keep apart the secondary structure of green wood and prevent any significant tangential shrinkage. Moreover, mixed organicwater solvents bring about a significant non-linear swelling of green wood, which is typical behavior of organic gels.

Fig. 1 .

 1 Fig.1. Mean longitudinal strain (ε L ) after serial organic solvents substitution in poplar normal and tension wood. Filled lines (filled symbols): tension wood; broken lines (void symbols): normal wood; square: acetonitrile; circle: acetone; triangle: isopropanol; diamond: ethanol

Fig. 2 .

 2 Fig. 2. (a) Mean tangential strain (ε T ) and (b) estimated volume strain during serial organic solvents substitution in poplar normal and tension wood. Filled lines (filled symbols): tension wood; broken lines (void symbols): normal wood; square: acetonitrile; circle: acetone; triangle: isopropanol; diamond: ethanol

Fig. 3 .Fig. 4 .

 34 Fig. 3. Nitrogen adsorption-desorption isotherms at 77K of solvent-substituted poplar tension wood. (a) acetone; (b) ethanol; (c) acetonitrile; (d) isopropanol. Filled shape: adsorption; void shape: desorption

Table 1 .

 1 Properties of the Solvents Used

	Solvent	Boiling point / °C	Octanol/water partition coefficient (log)	Hildebrandt solubility parameter / MPa 0.5	Hansen hydrogen bond parameter / MPa 0.5
	water	100	-	47.9	42.3
	ethanol	79	0.07	26.4	19.4
	isopropanol	82	0.39	23.5	16.4
	acetone	56	0.67	20.2	7
	acetonitrile	82	0.66	24.1	6.1

ACKNOWLEDGMENTS

The authors thank P. Brunier from Domaine Maspiquet in Grabels (Lycee Agropolis Montpellier) for the poplar tree used for this study. This work was partly funded by the Conseil Scientifique-Université Montpellier 2 (CS-UM2).

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 [START_REF] Meyer | Swelling green wood in solvents[END_REF]. "Swelling green wood in solvents," BioResources 7(2), 2418-2430. 2428 3. Among the four organic solvents selected (ethanol, isopropanol, acetonitrile, and acetone), the more polar ones (ethanol and isopropanol) bring the lowest effect on the dimensional deformations both in longitudinal and tangential directions. Ethanol, which is still the primary solvent for sample preparation in research activities, provides a gentle process that can better keep the structure of samples and also can be readily carried out practically. On the other hand, the less polar solvents, especially acetonitrile, are highly effective for stress release of tension wood, a pre-condition for the use of this kind of wood in structural applications. In general, solvent exchange could be a promising way to reduce internal stress in wood product, similar to hygrothermal recovery induced by heating treatments. The development of effective methods of wood treatments will require a more detailed knowledge of the way in which solvents interact with each part of the wood structure.


Keywords: Time-delay systems, stability analysis, Lyapunov

The paper addresses the stability problem of linear time delay system. In the literature, the most popular approach to tackle this problem relies on the use of Lyapunov-Krasovskii functionals. Many results have proposed new functionals and techniques for deriving less and less conservative stability conditions. Nevertheless, all these approaches use the same trick, the well-known Jensen's inequality which generally induces some conservatism difficult to overcome. In light of those observations, we propose to reduce the conservatism of Lyapunov-Krasovskii functionals by introducing new classes of integral inequalities called Wirtinger inequalities. This integral type inequality is firstly shown to encompass Jensen's inequality and is then employed to derive new stability conditions. To this end, a slightly modified Lyapunov functional is proposed. Several examples illustrate the effectiveness of our methodology.

On the use of the Wirtinger inequalities for time-delay systems

A. Seuret * F. Gouaisbaut * * , * * *

INTRODUCTION

The last decade has shown an increasing research activity on time-delay systems analysis and control due to both emerging adapted theoritical tools and also practical issues in the engineering field and information technology as well as for biology, economics or ecology (see [START_REF] Sipahi | Stability and stabilization of systems with time delay[END_REF] for a recent survey).

In the case of linear time delay system with a constant delay, three different techniques allow to derive efficient criteria proving the stability of such a system. The first classical methodology relies on the study of the roots of the associated characteristic equation, a quasi-polynomial in s and e -hs . Even very effective in practice (see the monograph [START_REF] Gu | Stability of timedelay systems[END_REF] and the survey [START_REF] Sipahi | Stability and stabilization of systems with time delay[END_REF]), these technics generally lead to a perfect description of the stability with respect to the delay. Nevertheless, these methods reveal themselves restrictive in the sense that they cannot be extended straightforwardly to the robust case and to the case of time varying delay.

Another important technic comes from the robust analysis framework. In that case, the basic idea is pull out the delay element from a nominal system and to merge it into a unstructured uncertainty. The original time delay system is then modeled as nominal system without delay submitted to a perturbation. The use of classical robustness tools like Small Gain theorem [START_REF] Niculescu | Frequency sweeping tests for asymptotic stability: a model transformation for multiple delays[END_REF], [START_REF] Zhang | Stability of time-delay systems: Equivalence between Lyapunov and scaled small-gain conditions[END_REF]), IQC approach [START_REF] Kao | Stability analysis of systems with uncertain time-varying delays[END_REF]) or quadratic separation approach [START_REF] Ariba | Stability interval for time-varying delay systems[END_REF]) allow then to develop effective criteria. The sources of conservatism are then twofold: the way to model the interconnection between the nominal system and the delay uncertainty and the accuracy of the delay covering set (the more precise the uncertainty set is, the less conservative is the criterion).

The third approach and the most popular remains the use a Lyapunov-Krasovskii functional. In the linear case, its general structure is perfectly known but is numerically difficult to handle (see [START_REF] Gu | Stability of timedelay systems[END_REF] for a concise introduction). According to the important literature devoted to this subject [START_REF] Ariba | An augmented model for robust stability analysis of time-varying delay systems[END_REF], [START_REF] Kim | Note on stability of linear systems with timevarying delay[END_REF], [START_REF] Shao | New delay-dependent stability criteria for systems with interval delay[END_REF], [START_REF] Sun | Improved delay-range-dependent stability criteria for linear systems with time-varying delays[END_REF]), conservative results are then provided often expressed in terms of LMIs if some additional hypothesis are formulated on the Lyapunov functional. The challenge is then to reduce the conservatism of such approaches by either choosing extended state based Lyapunov-Krasovskii functional [START_REF] Ariba | An augmented model for robust stability analysis of time-varying delay systems[END_REF], [START_REF] Kim | Note on stability of linear systems with timevarying delay[END_REF]) and/or discretized Lyapunov functional [START_REF] Gu | Stability of timedelay systems[END_REF]). Furthermore, apart the choice of an appropriate Lyapunov-Krasosvkii V , an important source of conservatism is the way to bound some cross terms arisen when manipulating the derivative of V . According to the literature on this subject (see [START_REF] Park | Reciprocally convex approach to stability of systems with time-varying delays[END_REF], [START_REF] He | Further improvement of free-weighting matrices technique for systems with time-varying delay[END_REF], [START_REF] Shao | New delay-dependent stability criteria for systems with interval delay[END_REF] for some recent papers), a common feature of all these techniques is the use of slack variables [START_REF] Jiang | Delay-dependent robust stability for uncertain linear systems with interval time-varying delay[END_REF]) and Jensen inequality [START_REF] Shao | New delay-dependent stability criteria for systems with interval delay[END_REF], [START_REF] Sun | Improved delay-range-dependent stability criteria for linear systems with time-varying delays[END_REF]). At the price of an increasing conservatism, this last inequality allows to get some LMIs which may be solved efficiently with Semi-Definite Programming (SDP) solvers.

In this preliminary work, we aim at reducing the conservatism of Lyapunov-Krasovskii techniques by considering an accurate integral inequality which includes the Jensen's one as a special case : the Wirtinger inequality. This new class of inequalities has been already employed in the stability of sampled-data systems by [START_REF] Liu | Wirtinger's inequality and lyapunov-based sampled-data stabilization[END_REF]. In this paper, its use combined with some special properties of sampled data systems has led to some inter-

PRELIMINARIES

Looking at the huge literature devoted to the stability analysis of time-delay systems, the use of Lyapunov-Krasovskii functionals generally leads to some matrix inequalities quite difficult to handle. In order to get numerically tractable inequalities (generally some LMIs to be optimized), researchers have been extensively used the well-known Jensen's inequality which is recalled in the following lemma. Lemma 1. For given symmetric positive definite matrices R > 0 and for any differentiable signal ω in [a, b] → R n , the following inequality holds:

b a ω(u)R ω(u)du ≥ 1 b -a ω(b) ω(a) T R -R * R ω(b) ω(a) (1)
In the context of time-delay systems, this inequality has been the core of several important contributions [START_REF] Gu | Stability of timedelay systems[END_REF], [START_REF] He | Further improvement of free-weighting matrices technique for systems with time-varying delay[END_REF]) although an inherent conservatism which has studied since several years in a Lyapunov-Krasovskii or robust analysis context (see for instance [START_REF] Briat | Convergence and equivalence results for the Jensen's inequality -application to time-delay and sampled-data systems[END_REF] and [START_REF] Gouaisbaut | Robust stability of timedelay systems with interval delays[END_REF]). In the present article, we aim at improving the resulting stability tests by employing a new class of integral inequalities called Wirtinger inequalities, which encompass the Jensen's inequality. In control theory, these types of inequalities have been successfully employed in [START_REF] Liu | Wirtinger's inequality and lyapunov-based sampled-data stabilization[END_REF] in the case of sampled data systems and have shown to reduce drastically the conservatism. In this paper [START_REF] Liu | Wirtinger's inequality and lyapunov-based sampled-data stabilization[END_REF], the authors used the following inequality

Lemma 2. Let z ∈ W [a, b) and z(a) = 0. Then for any n × n matrix R > 0, the following inequality holds b a z T (s)Rz(s)ds ≤ 4(b -a) 2 π 2 b a żT (s)R ż(s)ds (2)
There exists also another Wirtinger inequality that can be found in [START_REF] Kammler | A first Course in Fourier Analysis[END_REF], which leads to a more precise inequality using a stronger assumption on the function z. Indeed, the previous inequality can be refined as follows:

Lemma 3. Let z ∈ W [a, b) and z(a) = z(b) = 0. Then for any n × n matrix R > 0, the following inequality holds b a z T (s)Rz(s)ds ≤ (b -a) 2 π 2 b a żT (s)R ż(s)ds (3)
In this paper, we aim at understanding how these inequalities can help in the context of assessing stability of time-delay systems. In particular, in the next section, we propose a novel inequality which is proved to be less conservative than the Jensen's one.

APPLICATION OF THE WIRTINGER'S INEQUALITIES

3.1 How to encompass Jensen's inequality?

In the sequel, we present a first new inequality that allows to derive less conservative stability conditions for time delay systems. Lemma 4. For a given symmetric positive definite matrix R > 0 and any differentiable signal ω in [a, b] → R n , then the following inequality holds:

b a ω(u)R ω(u)du ≥ b a Ω T (u) 6W 0 (R) + π 2 W 1 (u) 6(b -a) 2 Ω(u)du (4) where Ω = ω T (b) ω T (a) ω T (u)
T and (5) Consider now the left-hand side of the Wirtinger inequality. Developing this expression leads to:

W 0 (R) = R -R 0 * R 0 * * 0 , W 1 (u) =      2R R -6 (u -a) (b -a) R * 2R -6 (b -u) (b -a) R * * 6R      . Proof :
a z T (u)Rz(u)du = b a (b -u) 2 ω T (a)Rω(a)du + b a (u -a) 2 ω T (b)Rω(b)du +2 b a (b -u)(u -a)ω T (a)Rω(b)du -2(b -a)ω T (b)R b a (u -a)ω(u)du -2(b -a)ω T (a)R b a (b -u)ω(u)du +(b -a) 2 b a ω T (u)Rω(u)du, (6) or equivalently, b a z T (u)Rz(u)du = (b -a) 3 3 ω T (a)Rω(a) + (b -a) 3 3 ω T (b)Rω(b) + 2 (b -a) 3 6 ω T (a)Rω(b) -2(b -a)ω T (b)R b a (u -a)ω(u)du -2(b -a)ω T (a)R b a (b -u)ω(u)du +(b -a) 2 b a ω T (u)Rω(u)du.
(7) This last equality can be rewritten as

b a z T (u)Rz(u)du = (b -a) 2 6 b a Ω T (u)W 1 (u)Ω(u)du.
(8) We conclude the proof by applying the Wirtinger inequality which leads to

π 2 6(b -a) 2 b a Ω T (u)W 1 (u)Ω(u)du ≤ b a ωT (u)R ω(u)du - 1 (b -a) 2 b a (ω(b) -ω(a)) T R(ω(b) -ω(a))du, (9)
which is equivalent to the inequality proposed in Lemma 4. ♦

The previous lemma presents an improved version of the Jensen's inequality. Effectively, by splitting the integral given in the right-hand side of the inequality (4), we can highlight the difference with the Jensen's inequality:

b a ωT (u)R ω(u)du ≥ 1 (b -a) ω(b) ω(a) T R -R * R ω(b) ω(a) + π 2 6(b -a) 2 b a Ω T (u)W 1 (u)Ω(u)du.
(10) As equation ( 8) ensure that the last term of the previous inequality is positive, Lemma 4 gives a more accurate results than the Jensen's inequality.

However, a simple inspection of the right-hand side of the inequality (10) shows that this new inequality is difficult to handle since the matrix W 1 is polynomial of the integration variable. It is therefore not directly suitable for assessing the stability of time delay systems via numerical tools like LMIs. In the following, a more practical inequality is derived using a method based on integral manipulations introduced in the discretization method from [START_REF] Gu | Stability of timedelay systems[END_REF].

An appropriate inequality

As mentioned above, we cannot use the Lemma 4 to derive directly a numerically tractable stability criterion. In this subsection, we propose a more suitable formulation by finding a lower bound expressed only in terms of signals

a ωT (u)R ω(u)du ≥ 1 b -a ω(b) ω(a) ν T W 2 (R) ω(b) ω(a) ν , (11)
where

ν = 1 b -a b a ω(u)du, W 2 (R) = W 0 (R) + π 2 4 R R -2R * R -2R * * 4R
.

Proof : Consider the inequality (10), which is satisfied for any symmetric positive definite matrix R and any function ω. We aim at finding an appropriate lower bound of the right-hand side of this inequality by studying only the last integral of inequality (10), b a Ω T (u)W 1 (u)Ω(u)du, which can be rewritten as follows

I = b a ω(b) ω(a) ω(u) T W 1 1 W 2 1 (u) * 6R ω(b) ω(a) ω(u) du ( 12
)
where

W 1 1 = 2R R * 2R , W 2 1 (u) = - 6 (b -a) (u -a)R (b -u)R
In order to find a lower-bound, we apply a similar transformation proposed in [START_REF] Gu | Stability of timedelay systems[END_REF] (Proposition 5.21). For any > 0, consider the integral

I * = b a ξ T (u) 1 + 6 R -1 I * 6R ξ(u)du,
where

ξ(u) =   W 2 1 T (u) ω(b) ω(a) ω(u)   .

By virtue of the Schur complement, this choice of ensures

that 1 + 6 R -1 I * 6R > 0.
The integral I * is then strictly positive for all > 0. Developing the previous integral we get

I * = 1 + 6 ω(b) ω(a) T b a W 2 1 (u)R -1 W 2 1 T (u)du ω(b) ω(a) +2 ω(b) ω(a) T b a W 2 1 (u)ω(u)du +6 b a ω T (u)Rω(u)du.
It yields that

I = I * + (b -a) ω(b) ω(a) T W 1 1 ω(b) ω(a) - 1 + 6 ω(b) ω(a) T b a W 2 1 (u)R -1 W 2 1 T (u)du ω(b) ω(a) , Furthermore, simple computations show that b a W 2 1 (u)R -1 W 2 1 T (u)du = 36 (b -a) 2     b a (u -a) 2 duR b a (u -a)(b -u)duR * b a (b -u) 2 duR     = 36(b -a) 3 6(b -a) 2 W 1 1 = 6(b -a)W 1 1 ,
and consequently,

I = I * + (b -a) ω(b) ω(a) T W 1 1 ω(b) ω(a) -(1 + )(b -a) ω(b) ω(a) T W 1 1 ω(b) ω(a) I = I * -(b -a) ω(b) ω(a) T W 1 1 ω(b) ω(a) .
Following the same method proposed in [START_REF] Gu | Stability of timedelay systems[END_REF], it is possible to apply the Jensen's inequality to the integral I * . This leads to

I * ≥ 1 b -a b a ξ(u)du T 1 + 6 R -1 I * 6R b a ξ(u)du and noting that b a ξ(u)du =     -3(b -a) [ R R ] ω(b) ω(a) b a ω(u)du     ,
we have

I * ≥ 9(1 + )(b -a) 6 ω(b) ω(a) T R R R R ω(b) ω(a) -6 ω(b) ω(a) T R R b a ω(u)du + 6 b -a b a ω(u)du T R b a ω(u)du .

Now, introducing the vector

Ξ = 1 b -a b a Ω(u)du =     ω(b) ω(a) 1 b -a b a ω(u)du     ,
we have that, for all > 0

I - 3(b -a) 2 Ξ T R R -2R * R -2R * * 4R Ξ ≥ (b -a) 2 ω(b) ω(a) T 3 R R * R -2W 1 1 ω(b) ω(a) ≥ - (b -a) 2 (ω(b) -ω(a)) T R(ω(b) -ω(a))
Note that the left-hand side of the previous inequality does not depend on the parameter while the right-hand does.

The previous inequality holds for all > 0. Then taking → 0 leads to the following inequality.

I ≥ 3(b -a) 2 Ξ T R R -2R * R -2R * * 4R Ξ
Replacing I by its lower bound into (10) proves the results. ♦ At this stage, we have provided a useful inequality which has been proved to be less conservative than Jensen'inequality. The next section is devoted to the construction of a new stability criterion.

STABILITY ANALYSIS OF SYSTEMS WITH CONSTANT DELAY

A Lyapunov-Krasovskii functional approach

We present in this sub-section the main result of the article which is based on the use of the Wirtinger's inequality developed in section 3. This approach is based on a modified Lyapunov-Krasovskii functional and allows us to establish the main theorem for the robust delay range stability analysis. Consider a linear time-delay system of the form:

ẋ(t) = Ax(t) + A d x(t -h) ∀t ≥ 0, x(t) = φ(t) ∀t ∈ [-h max , 0], (13)
where x(t) ∈ R n is the state vector, φ is the initial condition and A, A d ∈ R n×n are constant matrices. The delay h is a positive scalar ans satisfies the constraints: h ∈ [h min , h max ] (14) where h min , h max are given positive constants. In the following, we aim at assessing the stability of system (13) with delay constraints (14) via the an appropriate Lyapunov-Krasovskii functional. Based on the previous inequality, the following theorem is provided. Theorem 6. For a given constant delay h, assume that there exist n × n matrices

P = P T > 0, S = S T > 0 and R = R T > 0 Π 1 (h) > 0, Π 2 (h) < 0 (15) with Π 1 (h) = P Q * Z + S/h , (16)
Π 2 (h) = Π 0 2 (h) - 1 h W 2 (R), (17)
Π 0 2 =   ∆ 0 2 P A d -Q hA T Q + hZ * -S hA T d Q -hZ * * 0   +   A T A T d 0   hR   A T A T d 0  
where ∆ 0 2 = P A+A T P +S+Q+Q T and the matrix W 2 (R) is given in Lemma 5, then the system (13) is asymptotically stable for the constant delay h.

Proof : Consider a Lyapunov-Krasovskii functional of the form

V (x t , ẋt ) =   x(t) t t-h x(s)ds   T P Q Q T Z   x(t) t t-h x(s)ds   + t t-h x(s)Sx(s)ds + t t-h (h -t + s) ẋ(s)R ẋ(s)ds (18)
The previous functional is the one of the simplest type of functionals to derive delay-dependent stability conditions. First of all, following [START_REF] Gu | Stability of timedelay systems[END_REF] and using Jensen inequality, a lower-bound for V can be easily found:

V (x t , ẋt ) ≥   x(t) t t-h x(s)ds   T Π 1 (h)   x(t) t t-h x(s)ds   + t t-h (h -t + s) ẋ(s)R ẋ(s)ds, ( 19
)
and it is clear that the positive definiteness of the matrices P , S, R and Π 1 (h) implies the positive definiteness of the functional V . The derivative of the functional along the trajectories of the system (13) leads to

V (x t , ẋt ) = 2x T (t)P [Ax(t) + A d x(t -h)] +2x T (t)Q(x(t) -x(t -h)) +2[Ax(t) + A d x(t -h)] T Q t t-h x(s)ds +2(x(t) -x(t -h)) T Z t t-h x(s)ds +x T (t)Sx(t) -x T (t -h)Sx(t -h) +h[Ax(t) + A d x(t -h)] T R[Ax(t) + A d x(t -h)] - 0 -h ẋT (t + s)R ẋ(t + s)ds.
The previous equality can be reformulated as

V (x t , ẋt ) =     x(t) x(t -h) 1 h t t-h x(s)ds     Π 0 2 (h)    x(t) x(t -h) 1 h t t-h x(s)ds     - 0 -h ẋT (t + s)R ẋ(t + s)ds.
According to Lemma 5, the following upper-bound of the derivative of the functional is then obtained:

V (x t , ẋt ) =     x(t) x(t -h) 1 h t t-h x(s)ds     Π 2 (h)     x(t) x(t -h) 1 h t t-h x(s)ds     (20)
where Π 2 (h) is defined in (17). Then if the condition (15) from Theorem 6 are satisfied, the solutions of the system (13) are asymptotically stable. ♦

Delay range stability criterion

At this stage, the proposed criterion of Theorem 6 can perform the stability analysis only for a given delay h and not for a delay satisfying the constraints (14) h ∈ [h min , h max ]. Nevertheless, this can be done by modifying slightly the LMIs conditions as proposed in the following theorem.

Theorem 7. For a given constant delay h satisfying the delays constraints ( 14) h ∈ [h min , h max ], assume that there exist n × n matrices P = P T > 0, S 1 = S T 1 > 0 and

R 1 = R T 1 > 0 Π1 > 0, Π2 (h min ) < 0, Π2 (h max ) < 0, (21) with Π1
= P Q * Z + S 1 , Π2 (h) = Ũ -W 2 (R 1 ) Ṽ * -R 1 , Ũ =   Υ 0 2 P A d -Q hA T Q + hZ * -hS 1 hA T d Q -hZ * * 0   , Υ 0 2 = P A + A T P + hS 1 + Q + Q T , Ṽ =   hA T R 1 hA T d R 1 0   ,
then the system ( 13) is asymptotically stable for the all constant delays h ∈ [h min , h max ].

Proof : Applying Theorem 6 by choosing

S 1 = 1 h S, R 1 = 1 h R yields to Π1 > 0 and Π 2 (h) = Π 0 2 (h) -W 2 (R 1
) < 0, where the matrix W 2 (R 1 ) is given in Lemma 5 and where

Π 0 2 =   Υ 0 2 P A d -Q hA T Q + hZ * -hS 1 hA T d Q -hZ * * 0   +   hA T R 1 hA T d R 1 0   R -1 1 [ hR 1 A hR 1 A d 0 ] .
Using Schur's complement to the inequality Π 2 (h) = Π 0 2 (h) -W 2 (R 1 ) < 0 leads to Π2 (h) < 0. Hence, the system (13) is stable for a given delay h if Π1 > 0 and Π2 (h) < 0. Therefore, simple inspection shows that Π2 (h) is affine and thus convex, in h. So, the condition Π2 (h) has to be assessed only at the two vertices of the polytope generated by the interval of h which concludes the proof. ♦

EXAMPLES

First example: delay dependent case

Consider the linear time-delay systems (13) with the matrices

A = -2 0 0 -0.9 , A d = -1 0 -1 -1
This system is a well-known delay dependent stable system, that is the delay free system is stable and the maximum allowable delay h max = 6.1721 can be easily computed by delay sweeping techniques. To demonstrate the effectiveness of our approach, results are compared to the literature and are reported in burden, compared to our proposal. Notice that the robust approach [START_REF] Kao | Stability analysis of systems with uncertain time-varying delays[END_REF] give a very good upperbound with a similar computational complexity than our result.

Theorem 7 addresses also the stability of systems with interval delays, which may be unstable for small delays (or without delays) as it is illustrated with the second example.

Second example: delay range case

Consider the linear time-delay systems (13) with the matrices A = 0 1 -2 0.1 , A d = 0 0 1 0 As Re(eig(A + Ad)) = 0.05 > 0, the delay free system is unstable and in this case, the results to assess the stability of this system are much more scarce. They are often related to robust analysis [START_REF] Ariba | Stability interval for time-varying delay systems[END_REF] or discretized Lyapunov-Krasovskii functionals [START_REF] Gu | Stability of timedelay systems[END_REF]. The results are reported in Table 2. In this example, Theorem 6 delivers better result than [START_REF] Gu | Stability of timedelay systems[END_REF] and [START_REF] Ariba | Stability interval for time-varying delay systems[END_REF] with a fewer number of variables to be optimized. Notice that with the discretization technique from [START_REF] Gu | Stability of timedelay systems[END_REF], increasing N yields to a better result approaching the analytical bound.

CONCLUSIONS

In this paper, we have provided a new useful inequality which encompass the Jensen's inequality. In combination with a simple Lyapunov-Krasovskii functional, this inequality leads to new stability criteria for linear time delay system. This new result has been expressed in terms of LMIs and has shown on numerical examples a large improvement of existing results using only a limited number of matrix variables. More generally, this preliminary result can be extended to all existing results which are using the Jensen's inequality, especially the case of time varying delay or sampled data systems.

  The proof is based on the second Wirtinger inequality. Consider the function z(u) = (b -u)ω(a) + (u -a)ω(b) -(b -a)ω(u) where u ∈ [a, b]. As the function z(u) satisfies the conditions of Lemma 3, i.e. z(a) = z(b) = 0, we apply the second Wirtinger inequality to z(u). The derivative of z with respect to u is given by: d du z(u) = ω(b) -ω(a) -(b -a) ω(u), and the right-hand-side of the Wirtinger inequality can be rewritten as: b a żT (u)R ż(u)du = (b -a) 2 b a ωT (u)R ω(u)du b a (ω(b) -ω(a)) T R(ω(b) -ω(a))du.

  b

  For given symmetric positive definite matrices R > 0 and for any differentiable signal ω in [a, b] → R n , the following inequality holds: b

Table 1 .

 1 All papers except[START_REF] Kao | Stability analysis of systems with uncertain time-varying delays[END_REF] use Lyapunov theory in order to derive stability criteria. Many recent papers give the same result since they are intrinsically based on the same Lyapunov functional and use the same bounding cross terms technique i.e. Jensen inequality. Some papers[START_REF] Ariba | Stability interval for time-varying delay systems[END_REF],[START_REF] Sun | Improved delay-range-dependent stability criteria for linear systems with time-varying delays[END_REF], which use an augmented Lyapunov can go further but with a numerically increasing

	Theorems	hmax	number of variables
	Gouaisbaut and Peaucelle [2006]	4.472	1.5n 2 + 1.5n
	He et al. [2007]	4.472	3n 2 + 3n
	Shao [2009]	4.472	2.5n 2 + 1.5n
	Sun et al. [2010]	4.472	3n 2 + 3n
	Kao and Rantzer [2007]	6.1107	1.5n 2 + 9n + 9
	Ariba et al. [2010]	5.120	7n 2 + 4n
	Sun et al. [2010]	5.02	18n 2 + 18n
	Kim [2011]	4.97	69n 2 + 5n
	Gu et al. [2003] (N=1)	6.059	5.5n 2 + 2.5n
	Th.6	5.901	3n 2 + 2n
	Th.6 with Q = Z = 0	4.472	1.5n 2 + 1.5n

Table 1 .

 1 Results for Example 1.

	Theorems	h min	hmax	number of variables
	He et al. [2007]	∅	∅	3n 2 + 3n
	Ariba et al. [2010]	0.102	1.424	7n 2 + 4n
	Gu et al. [2003] (N=1) 0.1006 1.4272	5.5n 2 + 2.5n
	Th.6	0.1006	1.473	3n 2 + 2n
	Th.6 with Q = Z = 0	∅	∅	1.5n 2 + 1.5n

Table 2 .

 2 Results for Example 2.


François Maupetit

Olivier Ramalho

Nathalie Bonvallot

Antoine Villa

Dominique Gombert

Characterization of glycol ether emissions from a water-based paint: chamber tests and field experiments

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Mélanie Nicolas

Olivier Ramalho

François Maupetit

Jean-Francois Doussin

Jean-Louis Colin

Predicting indoor ozone and NOx concentrations

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Raymond Ghandour
email: rghandou@hds.utc.fr

Alessandro Victorino

Ali Charara
email: acharara@hds.utc.fr

Daniel Lechner
email: daniel.lechner@inrets.fr

Evaluation du risque de dérapage basée sur l'estimation du coefficient d'adhérence maximal

Keywords: Dynamique de véhicule, indicateurs de risque, coefficient d'adhérence maximal, modèle de Dugoff, Optimisation non-linéaire

La sécurité routière nécessite un diagnostic précis du comportement dynamique d'un véhicule routier. Ce diagnostic peut être décrit par l'évaluation des indicateurs de risque. L'estimation de ces indicateurs est d'une grande importance pour prévenir le conducteur d'un danger probable ou pour agir sur la dynamique du véhicule pour éviter le danger. Nous proposons dans cet article, une méthode de determination d'un indicateur de risque de dérapage basée sur l'estimation du coefficient d'adhérence maximal par le modèle quasi-statique de Dugoff, un observateur des efforts latéraux et verticaux et la méthode d'optimisation non linéaire de Levenberg-Marquardt.

I. Introduction

Le risque est la probabilité d'une exposition à un danger potentiel, attaché à une situation ou une activité dans une période donnée ou dans des circonstances déterminées. L'accident représente la réalisation de ce risque. Plusieurs variables peuvent caractériser des situations à risques et donc peuvent être utilisées pour l'avertissement.

Une des tendances les plus remarquables de l'industrie automobile est l'émergence de technologies de sécurité active. Au lieu de la protection passive et de la résistance aux collisions, l'accent a été déplacé vers la prévention des accidents et la réduction des dommages en utilisant des techniques de sécurité active [START_REF] Aga | Analysis of vehicle stability control (VSC)'s effectiveness from accident data[END_REF]. Pour développer des dispositifs efficaces de sécurité active, il est nécessaire d'estimer avec précision le comportement dynamique du véhicule. A base de cette dynamique, plusieurs indicateurs de risques ont été conçu pour identifier les situations de risque et avertir le conducteur sur ces situations.

Parmi les indicateurs les plus traités dans la littérature, on cite : le transfert de charge latérale (LTR) [START_REF] Kamnik | Roll dynamics and lateral load transfer estimation in articulated heavy freight vehicles[END_REF], [START_REF] Doumiati | Lateral load transfer and normal forces estimation for vehicle safety : experimental test[END_REF], l'excès de roulis [START_REF] Yoon | Design of a rollover index-based vehicle stability control scheme[END_REF], [START_REF] Bouton | A rollover indicator based on the prediction of the load transfer in presence of sliding : application to an All Terrain Vehicle[END_REF], la possible sortie de voie (Sous-virage et survirage) [START_REF] Ewards | Sliding mode observers for vehicle mode detection[END_REF], l'excès de vitesse lors d'un virage [START_REF] Sentouh | Analyse du risque et détection de situations Limites -Application au développement des systèmes d'alerte au conducteur[END_REF] et le manque d'adhérence [START_REF] Ahn | Robust Nonlinear Observer To Estimate Road Friction Coefficient and Tire Slip Angle[END_REF] - [START_REF] Villagra | An algebraic approad for maximum friction estimation[END_REF]. Certains sont liés au mode longitudinal et d'autres sont liés au mode latéral. Ces indicateurs ont été traités dans la littérature en utilisant différentes méthodes d'estimation et pour différents types de véhicules.

Dans cet article nous proposons un indicateur de risque de dérapage basé sur l'observation robuste du coefficient d'adhérence latérale maximal (appelé ici adhérence maximale). L'adhérence maximale est un paramètre de la dynamique du véhicule étroitement lié à l'interaction pneumatique-chaussée, nécessaire au calcul ou à l'observation des forces au niveau des pneus, et qui caractérise l'état de la chaussée (sèche, mouillée, enneigée ou glacée). L'adhérence maximale créée au point de contact pneuchaussée est fonction de plusieurs paramètres, tels que les forces latérales et verticales au niveau des pneus, l'état de la chaussée, la rigidité de dérive. Une estimation du coefficient d'adhérence maximal en temps réel, pendant le déplacement du véhicule, en utilisant des mesures faciles d'accès (accélérations, vitesses, angle de braquage, etc), permet de fournir au conducteur (ou à un système de pilotage automatique) un moyen d'avertissement d'une possible perte d'adhérence, qui éviterait une sortie de route imminente.

Une motivation pour l'estimation de l'indicateur de risque de dérapage, est donnée par le nombre important d'accidents liés aux conditions de la route, qui pourrait être réduit si le conducteur était correctement informé sur les conditions limites du véhicule en temps réel. La sécurité peut aussi être augmentée, puisque la performance de nombreuses applications dans le véhicule peut être améliorée par la connaissance de l'état actuel de l'adhérence et donc par l'information sur les possibles situations de dérapage.

L'article est organisé comme suit : Dans la section II, on définit un état de l'art sur les indicateurs de risques traitées dans la littérature ainsi que sur les travaux d'estimation du coefficient d'adhérence maximal. Dans la section III, on présente la méthode d'estimation itérative du coefficient d'adhérence latérale maximale. Dans la section IV, on définit l'indicateur de dérapage (IDD). Dans la section V, on présente les résultats qui valident la performance de cette méthode, suivie de la conclusion et des perspectives.

II. Etat de l'art

L'estimation des indicateurs de risque est de plus en plus traitée dans la littérature en se basant sur plusieurs paramètres de la dynamique du véhicule.

Dans [START_REF] Kamnik | Roll dynamics and lateral load transfer estimation in articulated heavy freight vehicles[END_REF] le transfert de charge latéral (LTR) est estimé pour les véhicules lourds en se basant sur le coefficient d'adhérence utilisé et l'angle de dérive des pneus et sans la connaissance des forces normales, alors qu'en [START_REF] Doumiati | Lateral load transfer and normal forces estimation for vehicle safety : experimental test[END_REF] le LTR est estimé en se basant sur l'estimation des efforts verticaux appliqués sur chaque roue. Ces efforts sont estimés par un algorithme embarqué. Dans [START_REF] Yoon | Design of a rollover index-based vehicle stability control scheme[END_REF] un indicateur de risque de renversement causé par l'excès de roulis est estimé en utilisant des estimations de l'angle de roulis et la mesure de l'accélération latérale, alors qu'en [START_REF] Bouton | A rollover indicator based on the prediction of the load transfer in presence of sliding : application to an All Terrain Vehicle[END_REF] cet indicateur est estimé pour les véhicules lourds, basé sur la prévision du transfert de charge latérale en s'appuyant sur les modèles de véhicules y compris les modèles à effets de glissement. Dans [START_REF] Ewards | Sliding mode observers for vehicle mode detection[END_REF], un algorithme qui permet de détecter et prédire les manoeuvres dangereuses est utilisé pour éviter les situations de sous-virage et de survirage.

Enfin dans [START_REF] Sentouh | Analyse du risque et détection de situations Limites -Application au développement des systèmes d'alerte au conducteur[END_REF] le cas de la vitesse excessive à l'approche d'un virage est traité, ainsi que les accidents de sortie de voie.

Dans cet article, nous présentons une méthode, pour estimer un indicateur de dérapage, basée sur l'estimation du coefficient d'adhérence maximal. Cette dernière est basée sur l'estimation robuste des efforts pneumatique chaussée, et en utilisant la méthode itérative d'optimisation nonlinéaire de Levemberg-Marquardt. L'estimation du coefficient d'adhérence latérale maximale est elle aussi de plus en plus traitée dans la littérature en utilisant plusieurs modèles de pneumatique et pour plusieurs états des chaussées. Plusieurs publications, estiment ce coefficient mais pour un état de route bien déterminé [START_REF] Erdogan | Friction coefficient measurement for autonomous winter road maintenance[END_REF]. D'autres [START_REF] Haffner | Comparison of two Methods for the Estimation of the Maximum Coefficient of Friction in a Cornering Maneuver of a Passenger Vehicle[END_REF], [START_REF] Liu | Road Friction Coefficient Estimation For Vehicle Path Prediction[END_REF], [START_REF] Ahn | Robust Nonlinear Observer To Estimate Road Friction Coefficient and Tire Slip Angle[END_REF], et [START_REF] Muller | Estimation of the Maximum Tire-Road Friction Coefficient[END_REF], estiment ce coefficient pour plusieurs types de chaussées. Les méthodes d'estimation développées sont, principalement basées sur des méthodes d'optimisation de l'erreur, comme la méthode de gradient descendant [START_REF] Haffner | Comparison of two Methods for the Estimation of the Maximum Coefficient of Friction in a Cornering Maneuver of a Passenger Vehicle[END_REF] et la méthode des moindres carrées récursives [START_REF] Liu | Road Friction Coefficient Estimation For Vehicle Path Prediction[END_REF]. Dans ces travaux, les efforts au point de contact pneu-chaussée sont considérés connus, mesurés par des capteurs extrêmement coûteux, ou calculés par des méthodes en boucle ouverte.

Dans [START_REF] Haffner | Comparison of two Methods for the Estimation of the Maximum Coefficient of Friction in a Cornering Maneuver of a Passenger Vehicle[END_REF], deux méthodes pour estimer le coefficient d'adhérence maximale sont comparées. La première méthode est basée sur la méthode du gradient alors que la deuxième méthode est basée sur une classification stochastique avec des intervalles de confiance. Les deux méthodes se concentrent sur la dynamique transversale. Dans [START_REF] Ahn | Robust Nonlinear Observer To Estimate Road Friction Coefficient and Tire Slip Angle[END_REF], un observateur non linéaire pour estimer le coefficient d'adhérence maximal et l'angle de dérive est présenté. Il est basé sur un état simultané et sur un cadre d'estimation des paramètres.

Dans [START_REF] Muller | Estimation of the Maximum Tire-Road Friction Coefficient[END_REF], l'adhérence maximale pneumatique chaussée disponible pour identifier les états de route sèche et lubrifiée est estimée. Cette méthode repose sur l'hypothèse de faible glissement, et d'une faible adhérence utilisée, en conduite normale. Dans [START_REF] Lechner | On-board estimation of friction potential[END_REF], un système pour estimer en temps réel le coefficient d'adhérence maximale est developpé. Ce système est basé sur une méthode de modèle d'appariement et nécessite environ 16 capteurs pour estimer l'adhérence maximale sur chaque roue. Les meilleurs résultats sont obtenus dans le sens longitudinal et pour un minimum d'accélération (2m/s 2 ) et de vitesse (15 km/h). Dans [START_REF] Li | Integrated longitudinal and lateral tire/road friction modeling and monitoring for vehicle motion control[END_REF] une recherche sur l'évolution de la modélisation pneumatique chaussée et sur les différentes techniques de contrôle associées est présentée, pour estimer le coefficient d'adhérence maximal et identifier les conditions de la route. Ces auteurs ont successivement étudié les recherches antérieures sur la modélisation longitudinale, latérale, pneumatique chaussée et les ont appliqué dans la conception des contrôleurs/observateurs de traction, de braquage et de freinage du véhicule. Dans [START_REF] Rabhi | Estimation of contact forces and tire road friction, 15th Mediterranean conference on control and automation[END_REF] une estimation du coefficient d'adhérence maximal est proposée en se basant sur l'estimation des forces de contact pneumatique chaussée et trois observateurs en cascade. Dans [START_REF] Bouteldja | Estimation des efforts latéraux à l'aide d'un observateur pour la détection du risque de mise en portefeuille des véhicules articulés[END_REF], la detection du risque de mise en portefeuille est acquise en se basant sur l'estimation des forces de contact pneumatique chaussée et en considérant le µ max mesuré par un capteur. Enfin, dans [START_REF] Villagra | An algebraic approad for maximum friction estimation[END_REF], une nouvelle approche pour estimer l'adhérence maximale est présentée. Elle est basée sur l'utilisation des méthodes algébriques pour l'estimation et le diagnostic par différentiation numérique des signaux bruités.

Comme mentionné précédemment, l'adhérence maximale est liée aux efforts développés au point de contact pneumatique chaussée. Cela est donc un paramètre important dans le calcul de ces efforts, et de l'angle de dérive. Dans nos travaux précédents, nous avons développé des observateurs pour estimer les forces et l'angle de dérive [START_REF] Doumiati | Estimation embarquée des efforts latéraux et de la dérive d'un véhicule : validation expérimentale[END_REF]. En revanche, l'adhérence maximale a été considérée comme un paramètre constant et supposé connu. La méthodologie utilisée pour l'estimation de l'adhérence maximale, présentée dans cet article, est ainsi une continuation des travaux antérieurs sur les observateurs des forces, pour les rendre robustes aux différents types de chaussées. Cette méthodologie sera présentée en détails dans la section III.

III. Estimation du coefficient d'adhérence maximale

A. Méthode d'estimation

Le schéma bloc de la figure 1 représente la méthode utilisée pour calculer l'indicateur du risque de dérapage en se basant sur l'estimation du coefficient d'adhérence maximale.

Le premier bloc (Observateurs(Données)) est constitué des observateurs pour estimer les forces latérales (F yij ), les forces verticales (F zij ), et l'angle de dérive (α ij ). Cette méthode est présentée en détails dans [START_REF] Ghandour | Tire/road friction coefficient estimation applied to road safety[END_REF] et [START_REF] Doumiati | Estimation embarquée des efforts latéraux et de la dérive d'un véhicule : validation expérimentale[END_REF]. Ces observateurs sont basés sur un modèle de véhicule 4 roues et utilisant la méthode d'estimation non linéaire UKF (Unscented Kalman Filter). La méthode d'estimation de µ max , décrite

F yij = -C αi tanα ij .f (λ) (1)
où C αi est la rigidité de dérive et f (λ) est donnée par : Le dernier bloc (Indicateur de dérapage (IDD)), constitue la partie du calcul de l'indicateur de dérapage. Ce bloc sera présenté en détails dans la section IV.

f (λ) = (2 -λ)λ, if λ < 1 1, if λ ≥ 1 (2)
λ = µ max F zij 2C αi |tanα ij | (3)
La formalisation de l'algorithme d'optimisation est basée sur trois hypothèses, qui seront validées plus loin dans cette section : Hypothèse 1 : Le modèle quasi-statique de Dugoff, peut être utilisé dans l'estimation de la force latérale pour chaque pneu.

Hypothèse 2 : Si l'hypothèse 1 est vérifiée, nous supposons à chaque instant k de la trajectoire, il est possible de calculer une erreur e ij = F yij -F yij où F yij est la force latérale calculée avec le modèle de Dugoff et F yij la force latérale estimée par l'observateur.

Hypothèse 3 : Nous supposons que : -La rigidité de dérive est connue ; -Les forces latérales (F yij ) et verticales (F zij ) et l'angle de dérive (α ij ) sont estimés par les observateurs présentés dans [START_REF] Ghandour | Tire/road friction coefficient estimation applied to road safety[END_REF] et [START_REF] Doumiati | Estimation embarquée des efforts latéraux et de la dérive d'un véhicule : validation expérimentale[END_REF].

B. Validation des hypothèses

Avant de développer l'algorithme pour l'estimation de l'adhérence maximale et donc l'indicateur de dérapage, il nous a paru indispensable de vérifier la validité des hypothèses, établies au paragraphe précédent. Pour cela, nous avons validé le modèle en utilisant les données d'une trajectoire effectuée par la Peugeot 307 de l'INRETS-MA sur le site public de Bourbriac (sur une chaussée sèche). La figure 2, montre l'évolution de la force latérale F y11mesure développée au niveau du pneu avant droit et mesurée par des capteurs (ligne continue en rouge dans la figure).La validation des hypothèses 1 et 2 consiste à calculer Fy en utilisant le modèle de Dugoff (équations 1-3), pour un coefficient d'adhérence latérale maximal µ max = 1.1 (cette valeur n'est pas connue dans la pratique, mais elle correspond à un µ max qui caractérise une chaussée sèche), et la comparer avec la Fy mesurée par les capteurs. Dans la figure 2, Fy calculée par Dugoff est montrée par la ligne poitillée en bleu. On voit que les courbes se superposent et que l'erreur e 11 = F ydugof f11 -F y11 est petite (avec une moyenne de 46.8015 N). Cela indique que les hypothèses 1 et 2 ci-dessus ne sont pas rejettées.

C. Algorithme d'optimisation

La méthode d'optimisation utilisée est la méthode de Levenberg-Marquardt. L'algorithme de Levenberg-Marquardt (LM) est une technique itérative pour localiser le minimum d'une fonction à plusieurs variables, qui est exprimée comme la somme des carrés des fonctions à valeurs réelles non-linéaires. Elle est devenue une technique standard pour résoudre les problèmes de moindres carrés non linéaires, et largement adoptée dans un grand nombre de disciplines [START_REF] Press | Numerical Recipes in C (3rd Edition)[END_REF], [START_REF] Hartley | Multiple View Geometry in computer vision[END_REF]. La méthode LM peut être considérée comme une combinaison entre la méthode du gradient descendant et la méthode de Gauss-Newton. Quand une La description de l'algorithme de Levenberg-Marquardt basée sur la minimisation de la fonction d'erreur développée entre les forces latérales des observateurs et ceux du modèle de Dugoff, est présentée en détails dans [START_REF] Ghandour | Tire/road friction coefficient estimation applied to road safety[END_REF] et [START_REF] Lourakis | A brief description of the Levemberg-Marquardt Algorithm Implemented by levmar[END_REF].

IV. Indicateur de dérapage

Après l'application de la méthode décrite dans la section III, le coefficient d'adhérence maximal est estimé. Ayant ce coefficient estimé, le risque associé peut être identifié en se basant sur la valeur maximale du coefficient d'adhérence. Le risque associé est la perte du contrôle du véhicule. Nous avons développé un indicateur pour estimer une possible perte de contrôle du véhicule. L'indicateur de dérapage (IDD) est donné par l'équation suivante :

IDD = (1 -|(µ max -µ ij ) /µ max | , if µ ij ≤ µ max 1, if µ ij > µ max (4)
Avec µ max le coefficient d'adhérence maximale, et µ ij le coefficient d'adhérence utilisé du pneumatique pendant le mouvement du véhicule.

Les valeurs possibles que IDD peut avoir sont définies par : 0 ≤ IDD ≤ 1. Si la valeur de IDD dépasse 0.8, alors le véhicule est soumis à une possible situation de dérapage.

Les résultats de l'indicateur de dérapage, sont présentés dans la section V.

V. Résultats expérimentaux et validation

L'efficacité de la méthode d'estimation de l'indicateur de risque de dérapage basée sur l'estimation du coefficient d'adhérence maximal, a été testée en utilisant des données expérimentales. Les essais sur la route ont été effectués avec la Peugeot 307 de l'INRETS-MA ( [START_REF] Lechner | Embedded laboratory for vehicle dynamic[END_REF]). ). C'est une Peugeot 307 équipée de nombreux capteurs parmi lesquels des accéléromètres, des gyromètres, des capteurs de mesure de l'angle de braquage, des capteurs linéaires de suspension, et des moyeux dynamométriques. Ces moyeux mesurent en temps réel les forces et les moments agissant au centre de la roue. La fréquence d'échantillonnage pour ces différents capteurs est de 100 Hz.

A. Véhicule de l'INRETS MA

Moyeux dynamométriques

Les données acquises par ce véhicule ont été utilisées pour tester la performance de la méthode présentée dans la section III. Les données expérimentales correspondent à une route sèche. Les test ont été fait sur table en utilisant des données acquises pendant le déplacement de notre véhicule expérimental.

B. Validation de la méthode

La voiture a effectué différentes manoeuvres sollicitant notamment la dynamique transversale. Trois différentes manoeuvres sur chaussée sèche sont effectuées sur trois voies publiques en France : Bourbriac, Callac et Rostronen.

La méthode proposée est testée sur les données acquises expérimentalement, en temps différés sur table, pour chaque manoeuvre avec l'application d'une fenêtre glissante sur les données. Chaque fenêtre correspond à 20 secondes de données avec un pas de 2 secondes au cours de la trajectoire. C'est-à-dire, on considère les 20 premiers secondes (≃ 2000 points de mesure à une fréquence de 100 Hz), on applique les méthodes décrites dans les sections 3 et 4 pour l'évaluation de l'IDD dans cette fenêtre. Ce processus continue en glissant la fenêtre en avant d'un pas de 200 points (2 secondes à 100 Hz).

Le premier essai est effectué par le véhicule sur la voie publique de Bourbriac avec une fréquence de 100 Hz. Les résultats sont présentés à la figure 4. La première partie (en La marque en rouge sur la trajectoire représente la partie qui correspond à un indicateur du dérapage supérieur à 0.8 et donc une valeur du coefficient d'adhérence mobilisé (utilisé) supérieure à 80% de la valeur du coefficient d'adhérence maximal. Dans cette partie, il y aura une alerte de danger.

Le coefficient d'adhérence mobilisé (utilisé) est également affiché avec le coefficient d'adhérence latérale maximale pour montrer le dépassement de 80% de µ max et donc justifier de plus l'occurrence d'une situation dangereuse. On constate pour cet essai que l'adhérence maximale pneumatique chaussée est bonne, sur presque toutes les fenêtres de la trajectoire. Le IDD dépasse la valeur 0.8 à cause d'une chute brutale de la vitesse de conduite juste avant un virage à la fin de la trajectoire.

Le second essai est effectué par le véhicule sur la route publique de Callac avec une fréquence de 100 Hz. Les résultats sont présentés à la figure 5.

La première partie (en haut) de la figure 5 présente la Le dernier essai est effectué par le véhicule sur la route publique de Rostronen avec une fréquence de 100 Hz. Les résultats sont présentés à la figure 6.

La première partie (en haut) de la figure 6 présente la trajectoire effectuée par le véhicule. Les autres parties de la figure indiquent que l'adhérence maximale pneumatique chaussée est bonne, sur toutes les fenêtres de la trajectoire. Le IDD dépasse la valeur 0.8 en virage et pour une conduite à haute vitesse (V itesse > 100Km/h).

Pour les trois essais, il y avait une grande correspondance entre les estimations et les valeurs réelles de la force latérale, et donc les valeurs estimées et la valeur réelle de µ max . Ces bons résultats confirment la validité de la méthode proposée.

VI. Conclusion et perspectives

Dans cet article, nous avons présenté une nouvelle méthode pour estimer un indicateur de dérapage basé sur l'es-

Fig. 1 .

 1 Fig. 1. Méthode d'estimation de l'indicateur de dérapage.

Fig. 2 .

 2 Fig. 2. Validation du modèle

Fig. 3 .

 3 Fig. 3. Véhicule d'essai de l'INRETS-MA.

Fig. 4 .

 4 Fig. 4. Essai Bourbriac

Fig. 5 .

 5 Fig. 5. Essai Callac

Fig. 6 .

 6 Fig. 6. Essai Rostronen

  où µ max représente le coefficient d'adhérence maximal et F zij est la charge normale appliquée au pneu ij. Ce modèle simplifié néglige l'effet des efforts longitudinaux. L'erreur entre cette valeur fournie par le modèle et la valeur de la force estimée est évaluée et constitue le critère d'arrêt de la boucle. Si cette erreur est inférieure à un cer-

tain seuil, alors le coefficient d'adhérence maximal utilisé par le modèle de Dugoff, est à un seuil près, celui développé au point de contact pneu-chaussée. La valeur du seuil est choisie de manière empirique issue d'un compromis entre la précision souhaitée et le temps d'exécution de la boucle de minimisation.