Environmental control of terpene emissions from Cistus monspeliensis L. in natural Mediterranean shrublands

Keywords: Cistus monspeliensis L., terpene emissions, soil nature, age, season

The large amount of volatile organic compound (VOC) emitted by vegetation modifies air quality contributing to both tropospheric ozone and secondary organic aerosol production. A better understanding of the factors controlling VOC emissions by vegetation is mandatory in order to improve emission estimates derived from tropospheric chemistry models. Although the Mediterranean shrublands are particularly abundant and rich in emitting species, their emission potential is poorly known. Focusing on a VOC-emitting shrub species widespread in the Mediterranean area (Cistus monspeliensis L.), we measured and analysed its emissions of terpenes taking into account the age of individuals, the season of sampling and the soil type.

Sampling was done under natural environmental conditions. Species of the genus Cistus are frequently reported to be storing species, although we found only one stored monoterpene and three sesquiterpenes in very low amount. Major emitted compounds were α-pinene and β-myrcene. Total terpene emissions were not influenced by plant age but emission of some individual terpenes was positively correlated with age. A strong seasonal effect was evidenced. A larger amount of terpenes was emitted during spring and summer than during fall and winter. Summer emission rates were nearly 70 times higher than winter emission rates. Total and individual terpene emissions were influenced by soil type; emissions on siliceous substrate were ca. 7 times higher than those on calcareous substrate. In conclusion, it appears clearly that environmental factors such as soil nature and season should be taken into account in order to achieve improved modelling of terpene emissions by shrub species.

Introduction

The chemical breakdown of BVOC in the atmosphere affects air quality and climate forcing by influencing the formation and life time of greenhouse gases and air pollutants such as ozone and methane as well as the formation and growth of secondary organic aerosols [START_REF] Monson | Biospheric trace gas fluxes and their control over tropospheric chemistry[END_REF]. The currently available air chemistry models are unable to reproduce some pollution events and one of the main problems seems to be related to the uncertainties linked to the biogenic emission inventory [START_REF] Moukhtar | Monoterpene emissions from Beech (Fagus sylvatica) in a French forest and impact on secondary pollutants formation at regional scale[END_REF]. Annual global emissions of non-methane VOC are estimated at around 1.3 Pg (10 15 g) of carbon and about half of this total is associated with emissions from terrestrial vegetation [START_REF] Guenther | The contribution of reactive carbon emissions from vegetation to the carbon balance of terrestrial ecosystems[END_REF]. The large amount of VOC emitted from vegetation modifies air quality contributing to both ozone [START_REF] Fehsenfeld | Emissions of volatile organic compounds from vegetation and the implications for atmospheric chemistry[END_REF] and secondary organic aerosol production [START_REF] Guenther | The contribution of reactive carbon emissions from vegetation to the carbon balance of terrestrial ecosystems[END_REF].

Among the large variety of VOC synthesised and emitted by plants, terpenes are a dominant class that plays a role in several ecological processes. They are involved in inter and/or intra-species communication (e.g. attraction of pollinators and defence against herbivores) as well as in protection mechanisms against oxidative stress [START_REF] Laothawornkitkul | Biogenic volatile organic compounds in the Earth system[END_REF]. Although our understanding of the sources, controls and effects of VOC has increased significantly over the past few decades, prediction of them still remains uncertain [START_REF] Laothawornkitkul | Biogenic volatile organic compounds in the Earth system[END_REF].

Terpene emissions are species-specific and influenced by various biotic and abiotic factors, in particular light and temperature. They show qualitative and quantitative variability according to seasons with generally maximum emissions in spring and summer [START_REF] Llusia | Seasonal patterns of terpene content and emission from seven Mediterranean woody species in field conditions[END_REF][START_REF] Angelopoulou | Diurnal and seasonal variation of the essential oil labdanes and clerodanes from Cistus monspeliensis L. leaves[END_REF]. The range of this variability differs from one species to another and therefore has to be integrated in a specific way into biogenic emission inventories.

Whereas the effect of climate-related factors on emissions is welldocumented, other factors such as plant age and soil nature constraints have been less thoroughly studied. It has been reported that plant age is related to terpene emission rate but the literature still remains inconclusive on this subject: whether in favour of a positive correlation [START_REF] Street | Effect of habitat and age on variations in volatile organic compound (VOC) emissions from Quercus ilex and Pinus pinea[END_REF][START_REF] Kim | Seasonal variations of monoterpene emissions from coniferous trees of different ages in Korea[END_REF] or a negative correlation [START_REF] Kim | Factors controlling natural VOC emissions in a southeastern US pine forest[END_REF][START_REF] Funk | Influence of nutrient availability, stand age, and canopy structure on isoprene flux in a Eucalyptus saligna experimental forest[END_REF]. In addition, many species occur on a wide range of substrate types with different chemical and structural properties. These soils, which differ in their pH and nutrient contents, can modify emissions, also in a species-specific way [START_REF] Ormeño | Production and diversity of volatile terpenes from plants on calcareous and siliceous soils: Effect of soil nutrients[END_REF]. Identifying such factors inducing variations in emission capacity is necessary to improve emission estimates and tropospheric chemistry modelling.

In the French Mediterranean region, strong anthrogenic NOx sources [START_REF] Cros | The ESCOMPTE program: an overview[END_REF] combined with the abundance of vegetation with high terpene emission rates favour fast ozone accumulation [START_REF] Simon | The biogenic volatile organic compounds emission inventory in France -Application to plant ecosystems in the Berre-Marseilles area (France)[END_REF]. Moreover cloudless sky conditions, intense solar radiation and high temperatures amplify this phenomenon.

Although emissions of Mediterranean tree species are well documented (Pinus halepensis L. (Ormeño et al., 2007b), Quercus ilex L. and Quercus suber L. [START_REF] Staudt | Isoprenoid emissions of Quercus spp. (Q. suber and Q. ilex) in mixed stands contrasting in interspecific genetic introgression[END_REF]), shrub emissions have been less widely studied. Yet, in the South of France, shrublands are the second most extensive land cover type (23% of landscape covered) after forest areas (31% of landscape covered).

This study focused on a widespread shrub species Cistus monspeliensis L. In spite of its abundance among Mediterranean vegetation, only few authors have studied this species under experimental conditions and their conclusions were inconsistent: Llusia and Peñuelas (1998) recorded low monoterpene emissions from C. monspeliensis whereas [START_REF] Owen | Light dependency of VOC emissions from selected Mediterranean plant species[END_REF] classified this species as a high monoterpene emitter. Thus, in order to improve biogenic emission inventories, we characterised branch terpene emissions of this species under field conditions. Our first objective was to determine the seasonal variability range of the emissions, and the second was to test the impact of plant age and soil nature (calcareous and siliceous).

Material and methods

Study area and species

Our study took place in South-Eastern France, under Mediterranean climate with strong seasonality (cool winters, warm dry summers), characterised by rather irregular precipitation with an annual average around 800 mm and mean annual temperature around 13.8 °C [START_REF] Rippert | Chapitre 2bis: Evaulation des potentialités forestières[END_REF].

Sampling plots were located in shrublands dominated by C. monspeliensis.

The genus Cistus (Cistaceae) comprises 21 species, distributed primarily in the Mediterranean basin. C. monspeliensis is native to the Mediterranean region and has a distinct resinous fragrance with hairy, lanceolate leaves that are sticky to the touch. This species is a pioneer shrub that can colonise wide areas after fire on both calcareous and siliceous substrates, where it can establish even-aged stands and persist over long periods. C. monspeliensis is one of the most widespread species in the circum-Mediterranean area [START_REF] Guzman | Long-distance colonization of the Western Mediterranean by Cistus ladanifer (Cistaceae) despite the absence of special dispersal mechanisms[END_REF].

Experimental protocol

Terpene sampling took place every 3-4 months between 11:00 a.m. and 03:30 shown in Table 1.

Age and soil effect were analysed in spring which is supposed to be an optimal season for emission with no limitation due to either low temperature [START_REF] Tingey | Influence of light and temperature on monoterpene emission rates from slash pine[END_REF] or drought [START_REF] Lavoir | Drought reduced monoterpene emissions from the evergreen Mediterranean oak Quercus ilex: results from a throughfall displacement experiment[END_REF]. Age effect was studied on siliceous substrate. Sampling involved young individuals (Y) from 4 to 7 years old and old individuals (O) from 6 to 18 years old. Age determination of each individual was done following the classical method of dendrochronology [START_REF] Schweingruber | Tree-rings, basics and applications of dendrochronology[END_REF]. Soil effect was analysed on calcareous (C) and siliceous (S) substrates, selecting only old individuals because they were more abundant on calcareous soil than younger ones.

In fine, three modalities with three replicates each were sampled: young shrubs on siliceous soil (YS 1 YS 2, YS 3 ,), old shrubs on siliceous soil (OS 1 , OS 2, OS 3 ) and old shrubs on calcareous soil (OC 1 , OC 2, OC 3 ). Characteristics of sampled plots are summarised in Table 2 (see section 2.5 for experimental details). No significant difference of elevation, aspect or slope between modalities were observed (Kruskal-Wallis, H = 0.36 and p = 0.837 for altitude, H = 3.47 and p = 0.177 for aspect, and H = 1.16 and p = 0.561 for slope).

2.3. Terpene emissions 2.3.1. Terpene sampling Terpene emissions were sampled using a dynamic bag enclosure system following dynamic headspace sampling techniques [START_REF] Tholl | Practical approaches to plant volatile analysis[END_REF]. Dynamic enclosure technique keeps environmental parameters relatively constant and closer to ambient level, thus we can obtained realistic emission rates [START_REF] Ortega | Approaches for quantifying reactive and low-volatility biogenic organic compound emissions by vegetation enclosure techniques -Part A[END_REF]. The bag enclosure system, made of Teflon film (FEP), was carefully applied avoiding contact with leaves. We enclosed one healthy branch per plant. Healthy state of leaves was determined on the basis of a visual check [START_REF] Llusia | Seasonal patterns of terpene content and emission from seven Mediterranean woody species in field conditions[END_REF]. Enclosed branch carried a range from 3 to 15 g of leaf dried weight including sun and shade exposed leaves.

The system consisted of 7 bag enclosure units (5-10 L): 6 to sample Cistus emissions and 1 for blank sampling with no branch. Each bag enclosure was designed with inlet and outlet air streams. For this purpose, they were continuously flushed with clean air supplied by a zero-air generator (Perma Pure, Zero-Air™ Generator) through an inert Tygon tube (Tygon® Fuel and lubricant tubing, i.d: 8 mm). After enclosing the branch within the bag system, we performed an equilibration step for 30 min during which air in the bag is renewed twice. This time made it possible to achieve 80-90% of steady state for concentrations in the enclosure according to the equation given by [START_REF] Ortega | Approaches for quantifying reactive and low-volatility biogenic organic compound emissions by vegetation enclosure techniques -Part A[END_REF]. For this adaptation/renewal step and sampling, inflowing air (Q i ) was kept constant at 385 ± 12 ml min -1 (mean ± SE). Q i was higher than Q o (outflowing air) which allowed the air bag to be slightly over-pressurised, preventing leaf contact with the Teflon film and outside ambient air penetration. Inflowing air (Q i ) was accurately measured with a digital mass-flow controller (Aalborg® CFC17, 0-1000 ml) and Q o was measured with a bubble flow meter (0-280 ml min -1 , GPE Meterate 314-140/084), placed immediately after each glass sorbent tube. The outgoing terpenes were collected on glass sorbent tubes filled with preconditioned Tenax TA (Varian®,(20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30)(31)(32)(33)(34)(35)150 mg,l: 160 mm,o.d.: 5 mm,i.d.: 3 mm) using a pumping system (KNF Neuberger® N022AT.18), placed downstream from the adsorbent tubes. Terpenes were collected at a rate of 107 ± 6 ml min -1 for 15 min. The sampled air volume was calculated in order to optimise the signal/threshold ratio without exceeding the breakthrough volumes of each compound. During emission sampling, external PAR (Photosynthetically Active Radiation, EMS-7 Plant Canopy Transmission Meter, Surechem), such as air temperature and relative humidity (inside and outside the enclosure bag) was monitored with a time interval of 2 minutes. Internal and external air temperature were significantly correlated (r=0.940 and p<0.001), internal temperature being higher than external temperature about 3°C. Relative humidity inside the bag ranged from 23% to 86% depending on season which is in the range observed under natural conditions. After sampling, tubes were conserved in an icebox with liquid nitrogen (maximum 6 h) until being stored at -20 °C in the laboratory (analyses were performed within a maximum of three weeks).

Terpene analysis

For quantitative purposes, an internal standard (dodecane) was injected in adsorbent tubes (100 ng). Terpenes adsorbed on Tenax were analysed by a gas chromatography system (GC) fitted with a Flame Ionization Detector (FID) (HP®5890 series II). Prior to thermaldesorption, a preflush phase was run (3 min, 10 ml min -1 , 60 °C) to allow humidity in the Tenax to be evacuated. Then, thermaldesorption (Thermal Desorption Cold Trap injector, Varian®, CP4020-TCT, model) was carried out under nitrogen flow (10 min, 50 ml min-1, 250 °C) and cryogenic concentration in a silica capillary trap, cooled with liquid nitrogen at -100 °C. Compounds were then separated in the non-polar chromatographic column (Ultra 2, 50 m x 0.2 mm x•0.25 lm) through a temperature programme from 60 °C to 220 °C, at a rate of 3 °C min-1, then 220 °C (isothermal) for 5 min (Ormeño et al., 2007a). The identity of peaks was confirmed by matching the retention time of each compound with that of authentic references (Sigma-Aldrich, one standard for each identified compound listed in table 3 and4) and by comparison of calculated retention index with those found in the literature [START_REF] Adams | Identification of Essential Oil Components by Gas Chromatography/Mass Spectrometry[END_REF]. Internal standard dodecane, which did not mask any terpene, together with frequent calibration with terpene standards (for every Rivoal A., Fernandez C., Lavoir A.V., Olivier R., Lecareux C., Greff S., Roche P. and Vila B. (2010) Environmental control of terpene emissions from Cistus monspeliensis L. in natural Mediterranean shrublands, Chemosphere, 78, 8, 942-949. Author-produced version of the final draft post-refeering the original publication is available at www.elsevier.com/locate/chemosphere DOI: 0.1016/j.chemosphere.2009.12.047 compounds listed in table 3 and4), were used for quantification. Terpene calibration curves (N=4 different terpene concentrations) were always highly significant (R²>0.98) in the relationship between signal and terpene concentration.

Terpene content

At the end of terpene emission sampling, leaves enclosed in the bag were collected and immediately plunged into liquid nitrogen. Prior to terpene extraction, leaves were lyophilized for 48 h in a freeze drier (CHRIST, alpha 1-4 LD plus), in order to express results on the basis of leaf dry mass (DM). Freeze-drying is known to induce no terpene loss (Ormeño et al., 2007b). Leaves were mechanically ground with a grinder to obtain powder. This procedure allows better extraction of terpenes than manual procedures [START_REF] Ormeño | Production and diversity of volatile terpenes from plants on calcareous and siliceous soils: Effect of soil nutrients[END_REF]. The extraction consisted of dissolving 0.5 g of leaf DM in 10 ml of cyclohexane, for 30 min, under constant shaking at room temperature. Tightly closed glass vials wrapped in aluminium foil were used to avoid exposure to light and oxygen. A non-isoprenoid volatile internal standard (dodecane) was added to quantify measurements. Extracts were analyzed or stored at -20 °C until analysis.

Analyses were performed on a gas chromatograph (Hewlett Packard GC6890®) coupled to a mass selective detector (MSD; HP 5973N). Compound separation was achieved on a HP-5MS capillary column (30 m x 0.25 mm x 0.25 µm, J&W Scientific), in constant flow mode. 2 μl were injected through an automatic injector (ALS 7673). Helium (99.995%) was used as carrier gas. After sample injection, the initial temperature (40 °C), maintained for 5 min, was increased up to 270 °C at a rate of 4 °C min -1 , then maintained for 2.5 min. Identification of terpenes was established by comparison of the retention index and the mass spectrum of detected compounds with those of authentic reference samples (Sigma-Aldrich, one standard for each compound listed in table 3 and4) and literature [START_REF] Adams | Identification of Essential Oil Components by Gas Chromatography/Mass Spectrometry[END_REF].

Internal standard dodecane, which did not mask any terpene, together with frequent calibration with terpene standards (for every compounds listed in table 3 and4) were used for quantification. Terpene calibration curves (N=4 different terpene concentrations) were always highly significant (R²>0.98) in the relationship between signal and terpene concentration.

The same protocol was applied to leaves of a non-enclosed branch on the same individuals. Here no significant difference between enclosed and non-enclosed leaf content was observed (data not shown). Thus we considered leaf content of enclosed branch as representative of the individual.

Soil analyses

Main soil chemical parameters were measured (Table 2). For each sampling plot, five 200g-soil samples from the A1 horizon were collected. Aqueous pH (MeterLab PHM20,1 Radiometer analytical S.A.), total nitrogen (N) and total carbon (C) content (CHN elemental analyzer, FLASH-EA1112, Thermo Fisher Scientific Inc. controlled by the Eager 300 software), were determined in the laboratory. Cation exchange capacity (CEC, Metson) and available phosphorus (PA, Joret-Hébert method) were measured by an external analysis laboratory (LCA, La Rochelle, France) with a continuous-flow colorimetric analyzer (LCA instruments).

Data treatment

Temperature and light are currently agreed to be the main factors controlling emissions from non-storing species [START_REF] Staudt | Seasonal variation in amount and composition of monoterpenes emitted by young Pinus pinea trees -Implications for emission modeling[END_REF]. Analysis of leaf terpene content showed that emitted terpenes are not stored in C. monspeliensis leaves.

Therefore, all emissions data were standardised at 30 °C and 1,000 µmol s -1 m -2 using Guenther's algorithm [START_REF] Guenther | Seasonal and Spatial Variations in Natural Volatile Organic Compound Emissions[END_REF]. The standard emission factor (E s ) was calculated as follows:

T L s C C E E × =
where E is the measured emission and C L and C T are functions of temperature and light given by: where R (8.314 J K -1 mol -1 ) is the gas constant T S (303 K or 30 °C) is the standard temperature, T M (314 K or 41 °C), α (0.0027), C L1 (1.066), C T1 (95000 J mol -1 ) and C T2 (230000 J mol -1 ) are empirically defined parameters.

Data were log-transformed, when necessary, to meet normal requirements for statistical analyses. i) One-way analysis of variance (ANOVA) was applied to test the effect of season on terpene emissions. ii) The effect of age on terpene emissions was analysed through correlation analysis between individual age and emissions. iii) Two-sample t-test was performed to compare emissions on siliceous and calcareous substrate. Results are presented in mean ± SE throughout the paper. Statistical analyses were performed using the statistical software Minitab® for both overall terpene emissions and each compound separately.

Results

Terpene content and emissions

Terpene content analysis showed that only one oxygenated monoterpene and three sesquiterpenes were stored in C. monspeliensis leaves: β-cyclocitral (CAS number: 432-25-7), α-copaene (CAS n°: 3856-25-5), alloaromadendrene (CAS n°: 25246-27-9) and δ-cadinene (CAS n°: 483-76-1). These compounds were found in low concentration (<0.01 mg gDM -1 with DM means leaf dry mass, in average for each compounds, then frequently below the quantitative threshold) regardless of soil, age or season.

C. monspeliensis emitted several compounds but in small amount (Es=2.82 ±1.41 µg g -1 h -1 averaging for all seasons and modalities). α-pinene is the major released compound from C. monspeliensis leaves: it represented 31% to 70% of overall terpene emissions depending on the modality considered. β-myrcene is the second major compound and ranged between 7% and 25% of the overall terpene emissions. In addition, 7 other monoterpenes, 4 oxygenated monoterpenes, one monoterpene derivative and one sesquiterpene were determined in lower amounts (Table 4).

Seasonal variability

Despite temperature and light standardisation, season had a strong effect on overall terpene emission rate (ANOVA, p<0.001, Fig. 1). The overall terpene emission rate was significantly higher in spring and summer than in autumn and winter, ranging from 0.15 µg g -1 h -1 in winter to 9.87 µg g -1 h -1 in summer.

α-pinene, β-myrcene, β-pinene, 1,8 cineole and bornyl acetate were significantly more abundantly emitted in summer than in autumn and winter, spring emission rates being intermediate (ANOVA, p<0.05, table 3). Camphene follows the pattern of terpenes overall and had a higher emission rate in spring and summer than in autumn and winter (ANOVA, p<0.05, table 3). α-phellandrene had a very low emission rate throughout the year (Es<0.02 µg.g -1 .h -1 ) but spring emissions increased by a factor 20 to reach ca. 10% of the total emissions (ANOVA, p<0.001, Table 3). No significant seasonal effect was detected for α-terpinene and borneol although they were detected only in spring.

Age and soil effects

Temperature, PAR and relative humidity were not significantly different between modalities (YS, OS, and OC) during spring sampling.

Comparison of emission rate between age classes (young vs old) did not show an age effect (T-test, p>0.05, data not shown) although positive correlations between plant age and emission rate of β-myrcene, δ-3carene and borneol were observed (Table 4).

Overall terpene emission rates of C. monspeliensis were significantly higher on siliceous (Es = 5.28 ± 1.96 µg g -1 h -1 ) than on calcareous (Es = 0.78 ± 0.16 µg g - 1 h -1 ) substrate (T-test, p<0.05, Table 4). Likewise, emission rates of α-pinene and αterpineol were significantly modified by substrate (T-test, p<0.05, Table 4). Every analysed compound tended to have a higher emission rate on siliceous than on calcareous substrate.

Discussion

Species of the genus Cistus are generally reported to be terpene storing species [START_REF] Gulz | The Composition of Terpene Hydrocarbons in the Essential Oils from Leaves of 4 Cistus Species[END_REF]. They have leaf external glandular hairs allowing terpene storage [START_REF] Seufert | Volatile organics in Mediterranean shrubs and their potential role in a changing environment[END_REF]. Our results did not evidence a consistent pool of storing terpenes in C. monspeliensis leaves (0.02 mg gDM -1 in average): Compared to C. albidus [START_REF] Llusia | Seasonal patterns of terpene content and emission from seven Mediterranean woody species in field conditions[END_REF][START_REF] Ormeño | Production and diversity of volatile terpenes from plants on calcareous and siliceous soils: Effect of soil nutrients[END_REF], they stored 10-fold less. To our knowledge, only three studies have taken an interest in leaf terpene content in C. monspeliensis: Llusia and Peñuelas (1998) who extracted leaf terpenes in pentane and [START_REF] Robles | Infraspecific variability in the essential oil composition of Cistus monspeliensis leaves[END_REF] and [START_REF] Angelopoulou | Diurnal and seasonal variation of the essential oil labdanes and clerodanes from Cistus monspeliensis L. leaves[END_REF], both using a hydrodistillation extraction process which allows no comparison with our results.

In terms of composition, we only found one oxygenated monoterpene, which is consistent with the poor leaf storage of these compounds in the genus Cistus [START_REF] Llusia | Seasonal patterns of terpene content and emission from seven Mediterranean woody species in field conditions[END_REF][START_REF] Ormeño | Production and diversity of volatile terpenes from plants on calcareous and siliceous soils: Effect of soil nutrients[END_REF]. This compound was never reported in C. monspeliensis but has been reported in essential oil of other Cistus species [START_REF] Ogutveren | Composition of the essential oil of Cistus parviflorus L. from Turkey[END_REF][START_REF] Paolini | Detailed analysis of the essential oil from Cistus albidus L. by combination of GC/RI, GC/MS and C-13-NMR spectroscopy[END_REF]. [START_REF] Llusia | Changes in terpene content and emission in potted Mediterranean woody plants under severe drought[END_REF] reported stored α-pinene, but in very low quantity (<0.002 mg gDM -1 ). This compound is not found here probably due to very low quantity limit of detection capacity.

Conversely, sesquiterpenes are reported to be more numerous in Cistus sp., although we only found three of them. This observation could also be due to the very low quantity of compounds on the verge of detection capacity (<0.01 mg gDM -1 ). The three sesquiterpenes found here have always been reported in C. albidus leaf content and two (alloaromadendrene and δ-cadinene) are often considered as major compounds in this species. Although this study focused on volatile compounds, we noticed heavier compound storage, probably diterpenes. In particular, we identified manoyl oxide and its isomers (0.02 mg gDM -1 in average) which were quoted in previous studies on C. monspeliensis [START_REF] Robles | Infraspecific variability in the essential oil composition of Cistus monspeliensis leaves[END_REF][START_REF] Angelopoulou | Diurnal and seasonal variation of the essential oil labdanes and clerodanes from Cistus monspeliensis L. leaves[END_REF]. Rivoal A., Fernandez C., Lavoir A.V., Olivier R., Lecareux C., Greff S., Roche P. and Vila B. (2010) Environmental control of terpene emissions from Cistus monspeliensis L. in natural Mediterranean shrublands,Chemosphere,78,8,[942][943][944][945][946][947][948][949] Author-produced version of the final draft post-refeering the original publication is available at www.elsevier.com/locate/chemosphere DOI: 0.1016/j.chemosphere.2009.12.047 13 Although C. monspeliensis showed very few leaf stored terpenes, emissions have been detected. The major emitted compound found in this study was α-pinene which represents nearly 70% of emissions in summer. This result is consistent with [START_REF] Owen | Light dependency of VOC emissions from selected Mediterranean plant species[END_REF] who found that this compound represented nearly 80% of the overall emissions for the same species under standard conditions. However, the highest standard emission of α-pinene reported here (6.58 µg g -1 h -1 in summer) remains lower than that found by Owen et al. ( 10.30 µg g -1 h -1 ). The second major compound in this study, β-myrcene, has not been previously detected. The only compound reported by Llusià and Peñuelas (1998) for C. monspeliensis was αphellandrene. In our study this compound is minor except in spring which corresponds to their sample season. Like [START_REF] Owen | Light dependency of VOC emissions from selected Mediterranean plant species[END_REF], we found significant amounts of camphene, β-pinene and limonene but no sabinene. Other terpenes were detected here for the first time: δ-3 carene, α-terpinene, p-cymene, 1,8 cineole, camphor, borneol, α-terpineol, bornyl acetate and β-caryophyllene.

Composition variability of the emissions could be due to several causes: i) composition could be influenced by plant origin which can lead to various chemotypes [START_REF] Staudt | Intraspecific variability of monoterpene composition emitted by Quercus ilex leaves[END_REF]. ii) some compounds detected in very low amount in this study, such as camphor or β-caryophyllene, may not have been detected in other studies because of the low amount of material used. Indeed, [START_REF] Owen | Light dependency of VOC emissions from selected Mediterranean plant species[END_REF] used one leaf and Llusia and Peñuelas used only terminal leaves of the branch whereas we used an entire branch. iii) Season and age could also influence emissions: some compounds, such as β-myrcene, presented a positive correlation with age.

Consequently, emissions of these compounds could have been below detection level in previous studies which focused on younger plants or different seasons. iv) Moreover the studies of [START_REF] Owen | Light dependency of VOC emissions from selected Mediterranean plant species[END_REF] and Llusià and Peñuelas (1998) were conducted under controlled conditions (e.g. potted plants, irrigation) which differ from those of our in situ study.

The aim of this study was to document the seasonal effect on terpene emissions from C. monspeliensis. Our results showed a strong seasonal effect, with minima in winter and emission rates nearly 70 times higher in summer. This variation is in the range observed for other Mediterranean species. [START_REF] Pio | Diurnal and seasonal emissions of volatile organic compounds from cork oak (Quercus suber) trees[END_REF] found emissions about 100 times higher in summer than in winter for Quercus suber. [START_REF] Staudt | Intraspecific variability of monoterpene composition emitted by Quercus ilex leaves[END_REF] reported emissions about 40 times higher in August than in January for Quercus ilex. In this study, emissions of most of the analysed compounds follow the pattern of total emissions, increasing from winter to summer and then decreasing in autumn, even after standardisation. This pattern is influenced by long-term light and temperature effect, ontogeny and leaf age (see Grote and Niinemets 2008 for a review). Moreover, a previous work on Mediterranean species [START_REF] Lavoir | Drought reduced monoterpene emissions from the evergreen Mediterranean oak Quercus ilex: results from a throughfall displacement experiment[END_REF] reported a decrease of terpene emissions in summer due to severe drought. However, no drought episode was observed in summer 2008 in the study area.

Another objective of this study was to look at the impact of plant age on terpene emissions. Only few studies have paid interest to individual age effect on terpene emissions and they only concerned trees. [START_REF] Street | Effect of habitat and age on variations in volatile organic compound (VOC) emissions from Quercus ilex and Pinus pinea[END_REF] (2005) compared 20 and 60 year old conifers growing in forests. Both reported higher total terpene emission rates from older trees. In the present study, whereas total terpene emission rate was not linked to shrub age, some individual terpene emission rates were positively correlated with this parameter. Emission rate of β-myrcene (second major compound) was approximately three times higher in old individuals than in young individuals. Increase of emission rate was particularly substantial for borneol. This compound's emission rate was very low in young individuals but was about 30 times higher in old plants. Consequently, even if age did not significantly alter the total emissions, it greatly influenced emissions of some individual terpene.

Finally, this study revealed a soil effect on C. monspeliensis terpene emissions: higher emission rates (individual compounds and total) were observed on siliceous substrate. This result is in agreement with the carbon/nutrient balance hypotheses (CNBH [START_REF] Bryant | Carbon/nutrient balance of boreal plants in relation to vertebrate herbivory[END_REF]) and the growth differentiation balance hypothesis (GDBH, [START_REF] Lorio | Growth-Differentiation Balance -a Basis for Understanding Southern Pine-Beetle Tree Interactions[END_REF])) since siliceous substrate has lower nutrient content (lower N level and lower CEC) than calcareous substrate. These theories state that when all factors are favourable for growth, growth processes predominate over secondary compound production but when a nutrient such as nitrogen is scarce, the plant will allocate proportionally more resources to carbon-based secondary compounds such as terpenes. We found a negative correlation between soil nitrogen content and leaf terpene emissions (r=-0.428, p=0.016). This result is in agreement with [START_REF] Blanch | Sensitivity of terpene emissions to drought and fertilization in terpene-storing Pinus halepensis and non-storing Quercus ilex[END_REF] who reported that fertilization induced a decrease in terpene emissions of Pinus halepensis Mill. However, this hypothesis could not be extended to all species. Ormeño et al. (2007a) who studied terpene emissions on calcareous and siliceous substrate in natural conditions, reported higher emissions of Cistus albidus on siliceous substrate but an inverse trend for Pinus halpensis and Rosmarinus officinalis L. The authors justify these differences by pointing out the different growth state of the three species at the sampling period. Likewise, [START_REF] Lerdau | Ecological Controls over Monoterpene Emissions from Douglas-Fir (Pseudotsuga-Menziesii)[END_REF], working on Pseudotsuga menziensii Mirb., an evergreen coniferous tree, showed that higher nitrogen fertilization induced an increase in terpene emissions, except during leaf expansion when the correlation was negative. In the case of C. monspeliensis, leaf expansion occurs throughout the year and was only slightly slower during the cold and dry seasons [START_REF] Delillis | Comparative Phenology and Growth in Different Species of the Mediterranean Maquis of Central Italy[END_REF].

Conclusion

Field experimentation is essential in order to improve predictions of biogenic emissions and air quality by determining absolute and species-specific emission rates and by highlighting factors influencing terpene emissions. Terpene storage is a species-specific character which is integrated in models. Two major algorithms simulate emissions according to the occurrence of storage: the Tingey algorithm which involves only temperature is used for storing-species [START_REF] Tingey | Influence of light and temperature on monoterpene emission rates from slash pine[END_REF] whereas Guenther's algorithm involving light and temperature is used for non-storing species [START_REF] Guenther | Seasonal and Spatial Variations in Natural Volatile Organic Compound Emissions[END_REF]. In our study, we found only one monoterpene and three sesquiterpene stored in very low amount in C. monspeliensis leaves (total storage <0.02 mg g -1 ), although many species of the genus Cistus are frequently reported to be storing species. As a conclusion, the occurrence of storage could show interand/or intra-specific variability, which needs to be checked as a basis for prediction at local scale.

To illustrate factors influencing terpene emissions, this study focused on seasonal variability of leaf terpene emissions from C. monspeliensis as well as plant age and soil nature impact on this process. Whereas no age effect on total emissions was observed, soil and seasonal effects showed strong variability which could be integrated in simulations. Seasonal variability of emissions was characterised by minima in winter and maxima in summer (70 times higher). These high emission rates occurred during the most favourable period for ozone pollution events, and could strengthen them. Finally, terpene emissions on siliceous substrate were ca. 7 times higher than on calcareous substrate, therefore soil effect should be integrated in biogenic emission models. This study highlighted a strong influence of season and substrate on terpene emissions. Thus, the integration of these factors into the simulation models of VOC emissions will increase the accuracy of predictions of pollution events.

FIGURE CAPTIONS

Fig. 1: Total terpene standard emission rate (E S , mean ± SE in µg g -1 h -1 ) for each season from C. monspeliensis leaves, growing on siliceous substrate. Significativity was tested by ANOVA (F=8.43, p<0.001) and Tukey test (different letters indicate statistically significant differences).

TABLE CAPTIONS

Table 1: sampling date and micro-climatic conditions out of the enclosure bag during sampling: Temp is the air temperature (°C), PAR is the Photosynthetically Active Radiation (µmol m -2 s -1 ) and RH is relative humidity (%).

Table 2: Plot characteristics: YS for Young Siliceous, OS for Old Siliceous and OC for Old Calcareous. Geographical coordinates are given in the system of reference UTM31N-WGS84. Elevation in meters, aspect is given as W for west, E for east, S for south and N for north, slope in degrees, average of individual age in years, available phosphorus (P A ) in mg kg -1 of dry soil, cation exchange capacity (CEC) in Cmol+/kg -1 of soil, soil carbon and nitrogen in percentage and C/N is the carbonnitrogen ratio. Characteristics are expressed in mean ± SE.

Table 3: standard emission rate (Es, mean and SE in µg g -1 h -1 ) for each season and results of ANOVA. nd = not detected. 5.78 ± 0.06 6.21 ± 0.07 6.22 ± 0.11 5.35 ± 0.17 6.14 ± 0.12 6.17 ± 0.08 7.74 ± 0.05 7.77 ± 0.07 7.74 ± 0.03 PA (mg kg -1 ) 59 ± 6 24 ± 2 34 ± 3 21 ± 1 21 ± 1 20 ± 0 28 ± 5 20 ± 0 32 ± 3 CEC (Cmol+/kg -1 ) 15.8 ± 1.0 7.2 ± 2.1 9.9 ± 0.9 8.4 ± 0.5 10.4 ± 0.7 9.6 ± 0.7 17.5 ± 1.1 23.5 ± 3.2 23.5 ± 0.9 Nitrogen (%) 0.27 ± 0.04 0.09 ± 0.04 0.13 ± 0.02 0.06 ± 0.01 0.07 ± 0.02 0.07 ± 0.01 0.21 ± 0.04 0.15 ± 0.00 0.22 ± 0.02 Carbon (%) 4.52 ± 0.56 1.84 ± 0.78 2.68 ± 0.33 1.48 ± 0.26 1.08 ± 0.31 1.38 ± 0.20 3.37 ± 0.82 2.55 ± 0.11 3.69 ± 0.25 C/N 16.6 ± 0.3 18.9 ± 0.7 20. 7 ± 0.7 28.6 ± 3.7 14.8 ± 0.9 20.1 ± 1.2 15.1 ± 1.1 17.2 ± 0.7 16.9 ± 0.2

  p.m. (solar time) from November 2007 to November 2008 on three plots to determine emission seasonality. Selected plots were located on siliceous substrate, C. monspeliensis being more frequent on this kind of substrate. For each date, six individuals were sampled per plot. Sampling dates and microclimate conditions are

  Rivoal A., Fernandez C., Lavoir A.V., Olivier R., Lecareux C., Greff S., Roche P. and Vila B. (2010) Environmental control of terpene emissions from Cistus monspeliensis L. in natural Mediterranean shrublands, Chemosphere, 78, 8, 942-949. Author-produced version of the final draft post-refeering the original publication is available at www.elsevier.com/locate/chemosphere DOI: 0.1016/j.chemosphere.2009.12

  compared 5 and 40 year old Pinus pinea L. emissions growing in plantations, and Kim et al.

  Fig. 1: Total terpene standard emission rate (E S , mean ± SE in µg g -1 h -1 ) for each season from C. monspeliensis leaves, growing on siliceous substrate. Significativity was tested by ANOVA (F=8.43, p<0.001) and Tukey test (different letters indicate statistically significant differences).

Table 4 :

 4 standard emission rate (Es, mean and SE in µg g -1 h -1 ) for old individuals growing on calcareous and siliceous substrates and for young individuals on siliceous substrate. Results of T-test for comparison of emission rates of individuals according to soil nature and results of Pearson correlation between age and emission rate on siliceous substrate. nd = not detected.

Table 1 : sampling date and micro-climatic conditions out of the enclosure bag during sampling: Temp is the air temperature (°C), PAR is the Photosynthetically Active Radiation (µmol m -2 s -1 ) and RH is relative humidity (%).

 1 Fernandez C., Lavoir A.V., Olivier R., Lecareux C., Greff S., Roche P. and Vila B. (2010) Environmental control of terpene emissions from Cistus monspeliensis L. in natural Mediterranean shrublands, Chemosphere, 78, 8, 942-949. Author-produced version of the final draft post-refeering the original publication is available at www.elsevier.com/locate/chemosphere DOI: 0.1016/j.chemosphere.2009.12.047

	Season	Site	Date	Temp (°C)	PAR (µmol m -2 s	-1 )	RH (%)
		OS1 26/11/2007	16,9	480,7		36,0
	Autumn	YS1 13/11/2007	16,0	710,6		63,0
		YS2 12/11/2007	23,5	953,5		64,0
		OS1 14/03/2008	24,7	358,7		43,8
	Winter	YS1 06/03/2008	10,5	242,1		52,5
		YS2 06/03/2008	16,0	321,2		51,8
		YS1 21/05/2008	21,8	671,0		63,5
		YS2 03/06/2008	34,0	1319,2		49,8
		YS3 23/05/2008	34,9	1894,2		44,0
		OS1 15/05/2008	31,1	1162,1		40,9
	Spring	OS2 23/05/2008	36,8	1792,3		45,5
		OS3 03/06/2008	37,5	1411,8		91,0
		OC1 22/05/2008	39,0	1570,0		39,7
		OC2 22/05/2008	25,7	1196,5		51,2
		OC3 22/05/2008	33,0	1506,5		42,2
		OS1 28/08/2008	43,5	1602,1		23,4
	Summer	YS1 29/08/2008	31,7	1528,3		31,4
		YS2 29/08/2008	41,0	1453,7		18,5
		OS1 07/11/2008	19,4	488,3		44,0
	Autumn	YS1 07/11/2008	11,3	361,9		76,5
		YS2 30/10/2008	14,8	597,8		56,0

Table 2 : Plot characteristics: YS for Young Siliceous, OS for Old Siliceous and OC for Old Calcareous. Geographical coordinates are given in the system of reference UTM31N-WGS84. Elevation in meters, aspect is given as W for west, E for east, S for south and N for north, slope in degrees, average of individual age in years, available phosphorus (P A ) in mg kg -1 of dry soil, cation exchange capacity (CEC) in Cmol+/kg -1 of soil, soil carbon and nitrogen in percentage and C/N is the carbon-nitrogen ratio. Characteristics are expressed in mean ± SE.

 2

	Plot	YS1	YS2	YS3	OS1	OS2	OS3	OC1	OC2	OC3
	Geographical coordinates
	Latitude	43°18'23''	43°31'21''	43°28'07''	43°11'43''	43°33'43''	43°21'18''	43°22'50''	43°21'07''	43°13'26''
	Longitude	6°27'41''	6°35'28''	6°47'38''	6°17'17''	6°47'25''	6°24'13''	5°01'57''	5°10'11''	5°30'45''
	Elevation (m)	520	72	105	326	158	90	65	137	222
	Aspect	NW	S	W-SW	(E)-SE	E-NE	W	E-(SE)	E	(E	)-SE
	Slope(degree)	14	5	7	21	4	2	3	4		8
	Age (years)	5.3 ± 0.3	5.5 ± 0.5	5.3 ± 0.3	15.4 ± 0.9	11.3 ± 0.2	9.3 ± 1.0	12.5 ± 1.3 12.5 ± 1.1 12.7 ± 0.2
	Soils characteristics
	Soil nature	Siliceous	Siliceous	Siliceous	Siliceous	Siliceous	Siliceous Calcareous Calcareous Calcareous
	pH

Table 4 : standard emission rate (Es, mean and SE in µg g -1 h -1 ) for old individuals growing on calcareous and siliceous substrates and for young individuals on siliceous substrate. Results of T-test for comparison of emission rates of individuals according to soil nature and results of Pearson correlation between age and emission rate on siliceous substrate. nd = not detected.

 4

		old -	old -	young -	T-test			correlation
		calcareous	siliceous	siliceous	(soil effect)		(age effect)
		mean SE	mean SE	mean SE	T	p	r	p
	Monoterpenes
	α-pinene	0.336 0.106	2.422 1.073	1.139 0.492	-2.31 0.035	-0.016 0.933
	camphene	0.020 0.011	0.069 0.028	0.021 0.013	-1.67 0.113		0.091 0.637
	β-pinene	0.037 0.022	0.280 0.142	0.092 0.032	-1.93 0.075		0.090 0.642
	β-myrcene	0.106 0.035	0.473 0.203	0.141 0.058	-1.88 0.080		0.377 0.044
	α-phellandrene	0.054 0.022	0.356 0.163	0.123 0.071	-1.95 0.072		0.301 0.112
	δ-3 carene	0.008 0.003	0.020 0.011	0.000 0.000	-1.10 0.289		0.391 0.036
	α-terpinene	0.000 0.000	0.121 0.060	0.163 0.149	-2.03 0.063	-0.089 0.646
	p-cymene	0.011 0.009	0.011 0.011	0.035 0.021	0.02 0.984	-0.072 0.711
	limonene	0.074 0.020	0.454 0.217	0.138 0.040	-1.84 0.088		0.011 0.957
	Oxygenated
	monoterpenes
	1,8 cineole	0.031 0.010	0.232 0.105	0.038 0.018	-2.09 0.057		0.083 0.669
	camphor	0.012 0.005	0.021 0.019	0.012 0.006	-0.39 0.704	-0.104 0.592
	borneol	0.039 0.019	0.366 0.316	0.010 0.008	-1.01 0.330		0.422 0.023
	α-terpineol	0.016 0.008	0.205 0.083	0.161 0.142	-2.52 0.026	-0.028 0.887
	Monoterpene
	derivatives
	bornyl acetate	0.025 0.012	0.212 0.090	0.041 0.018	-2.12 0.054		0.132 0.493
	Sesquiterpenes
	β-caryophyllene 0.006 0.002	0.033 0.025	0.008 0.003	-1.10 0.293		0.123 0.527
	Overall terpenes 0.776 0.159	5.278 1.955	2.123 0.702	-2.55 0.022		0.024 0.904

Acknowledgements

This work was funded by AtmoPACA (contract n°01-11-18-04), the French environmental agency (ADEME) and the PACA (Provence-Alpes-Côte d'Azur) Regional Council. We thank Sylvie Dupouyet, Samira Marir and Hussain Rizvi for assistance during the field work and sample analysis.

Author-produced version of the final draft post-refeering the original publication is available at www.elsevier.com/locate/chemosphere DOI: 0.1016/j.chemosphere.2009.12.047


Michael Defoort
email: michael.defoort@univ-valenciennes.fr

Mohamed Djemaï
email: mohamed.djemai@univ-valenciennes.fr

Thierry Floquet

Wilfrid Perruquetti

On finite time observer design for multicellular converter

Keywords: order sliding mode, nonlinear observer, switched system, finite time stability. I

The aim of this paper is to estimate the capacitor's voltages by tacking the hybrid behavior of the multicellular converter into account. A nonlinear finite time observer is given in order to solve this problem. The stability and properties of the proposed homogeneous finite time observer are studied using Lyapunov theory. Our approach enables the stabilization of the observation errors in spite of the presence of perturbations. Simulations highlight the efficiency of the proposed strategy.

Index Terms-multicellular converter, higher

For the first order sliding modes, it is common to deal with the issues of stability, robustness and convergence rate of the equilibrium by means of a Lyapunov approach [START_REF] Utkin | Sliding modes in control and optimization[END_REF]. For higher sliding mode and finite time observers, the proof of stability is usually done using geometric homogeneity based methods. Hence, the estimation of the convergence time and the study of robustness properties looks problematic. In [4], a strong Lyapunov function for the super-twisting algorithm is proposed, allowing the study of stability, robustness and convergence rate of the equilibrium.

In this paper, a Lyapunov function will be designed, based on a quadratic like function, in order to deeply study the reaching time estimation and robustness of the homogenous finite time observer proposed in [START_REF] Perruquetti | Finite time observers: application to secure communication[END_REF]. This observer is then applied in order to estimate the capacitor voltages in a multicellular converter whose hybrid behavior, due to switches, increases the difficulty. The results are illustrated with some simulations.

II. MULTICELLULAR CONVERTER MODELING

The multicellular converter is based on the combination of elementary cells of commutation. Figure 1 depicts the topology of a converter with p independent commutation cells. The independency is due to the (p -1) internal capacitors and can be considered only for a few switching periods. As a matter of fact, the current flows from the source E to the output I through different converter switches. The multicellular converter shows, by its structure, a hybrid behavior due to discrete variables (i.e. switching or commutation logic) [START_REF] Barbot | Nonlinear observer for autonomous switching systems with jumps[END_REF]. Note that because of the presence of capacitors, there are also continuous variables (i.e. currents and voltages).

E S p V c p-1 c p-1 S j+1 V c j c j S 2 V c 1 c 1 S 1 V s I Fig. 1. Multicellular converter
It is important to highlight that the electrical switches constraints should be similar. This requirement implies a unique voltage switch constraint of E p . Thus, it is necessary to ensure an equilibrated distribution of the capacitor voltages. Under these conditions, the reference voltage of the capacitor j ( j = {1, . . . , p -1}) is:

V c j ,re f = j E p
The dynamics of the converter, with a load consisting in a resistance R and an inductance L, is given by the following differential equations:

         İ = - R L I + E L S p -∑ p-1 j=1 V c j L (S j+1 -S j ) Vc j = I c j (S j+1 -S j ), j = 1, . . . , p -1 (1)
where I is the load current, c j is the capacitance, V c j is the voltage in the j-th capacitor and E is the voltage of the source. Each commutation cell is controlled by the binary input signal S j ∈ {0, 1}. Signal S j = 1 means that the upper switch of the j-th cell is "on" and the lower switch is "off" whereas S j = 0 means that the upper switch is "off" and the lower switch is "on". The discrete control input can be described as follows:

u j = S j+1 -S j , j = 1 . . . p -1 u p = S p (2)
Assuming that only the load current I can be measured (i.e. y = I), the system (1) can be represented as follows:

                 İ = - R L I + E L u p -∑ p-1 j=1 V c j L u j Vc j = I c j u j , j = 1, . . . , p -1 y = I (3)
Using the hybrid system formalism [START_REF] Lygeros | Dynamical Properties of Hybrid Automata[END_REF], one can describe the system (3) as:

ẋ = f (x, u) y = h(x, u) (4) where x = [I,V c 1 , . . . ,V c p-1 ] T ∈ R p is the continuous state, u = [u 1 , . . . , u p ]
T is the applied control input which only takes discrete values. The functions f (x, u) = A(u)x + B(u) and h(x, u) = Cx are vector fields defined as:

A(u) =       -R L -u 1 L . . . - u p-1 L u 1 c 1 0 . . . 0 . . . . . . . . . . . . u p-1 c p-1 0 . . . 0       B(u) = E L u p , 0, . . . , 0 T C = [1, 0, . . . , 0] III. OBSERVABILITY ANALYSIS
This section is devoted to the observability analysis of the capacitor voltages V c j ( j = {1, . . . , p -1}) from the measurement of the current I and the knowledge of the control input sequence u. This analysis can be done using the two following approaches [START_REF] Benmansour | Realisation d'un banc d'essai pour la Commande et l'Observation des Convertisseurs Multicellulaires Serie: Approche Hybride[END_REF].

A. Approach 1: Nonlinear observation theory

In this approach, the converter is considered as a nonlinear continuous system and tools from nonlinear observation theory [11] are used to study the observability.

Considering the system (3), one can see that there are several operating switching modes for which the state x is not observable. For instance, if u 1 = u 2 = • • • = u p-1 = 0, the voltages V c j (∀ j = {1, . . . , p -1}) become completely unobservable. Nevertheless, these operating switching modes occur only during some part of the control sequence. Indeed, if it occurs for all the control sequences then the physical interest is low because the multicellular converter does not operate.

From the observability rank condition [13] of the system (3), it follows that at most two components of x are simultaneously observable. System (3) can be written as follows:

         İ = - R L I + E L u p +V s Vs = -∑ p-1 j=1 I Lc j u 2 j (5) with V s = -1 L ∑ p-1 j=1 u j V c j .
From the observability rank condition of system (5), one can deduce that I and V s can be simultaneously observable.

From the estimation of V s , one can deduce the capacitor voltages

V c 1 ,V c 2 , . . . ,V c p -1 . Indeed, since V c j ( j = {1, . . . , p - 1}) slowly varies during the measurement time interval, one can obtain p -1 measurements of V s (i.e. V 1 s ,V 2 s , ...V p-1 s ) such that:    V 1 s . . . V p-1 s    = -1 L    u 1 1 . . . u 1 p-1 . . . . . . . . . u p-1 1 . . . u p-1 p-1       V c 1 . . . V c p-1    is invertible.

B. Approach 2: Z-observability theory

Here, the observability properties of the state vector is studied from the concept of Z-observability for switched systems [14]. This approach uses the fact that the multicellular converter belongs to a particular class of switching hybrid systems. For such a class, the observation concept is linked to the switches. Hence, it is important to give the following definitions. Definition 1: [START_REF] Lygeros | Dynamical Properties of Hybrid Automata[END_REF] A hybrid time trajectory is a finite sequence T n = {I i } i=0,N such that:

• I i = [t i,0 ,t i,1 ), for all 0 ≤ i ≤ N • For all 0 ≤ i ≤ N, t i,1 = t i+1,0 • t 0,0 = t ini and t N,1 = t end
Furthermore, T N is defined as the ordered list of u associated to T N , i.e. {u i } i=0,N where u i is the value of u on the interval I i .

Definition 2:

[14] For system (4), the function z = Z(t, x) is said to be Z-observable with respect to the hybrid time trajectory T N and T N , if for all trajectories (t, x, u) and

(t, x ′ , u ′ ) defined in [t ini ,t end ], the equality h(x, u) = h(x ′ , u ′ ) implies Z(t, x) = Z(t, x ′ ).
Theorem 1: [14] Consider system (4) and a fixed hybrid time trajectory T N and T N . Suppose that z = Z(t, x) is always continuous under any admissible control input. If there is a sequence of linear projections {P i } i=0,N such that: (z) . Then, z is Z-observable with respect to the hybrid time trajectory T N and T N .

• for all 0 ≤ i ≤ N, P i Z(t, x) is Z-observable for t ∈ I i , • Rank [P T 0 , . . . , P T N ] = dim(z), • d Pi Z(t, x) dt = 0 for t ∈ I i where PT i , P T i has a full rank in R dim(z)×dim
Remark 1: The first condition of Theorem 1 implies that there is at least a time interval in which the variable P i Z(t, x) is Z-observable; whereas the second one implies that all the components of Z are observable in a given time interval of the hybrid time trajectory T N . The third condition requires that components of Z which are not observable in I i must remain constant within this time interval. It prevents from the loose of observability and from the "re-observation" of variables which have been already estimated.

The application of Theorem 1 to the multicellular converter yields the following result:

Lemma 1: Consider the system (3) and the function z = x. Then, z is Z-observable with respect to the hybrid time trajectory T N and T N = u i i=0,N , if the vectors u i i=0,N generate the space R p-1 , where

u i = u (6)
on the time interval I i .

Remark 2: Lemma 1 ensures that the current crosses through all the capacitors; roughly speaking, in a linear independent form. In other words, after p -1 time intervals (i.e. p -1 control sequences) the measurement of the load current I allows to obtain a set of (p -1) linearly independent equations with respect to the voltages in the (p -1) capacitors; which enables to estimate the capacitors of the converter. This result is, in essence, similar as the result presented in Approach 1. Indeed, it is needed at least p -1 time intervals whose corresponding sequences u i are linearly independent, in order to estimate V c j , ∀ j ∈ {1, . . . , p -1}.

IV. CAPACITOR VOLTAGES ESTIMATION

Hereafter, a finite time observer is designed in order to estimate the capacitor voltages of the multicellular converter by using only the measurement of the current I. Due to the particular structure of the observer, finite time stability of the observation error is shown using Lyapunov stability arguments.

A. Finite time observer design

Assuming that the current I is measured, the finite time observer for the system (5) is designed as follows:

         İ = - R L I + E L u p + Vs + k 1 ∑ p-1 j=1 |u j |⌈I -Î⌋ α Vs = -∑ p-1 j=1 I Lc j u 2 j + K 2 ⌈I -Î⌋ 2α-1 (7)
The positive constants k 1 , K 2 and α will be defined hereafter.

The continuous function ⌈a⌋ b is the function described as:

∀a ∈ R, ∀b ≥ 0, ⌈a⌋ b = |a| b sign (a) (8)
Define the observation errors as:

e 1 = I - Î e 2 = V s -Vs (9)
Equations ( 5)-( 7) yield the dynamics of the observation error as:

   ė1 = e 2 -k 1 ∑ p-1 j=1 |u j |⌈e 1 ⌋ α ė2 = -K 2 ⌈e 1 ⌋ 2α-1 (10)
In order to simplify the system (10), one can note:

K 1 = k 1 p-1 ∑ j=1 |u j | (11)
Hence, the dynamics (10) can be written as follows:

ė1 = e 2 -K 1 ⌈e 1 ⌋ α ė2 = -K 2 ⌈e 1 ⌋ 2α-1 (12)
Through this paper, it is assumed that:

• there is T N such that z = x is Z-observable with respect to the hybrid time trajectory of system (1). • there is a constant τ > 0 such that for any time interval I i , |t i,1 -t i,0 | is larger than τ.

Remark 3: Since u remains constant on the time interval I i , one should guarantee that the reaching time t i,0 < T i < t i,1 onto the manifold e = [e 1 , e 2 ] T | e 1 = 0, e 2 = 0 is lower than τ. According to the above assumptions, there is a hybrid time trajectory such that the capacitor voltages are observable. Then, since e 2 equals to zero on the time interval [T i ,t i,1 ], by the means of the procedure defined in Section III, one can obtain an estimation of the capacitor voltages, i.e. V c j , ∀ j = {1, . . . , p -1}.

According to Remark 3, the objective, i.e. the estimation of the capacitor voltages, is achieved when the finite time stabilization of the system (12) is guaranteed.

Consider some perturbations f (e) = f 1 (e 1 ), f 2 (e 1 ) T ∈ R 2 on the system (3) due to resistance uncertainties for instance. In this case, the dynamics of the observation errors (12) is replaced by:

ė1 = e 2 -K 1 ⌈e 1 ⌋ α + f 1 (e 1 ) ė2 = -K 2 ⌈e 1 ⌋ 2α-1 + f 2 (e 1 ) (13)
Theorem 2: Suppose that the perturbations of the system (13) satisfy:

     f 1 (0) = 0 | ˙ f 1 (e 1 )| ≤ δ 1 |e 1 | 2α-1 | f 2 (e 1 )| ≤ δ 2 |e 1 | 2α-1 (14)
for some constants δ 1 ≥ 0 and δ 2 ≥ 0.

Consider the matrix A 0 :

A 0 = -αK 1 α -K 2 0
where 1 2 ≤ α < 1 and the gains K 1 > 0, K 2 > 0 are chosen such that matrix A 0 is Hurwitz and are sufficiently large. Then, the system (13) is globally finite-time stable.

Proof: Consider the following vector:

ξ = ξ 1 ξ 2 = ⌈e 1 ⌋ α e 2 + f 1 ( 15
)
Define the candidate Lyapunov function as:

V (ξ ) = ξ T Pξ ( 16
)
P = P T = p 1 p 3 p 3 p 2
> 0 is the solution of the following algebraic Lyapunov equation:

A T 0 P + PA 0 = -Q (17)
where Q = Q T > 0 is a positive definite matrix. One can note that V (ξ ) is continuous, positive definite and radially unbounded in R 2 . Furthermore, it is differentiable everywhere except on the manifold {e 1 = 0}. Since the trajectory of system (12) cannot stay on this set, before reaching the origin, the derivative of V can be calculated as in the conventional way (see [12] for further details).

Since

ξ = α|e 1 | α-1 e 2 -K 1 ⌈e 1 ⌋ α + f 1 -K 2 ⌈e 1 ⌋ 2α-1 + ˙ f 1 + f 2 = |e 1 | α-1 α (ξ 2 -K 1 ξ 1 ) -K 2 ξ 1 + 0 ˙ f 1 + f 2 = |e 1 | α-1 A 0 ξ + 0 ˙ f 1 + f 2
the time derivative of V along the solutions of system (13) is given by:

V = - 1 |e 1 | 1-α ξ T Qξ + 2ξ T P 0 |e 1 | 1-α ˙ f 1 + f 2 ( 18
)
where

2ξ T P 0 ˙ f 1 + f 2 = 2( ˙ f 1 + f 2 )(p 3 ξ 1 + p 2 ξ 2 ) (19)
Using the bound on the perturbation ( 14), it can be shown that:

2ξ T P 0 |e 1 | 1-α ˙ f 1 + f 2 ≤ 2(δ 1 + δ 2 )(p 3 ξ 2 1 + p 2 |ξ 1 ||ξ 2 |) (20)
Since, for all ε > 0,

p 2 |ξ 1 ||ξ 2 | ≤ ε 2 ξ 2 1 + p 2 2 2ε ξ 2 2
One has:

2ξ T P 0 |e 1 | 1-α ˙ f 1 + f 2 ≤ µ 2 1 ξ 2 1 + µ 2 2 ξ 2 2 ( 21
)
with

µ 1 = (δ 1 + δ 2 )(2|p 3 | + ε) and µ 2 = (δ 1 + δ 2 ) p 2 2 ε . Thus, V ≤ -1 |e 1 | 1-α ξ T Qξ ≤ -1 |e 1 | 1-α λ min { Q}||ξ || 2 (22)
where

Q = Q + µ 1 0 0 µ 2 (23)
Hence, if the gains K 1 and K 2 are chosen such that Q > 0, then V is negative definite.

From the standard inequality for quadratic forms, one can deduce that:

λ min {P}||ξ || 2 ≤ V (ξ ) ≤ λ max {P}||ξ || 2 (24)
where ||ξ || is the Euclidean norm.

Using (24) and 0 < 1 -α 2α , one can obtain:

|e 1 | 1-α ≤ ||ξ || 1-α α ≤ V 1-α 2α (ξ ) λ 1-α 2α min {P} (25)
As α > 1 2 , one can deduce that V (ξ ) is a strong Lyapunov function for the system (13) since:

V ≤ -γ( Q)V 3α-1 2α (26)
with

γ( Q) = λ min { Q}λ 1-α 2α min {P} λ max {P}
Since the transformation ( 15) is continuous, it follows that ξ converges to zero in finite time, then, e 1 and e 2 converge to 0 andf 1 , respectively.

Using the property of perturbation f 1 given in ( 14), it follows that a trajectory of the system (13) starting from e 0 ∈ R 2 converges to the origin in finite time and reaches that point at most after:

T = 2α 3α -1 V 3α-1 2α (e 0 ) γ( Q) (27)
Remark 4: It is interesting to note that the limit case α = 1 corresponds to the Luenberger observer and allows an asymptotic convergence of the observation errors. The case α = 1 2 corresponds to the classical super-twisting observer.

V. SIMULATION RESULTS

In order to illustrate the performance of the proposed finite time observer, we consider here a 3 cells converter connected to an RL load. This converter can be modeled by the system (1) where p = 3. Its parameters are as follows: c 1 = c 2 = 40.10 -6 F, R = 131Ω, L = 0.001H.

The voltage of the source is E = 30V . The inputs of the switches u j , j = {1, 2, 3} are generated for the hybrid time trajectory given in Fig. 2. This sequence of control is periodic with a period of 0.2ms and satisfies the assumptions of Lemma 1. Hence, z = x is Z-observable with respect to the hybrid time trajectory for system [START_REF] Barbot | Nonlinear observer for autonomous switching systems with jumps[END_REF]. Moreover, one can see in Fig. 4 that V c 1 (V c 2 resp.) remains constant during a certain time interval. The chopping frequency and the sampled period are F = 5kHz and T = 5.10 -6 s, respectively. The available measurement is the current I. The problem of the capacitor voltages estimation can be viewed as the finite-time stabilization of the system (12). Applying Theorem 2, one can set the parameters of the observers as follows: K 2 = 1500 and K 1 = 500.

Figures 345depict the results obtained with the finite time observer when α = 3 4 is selected. One can note the high convergence rate of the observer errors. One can see the equal distribution of the voltage constraints on each cells (i.e. steady conditions E 3 = 10V for V c 1 and 2E 3 = 20V for V c 2 ). It is worth noticing that the high frequency oscillations are of low amplitude. It is interesting to note in Fig. 6 that the finite time observer gives good results in terms of robustness properties with respect to resistance variation (25% around its nominal value).

VI. CONCLUSION

In this paper, a nonlinear finite time observer has been designed for a multicellular converter. The problem of the capacitor voltages estimation is solved. A Lyapunov function has been introduced in order to study the stability and properties of the proposed homogeneous finite time observer.

Fig. 3 .Fig. 5 .

 35 Fig. 3. Observation error in V c 1 for the finite time observer using α = 3 4

Fig. 6 .

 6 Fig. 6. Observation errors for the finite time observer using α = 3 4 when

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	[6]


Michael Defoort
email: michael.defoort@gmail.com

Thierry Floquet

Annemarie Kokosy

Wilfrid Perruquetti

A novel higher order sliding mode control scheme

Keywords: Finite time stability, higher order sliding mode control, Geometric homogeneity

A higher order sliding mode control algorithm is proposed for a class of uncertain multi-input multi-output nonlinear systems. This problem can be viewed as the finite time stabilization of a higher order input-output dynamic system with bounded uncertainties. The developed control scheme is based on geometric homogeneity and sliding mode control. The proposed procedure provides explicit conditions on the controller parameters and guarantees robustness against uncertainties. An illustrative example of a hovercraft vessel control demonstrates the advantages of the strategy.

Introduction

Sliding mode control (SMC) [START_REF] Utkin | Sliding modes control in electromechanical systems[END_REF] is a robust technique to control nonlinear systems operating under uncertainty conditions [START_REF] Bloch | Stablization and tracking in the nonholonomic integrator via sliding modes[END_REF][START_REF] Edwards | Sliding mode control: theory and applications[END_REF][START_REF] Perruquetti | Sliding mode control in engineering[END_REF]. The conventional SMC design approach consists of two steps. First, a sliding manifold is designed such that the system trajectories along the manifold meet the desired performance. Then, a discontinuous control is designed such that the system trajectories reach and stay, after a finite time, on the manifold. In spite of claimed robustness properties, the resulting controller has a specific disadvantage. Its drawback is the chattering phenomenon, i.e. high frequency vibrations of the controlled system, which degrades the performance and may lead to instability.

Preserving the main advantages of the conventional SMC, an approach, called higher order SMC, has been proposed in order to reduce the chattering phenomenon [START_REF] Drakunov | Sliding mode control in dynamic systems[END_REF][START_REF] Emel'yanov | Higher order sliding modes in control system[END_REF][START_REF] Fridman | Higher order sliding modes[END_REF]. Instead of influencing the first sliding variable time derivative, the signum function acts on its higher order time derivative. This method can also achieve a better accuracy with respect to discrete sampling time than conventional SMC [START_REF] Fridman | Higher order sliding modes[END_REF].

Many papers are available in the case of second order SMC [START_REF] Bartolini | A survey of applications of second order sliding mode control to mechanical systems[END_REF][START_REF] Floquet | Higher order sliding mode stabilization for a class of nonholonomic perturbed systems[END_REF][START_REF] Fridman | Higher order sliding modes[END_REF]. Arbitrary order sliding mode controllers have been recently proposed in [START_REF] Laghrouche | igher order sliding mode control based on optimal approach of an electropneumatic actuator[END_REF][START_REF] Laghrouche | Higher order sliding mode control based on integral sliding mode[END_REF][START_REF] Levant | Universal SISO sliding-mode controllers with finite time convergence[END_REF][START_REF] Levant | Higher-order sliding modes, differentiation and output-feedback control[END_REF][START_REF] Levant | Quasi-continuous high-order sliding-mode controllers[END_REF]. The algorithm proposed in [START_REF] Levant | Universal SISO sliding-mode controllers with finite time convergence[END_REF] allows the higher order sliding mode by tuning only one gain parameter. However, the convergence rate cannot be arbitrarily selected. One of the main problems of algorithms [START_REF] Levant | Higher-order sliding modes, differentiation and output-feedback control[END_REF][START_REF] Levant | Quasi-continuous high-order sliding-mode controllers[END_REF] is parameter adjustment. Indeed, although the control parameters may be obtained using the proofs, no explicit condition for the gain tuning is derived. Hence, the convergence cannot be easily made arbitrarily fast or slow. The approach given in [START_REF] Laghrouche | igher order sliding mode control based on optimal approach of an electropneumatic actuator[END_REF] combines SMC with linear quadratic one. In spite of several advantages (constructive approach, practical applicability), this approach suffers of a major drawback: the higher order sliding mode is only practical, i.e. the control ensures that the system trajectories reach an arbitrary small neighborhood of the origin in finite time. In [START_REF] Laghrouche | Higher order sliding mode control based on integral sliding mode[END_REF], the controller is based on integral SMC [START_REF] Utkin | Integral sliding mode in systems operating under uncertainly conditions[END_REF] and guarantees the establishment of a higher order sliding mode. The advantages of this method are the easiness of its implementation and the robustness of the system during the entire response. However, two drawbacks appear in this approach: it directly depends on the initial conditions of the system (which may not be accurately known because of noised measurements) and complex off-line computations are needed before starting the control action.

The aim of this paper is to propose a new controller for which:

• the higher order sliding mode establishment is guaranteed in finite time,

• robustness of the system against uncertainties is ensured, • the approach is constructive, i.e. the tuning of the control parameters is formalized, • no complex off-line computation (for instance, resolution of an optimal control problem) is needed, • the accurate knowledge of the initial conditions of the system is eliminated.

The problem of higher order SMC is formulated in input-output terms (as in [START_REF] Levant | Universal SISO sliding-mode controllers with finite time convergence[END_REF]) through the differentiation of the sliding variable and is equivalent to the finite time stabilization of higher order integrator chains with bounded nonlinear uncertainties. Most of the finite time stabilization results available in the literature are only applicable to the second or third order integrator chain [START_REF] Bartolini | On the finite time stabilization of uncertain nonlinear systems with relative degree three[END_REF][START_REF] Bhat | Continuous finite-time stabilitization of the translational and rotational double integrators[END_REF][START_REF] Hong | On an output feedback finite-time stabilization problem[END_REF]. In [START_REF] Bartolini | On the finite time stabilization of uncertain nonlinear systems with relative degree three[END_REF], a robust discontinuous control scheme guarantees the finite time stabilization of a perturbed triple integrator when only the sign of the output and its first and second time derivatives are available. In the higher order dimension case, some controllers are proposed in [START_REF] Bhat | Geometric homogeneity with applications to finite time stability[END_REF][START_REF] Hong | Finite time stabilization and stabilizability of a class of controllable systems[END_REF], but the finite time stabilization is not ensured if the system is perturbed. In [START_REF] Huang | Global finite time stabilization of a class of uncertain nonlinear systems[END_REF], the authors give a recursive design algorithm for the construction of a global finite time controller for a class of uncertain nonlinear systems but the expression of the control input is not explicit. Here, the proposed constructive algorithm combines the finite time controller based on geometric homogeneity, which has been introduced in [START_REF] Bhat | Geometric homogeneity with applications to finite time stability[END_REF] for the finite time stabilization of an integrator chain without uncertainty, with a discontinuous controller based on SMC in order to ensure the robustness with respect to the uncertainties. This constructive procedure is especially important for higher relative degrees, when the number of control parameters is significant. Moreover, due to the large freedom and easiness of the controller, one can arbitrarily hasten or slow down the finite time transient process.

The outline of this paper is as follows. Section II states the problem and assumptions. In section III, the proposed higher order SMC is derived. In Section IV, simulations for the robust control of a hovercraft vessel demonstrate the efficiency and the advantages of the proposed algorithm compared to the procedure given in [START_REF] Levant | Universal SISO sliding-mode controllers with finite time convergence[END_REF].

Problem formulation

Consider the following general multi-input multi-output nonlinear uncertain system:

ẋ = f (x) + m i=1 g i (x)u i y 1 = σ 1 (x)
. . .

y m = σ m (x) (1)
where x ∈ R n and u = [u 1 , . . . , u m ] T ∈ R m are the state variable and the control input, respectively. f (x) and g(x) = [g 1 (x), . . . , g m (x)] T are uncertain smooth functions. σ(x) = [σ 1 (x), . . . , σ m (x)] T ∈ R m is a smooth measurable output vector. The uncertainties on f (x) and g(x) are due to parameter variations, unmodeled dynamics or external disturbances and do not satisfy the well-known matching condition. The control objective consists in the vanishing of output σ(x) in finite time.

Assumption 1 The relative degree vector r = [r 1 , . . . , r m ] T of system [START_REF] Bartolini | A survey of applications of second order sliding mode control to mechanical systems[END_REF] with respect to σ(x) is assumed to be constant and known. It means that the m × m matrix:

B(x) =        L g 1 L r 1 -1 f σ 1 (x) . . . L gm L r 1 -1 f σ 1 (x) . . . . . . L g 1 L rm-1 f σ m (x) . . . L g m L rm-1 f σ m (x)       
is nonsingular and

L g j L k f σ i (x) = 0, for 1 ≤ i ≤ m, 1 ≤ j ≤ m and 0 ≤ k < r i -1.
Moreover, it is supposed that the associated zero dynamics are asymptotically stable.

Remark 1 If the relative degree is not well defined, the addition of integrations on certain input channels can be helpful in obtaining a well defined relative degree (see [START_REF] Isidori | Nonlinear Control Sytems II[END_REF] for further details on the Dynamic Extension Algorithm).

Definition 1 [11] Consider nonlinear system (1) and sliding variable σ(x).

Assume that the time derivatives σ i , σi , . . . , σ (r i -1) i for all i = 1, . . . , m are continuous functions. The manifold defined as:

S r =              x : σ 1 (x) = σ1 (x) = . . . = σ (r 1 -1) 1 (x) = 0 . . . σ m (x) = σm (x) = . . . = σ (r m -1) m (x) = 0             
is called the r th order sliding set. If it is non empty and locally an integral set in the Filippov sense [START_REF] Filippov | Differential equations with discontinuous right hand sides[END_REF], the motion on S r is called r th order sliding mode with respect to the sliding variable σ.

The r th order SMC approach allows the finite time stabilization of each variable σ i and its r i -1 first time derivatives by defining a suitable discontinuous control law. The r th i time derivative of each function σ i yields:

σ (r 1 ) 1 (x), . . . , σ (rm) m (x) T = A(x) + B(x)u (2)
with

A(x) = L r 1 f σ 1 (x), . . . , L r m f σ m (x) T .
Assumption 2 Solutions of the state differential equation ( 2) with discontinuous right-hand side are defined in the sense of Filippov [START_REF] Filippov | Differential equations with discontinuous right hand sides[END_REF].

Assumption 3 Vector A(x) and matrix B(x):

     A(x) = A(x) + ∆ A (x) B(x) = B(x) + ∆ B (x) (3)
are partitioned into nominal part (i.e. A and B), known a priori, and uncertain bounded functions ∆ A and ∆ B . Matrix B is non-singular. Furthermore, there are an a priori known nonlinear function ρ(x) and an a priori known constant 1 ≥ α > 0 such that the uncertain functions satisfy the following inequalities:

     ∆ A (x) -∆ B (x)B -1 (x)A(x) ≤ ρ(x) ∆ B (x)B -1 (x) ≤ 1 -α (4) for x ∈ X ⊂ R n , X
being an open subset of R n within which the boundedness of the system trajectories is ensured.

Let us apply to system (2) the following preliminary feedback:

u = B -1 {-A + w} (5)
where w = [w 1 , . . . , w m ] T ∈ R m is the auxiliary control input. This feedback partially decouples the nominal system (i.e. without uncertainties). Thus, system (2) can be expressed as follows:

σ (r 1 ) 1 , . . . , σ (r m ) m T = I m + ∆ B B -1 w -∆ B B -1 A + ∆ A ( 6
)
where I m denotes the m × m identity matrix.

The r th order SMC of system (1) with respect to the sliding variable σ is equivalent to the finite time stabilization of the multivariable uncertain system:

                                ż1,i = z 2,i . . . żr i -1,i = z r i ,i , ∀i = {1, . . . , m} [ żr 1 ,1 , żr 2 ,2 , . . . , żrm,m ] T = I m + ∆ B B -1 w -∆ B B -1 A + ∆ A (7) with 1 ≤ i ≤ m, 1 ≤ j ≤ r i , z j,i = σ (j-1) i , z i = [z 1,i , z 2,i , . . . , z r i ,i ] T and z = z T 1 , . . . , z T m T .
3 Design of a higher order sliding mode controller

In practical terms, the resolution of the finite time stabilization is a delicate task which has generally been studied for homogeneous systems of negative degree with respect to a flow of a complete vector field. Indeed, for this kind of systems, finite time stability is equivalent to asymptotic stability (see [START_REF] Bhat | Continuous finite-time stabilitization of the translational and rotational double integrators[END_REF][START_REF] Bhat | Geometric homogeneity with applications to finite time stability[END_REF][START_REF] Hong | Finite time stabilization and stabilizability of a class of controllable systems[END_REF][START_REF] Moulay | Finite time stability and stabilization of a class of continuous systems[END_REF][START_REF] Moulay | Finite time stability and stabilization of time delay systems[END_REF][START_REF] Moulay | Finite time stability of non autonomous non linear systems[END_REF] for more details). However, the existing techniques are generally not constructive or robust with respect to uncertainties. To the authors' knowledge, a constructive feedback control law for finite time stabilization of all-dimension chain of integrators without uncertainty has only been proposed in [START_REF] Bhat | Geometric homogeneity with applications to finite time stability[END_REF]. Before designing our robust finite time controller, let us introduce the algorithm given in [START_REF] Bhat | Geometric homogeneity with applications to finite time stability[END_REF] and let us show its problem in terms of robustness.

Finite time stabilization of an integrator chain system

Let us consider the nominal system of (7) (i.e. ∆ A = 0 and ∆ B = 0), which is represented by m single-input single-output independent integrator chains, defined as follows:

∀i ∈ {1, . . . , m},

                   ż1,i = z 2,i . . . żr i -1,i = z r i ,i żr i ,i = w nom,i (8)
The following result, introduced in [START_REF] Bhat | Geometric homogeneity with applications to finite time stability[END_REF], proves the existence of a continuous finite time stabilizing feedback controller for system (8) by giving an explicit construction involving a small parameter. This controller renders the closedloop system asymptotically stable and homogeneous of negative degree with respect to a suitable dilatation so that finite time stability follows. Note that the proof of asymptotic stability along with a continuity argument is based on the existence of a nonempty compact set that is strictly positively invariant with respect to the closed-loop vector field. One can refer to [START_REF] Bhat | Geometric homogeneity with applications to finite time stability[END_REF] for further details.

Theorem 1 (Theorem 8.1 in [START_REF] Bhat | Geometric homogeneity with applications to finite time stability[END_REF]) Let the positive constants k 1,i , . . . , k r i ,i be such that polynomial p r i + k r i ,i p r i -1 + . . . + k 2,i p + k 1,i is Hurwitz. There is i ∈ (0, 1) such that, for every ν i ∈ (1i , 1), system ( 8) is stabilized at the origin in finite time under the feedback:

w nom,i (z i ) = -k 1,i sign (z 1,i ) |z 1,i | ν 1,i -. . . -k r i ,i sign (z r i ,i ) |z r i ,i | ν r i ,i (9)
where the standard notation sign (.) denotes the signum function and ν 1,i , . . . , ν r i ,i satisfy:

ν j-1,i = ν j,i ν j+1,i 2ν j+1,i -ν j,i , j ∈ {2, . . . , r i } with ν r i +1,i = 1 and ν r i ,i = ν i .
Example 1 Theorem 1 guarantees finite time stability of the triple integrator system under the feedback:

w nom,i (z i ) = -sign (z 1,i ) |z 1,i | 1 2 -1.5 sign (z 2,i ) |z 2,i | 3 5 -1.5 sign (z 3,i ) |z 3,i | 3 4 (10)
which is obtained from [START_REF] Filippov | Differential equations with discontinuous right hand sides[END_REF] with

r i = 3, k 1,i = 1, k 2,i = 1.5, k 3,i = 1.
5 and ν i = 3 4 . Figure 1(a) depicts the evolution of the system trajectories under feedback w nom,i (z i ). One can see that the triple integrator system is stabilized at the origin in finite time. However, when this system is perturbed, the finite time stabilization is not ensured. Indeed, considering the following uncertain system:

             ż1,i = z 2,i ż2,i = z 3,i ż3,i = w i + p(z i ) (11)
where p(z i ) = sin(10z 1,i ) is a bounded uncertainty, the control law [START_REF] Floquet | Higher order sliding mode stabilization for a class of nonholonomic perturbed systems[END_REF] does not guarantee finite time stability. The corresponding simulation results are shown in Fig. 1(b). Let us generalize the results given in [START_REF] Bhat | Geometric homogeneity with applications to finite time stability[END_REF] by designing a robust discontinuous control law which ensures the finite time stabilization of system (7).

Robust finite time controller design

The higher order SMC algorithm is designed in two steps:

(1) the design of a finite time controller w nom (z) which guarantees the finite time stabilization of nominal system (8) at the origin, (2) the design of discontinuous control law w disc (z) which enables to reject the uncertainties of system [START_REF] Edwards | Sliding mode control: theory and applications[END_REF] and ensures that control objectives are fulfilled.

In order to stabilize in finite time uncertain system [START_REF] Edwards | Sliding mode control: theory and applications[END_REF], let us define the following control law:

     w(z) = w nom (z) + w disc (z, z aux ) żaux = -w nom (z) (12)
Auxiliary function z aux ∈ R m will be used in the design of the sliding variable associated with the discontinuous control law w disc (z, z aux ) ∈ R m . The construction of control law w nom (z) = [w nom,1 (z 1 ), . . . , w nom,m (z m )] T ∈ R m is given in the previous section (i.e. equation ( 9)).

Let us define the sliding variable s(z) ∈ R m , associated with w disc , as follows:

s(z) = [z r 1 ,1 , z r 2 ,2 , . . . , z r m ,m ] T + z aux ( 13
)
The time derivative of s along the system trajectories, can be expressed as:

ṡ = [ żr 1 ,1 , żr 2 ,2 , . . . , żrm,m ] T + żaux = I m + ∆ B B -1 w -∆ B B -1 A + ∆ A -w nom = I m + ∆ B B -1 w disc -∆ B B -1 A + ∆ A + ∆ B B -1 w nom
The control law w disc is defined to ensure the sliding mode on {x ∈ X : s = 0} is guaranteed in spite of uncertainties and is of the following form:

w disc = -G(z) Sign (s) (14)
where the gain satisfies:

G(z) ≥ (1 -α) w nom (z) + ρ + η α ( 15
)
with η > 0. The notation Sign [s 1 , . . . s m ] T denotes [sign (s 1 ) , . . . , sign (s m )] T .

Theorem 2 Consider nonlinear system [START_REF] Bartolini | A survey of applications of second order sliding mode control to mechanical systems[END_REF] and assume that hypotheses 1-3 are fulfilled. Then, the control law

u = B -1 -A + w nom (z) + w disc (z, z aux ) ( 16
)
where żaux = -w nom (z), w nom (z) and w disc (z, z aux ) are given by equations ( 9) and ( 14), respectively, ensures the establishment of a higher order sliding mode with respect to σ in finite time.

Proof. Let us choose the following Lyapunov function:

V = 1 2 s T s
The time derivative of V along the system trajectories is expressed as:

V = s T I m + ∆ B B -1 w disc -∆ B B -1 A + ∆ A + ∆ B B -1 w nom = s T -G I m + ∆ B B -1 Sign (s) -∆ B B -1 A + ∆ A + ∆ B B -1 w nom
Under the bounding relations in Assumption 3 and the choice of gain ( 15), one obtains:

V ≤ -G s + (1 -α)G s + (1 -α) w nom s + ρ s ≤ (-αG + (1 -α) w nom + ρ) s ≤ -η s ≤ -η √ 2 √ V
Thus, system trajectories evolve on the manifold {x ∈ X : s = 0} in finite time and remains there in spite of the uncertainties. According to Assumption 3, matrix I m + ∆ B B -1 is uniformly invertible. In sliding mode, the equivalent control of w disc , denoted w eq disc , obtained by writing ṡ = 0 (see [START_REF] Utkin | Sliding modes control in electromechanical systems[END_REF] for further details), is given by:

w eq disc = I m + ∆ B B -1 -1 ∆ B B -1 A -∆ A -∆ B B -1 w nom ( 17
)
Substituting w = w nom + w eq disc into [START_REF] Edwards | Sliding mode control: theory and applications[END_REF], one obtains the equivalent closedloop dynamics, in sliding mode, similar as the nominal system [START_REF] Emel'yanov | Higher order sliding modes in control system[END_REF]. Since the control law w nom is designed using theorem 1, the system trajectories converge to zero in finite time. Therefore, a r th order sliding mode with respect to σ is established in finite time. [START_REF] Levant | Higher-order sliding modes, differentiation and output-feedback control[END_REF][START_REF] Shtessel | Aeronautical and space vehicule control in dynamic sliding manifold[END_REF].

Remark 2 The missing derivatives of σ i (1 ≤ i ≤ m) can be estimated online by means of the robust exact finite time convergent differentiator

Example 2 In order to illustrate the proposed control strategy, let us apply theorem 2 to the perturbed triple integrator system [START_REF] Fridman | Higher order sliding modes[END_REF]. One can show that the origin of system [START_REF] Fridman | Higher order sliding modes[END_REF] is stable in finite time under the control law:

w i (z i ) = w nom,i (z i ) + w disc (z i , z aux ) żaux = -w nom,i (z i ) w nom,i (z i ) = -sign (z 1,i ) |z 1,i | 1 2 -1.5 sign (z 2,i ) |z 2,i | 3 5 -1.5 sign (z 3,i ) |z 3,i | 3 4 w disc (z i , z aux ) = -1.5 sign (z 3,i + z aux )
The nominal part w nom,i (z i ) is the same as the control law of example 1. The discontinuous part, which enables to reject the perturbation p(z i ), ensures the sliding mode on the manifold {z 3,i + z aux = 0}. In this example, ρ(z i ) = 1 and α = 1. Using inequality [START_REF] Isidori | Nonlinear Control Sytems II[END_REF], one can deduce that G > 1. As it is theoretically expected, system trajectories are stabilized at the origin in finite time. The simulation results are depicted in Fig. 2.

Higher order sliding mode control of the hovercraft vessel

To illustrate the procedure given above, let us consider the higher order sliding mode control of a hovercraft vessel (see Fig. 3).

Hovercraft model

We consider the following uncertain hovercraft model, derived in [START_REF] Sira-Ramirez | Dynamic second order sliding mode control of the hovercraft vessel[END_REF]:

                                 ẋ = u cos ψ -v sin ψ ẏ = u sin ψ + v cos ψ ψ = w u = vw + τ u v = -uw -βv + ∆ v (x) ẇ = τ w + ∆ w (t) (18)
where x, y and ψ denote the position and orientation angle of the hovercraft. u, v and w represent its surge, sway and yaw angular velocity, respectively. τ u and τ w are the force and the torque applied to hovercraft. In order to test the robustness of the proposed controller, some additive perturbations are introduced in the dynamics. A matching perturbation ∆ w (t) affects the ship's yaw-rate dynamics. In the following simulations, ∆ w (t) is a gaussian noise of mean 0 and of variance 0.2. Furthermore, in the non actuated dynamics (i.e., in the sway acceleration equation), an external perturbation force is added. It simulates a rather strong "wave field" effect and is of the form:

∆ v (x) = 0.15 (sin(10x) + 0.2 cos(10x))
Assumption 4 The quantity βu + τ u is assumed to satisfy the following inequality:

βu + τ u ≥ C > 0 (19)
Therefore, we are interested in maneuvers for which this quantity is bounded away from zero, which somehow limits the class of reference trajectories (see [START_REF] Sira-Ramirez | Dynamic second order sliding mode control of the hovercraft vessel[END_REF] for further details).

Trajectory tracking for the hovercraft system using the proposed strategy

The control objective is to design a robust control law such that the hovercraft vessel system (18) tracks the following reference trajectory:

     x ref (t) = 10 cos π 6 t y ref (t) = 10 sin π 6 t
Let us define the following sliding variable:

σ =    σ 1 σ 2    =    x -x ref y -y ref   
Since the relative degree of system [START_REF] Levant | Universal SISO sliding-mode controllers with finite time convergence[END_REF] with respect to the sliding variable σ is not well defined (the second time derivatives ẍ and ÿ depend on τ u but not on τ w ), a dynamic extension of system ( 18) is needed before designing the control. Thus, a second order integrator chain is added on the input variable τ u . In the following, we propose a fourth order sliding mode controller in order to solve the trajectory tracking problem.

Remark 3 In this case, the discontinuous function acts on [τ u , τ w ] T . Thus, high-frequency commutation of control input τ u is removed. Furthermore, since the discontinuous function does not act on the first time derivative of the sliding variable (which would be the case with first order SMC), high frequency oscillations is reduced.

The fourth time derivative of σ is:

σ (4) = B + ∆ B    τu τ w    + A + ∆ A ( 20
)
with A and B are the known parts whereas ∆ A and ∆ B , which depend on the perturbations ∆ v and ∆ w are unknown. Under Assumption 4, matrix B is uniformly invertible. Thus, using the following preliminary feedback:

B =    cos ψ cos ψ(βv) -sin ψ (βu + τ u ) sin ψ sin ψ(βv) + cos ψ (βu + τ u )    , ∆ B =    0 -∆ v cos ψ 0 -∆ v sin ψ    , A = -x (4) ref -w cos ψ w (2βu + τ u ) + 2β 2 v + sin ψ β 3 v + w β 2 u -2 τu -βτ u -2w 2 βv -y (4) ref -w sin ψ w (2βu + τu) + 2β 2 v -cos ψ β 3 v + w β 2 u -2 τu -βτu -2w 2 βv ∆ A = ∆v w 2 + ∆v β -∆v -∆w τu -∆v β 2 -
   τu τ w    = B -1 τ -A (21)
where τ ∈ R 2 is the auxiliary control input, system (20) becomes:

σ (4) = I 2 + ∆ B B -1 τ + ∆ A -∆ B B -1 A (22)
One can note that ∆ B B -1 ≤ 2|∆v| C . From Theorem 2, one gets:

τ = τ nom + τ disc żaux = -τ nom τ nom = -24    sign (σ 1 ) |σ 1 | 3 7 sign (σ 2 ) |σ 2 | 3 7    -50    sign ( σ1 ) | σ1 | 1 2 sign ( σ2 ) | σ2 | 1 2    -35    sign (σ 1 ) |σ 1 | 3 5 sign (σ 2 ) |σ 2 | 3 5    -10    sign σ (3) 1 |σ (3) 1 | 3 4 sign σ (3) 2 |σ (3) 2 | 3 4    τ disc = -G Sign σ (3) + z aux

Simulation results

Note that for the chosen reference circular trajectory, the nominal value of the quantity βu + τ u , appearing in the denominator of the controller expression, is given by:

βu + τ u = 10 π 6 β 2 + π 6 2 = 0
The only system parameter β is set to 1.2. The outputs x and y are measured by appropriate sensors. The time derivatives of the sliding variable σ are estimated using a higher order robust differentiator [START_REF] Levant | Higher-order sliding modes, differentiation and output-feedback control[END_REF].

In order to highlight the advantages of the proposed controller, a comparative study between higher order SMC algorithms is carried out. Among these algorithms which allow the use of the robust differentiator (i.e. do not require the accurate knowledge of the initial high order time derivatives, up to the relative degree, of the sliding variable), do not need complex off-line computations and provide a constructive procedure for the control parameters, the Figures 4 and5 show the simulation results using the proposed controller and the controller given in [START_REF] Levant | Universal SISO sliding-mode controllers with finite time convergence[END_REF], respectively. Figures 4(a From these results, one can conclude that the chattering phenomenon is lower using the proposed strategy. Indeed, the control law [START_REF] Levant | Universal SISO sliding-mode controllers with finite time convergence[END_REF] has only a discontinuous part. Here, the proposed controller is divided into a homogeneous part and a discontinuous one. Furthermore, contrary to [START_REF] Levant | Universal SISO sliding-mode controllers with finite time convergence[END_REF], one can easily hasten the finite-time transient process (see the convergence rate of σ 2 for both algorithms). These results show the robustness and the advantages of the proposed finite time convergent controller.

Conclusion

A constructive algorithm for the design of a higher order sliding mode controller for a class of multi-input multi-output nonlinear uncertain systems is proposed. The main contribution of this paper is the design of a robust finite time convergent controller based on geometric homogeneity and sliding mode control which can be easily applied to systems with high relative degree with respect to the sliding variable. The proposed method does not require complex off-line computation and provides explicit conditions on the control parameters. A case study for finite time convergent robust control of a hovercraft vessel system is presented. Simulation results show the robustness against parameter variations and perturbation force inputs and the advantages in terms of convergence rate with respect to [START_REF] Levant | Universal SISO sliding-mode controllers with finite time convergence[END_REF]. (c) Fig. 5. Circular path tracking performance under unmodeled perturbations using the algorithm given in [START_REF] Levant | Universal SISO sliding-mode controllers with finite time convergence[END_REF].

Fig. 1 .

 1 Fig. 1. Trajectories of the triple integrator system under feedback (10) (a) without perturbation (b) with pertubations.

Fig. 2 .

 2 Fig.2. Trajectories of the uncertain triple integrator system using the proposed control strategy.

Fig. 3 .

 3 Fig. 3. Hovercraft vessel system.

Term τ nom is obtained by setting r = 4 ,

 4 k 1 = 24, k 2 = 50, k 3 = 35, k 4 = 10 and ν = 3 4 (see Theorem 1). Gain G is tuned as G = 0.2 τ nom + 11 (condition (15).

  Figures4 and 5show the simulation results using the proposed controller and the controller given in[START_REF] Levant | Universal SISO sliding-mode controllers with finite time convergence[END_REF], respectively. Figures4(a) and 5(a) depict the tracking errors x -x ref and y -y ref and show the finite time convergence of σ 1 and σ 2 in spite that the vessel motions are started significantly far away from the reference trajectory (see Fig. 4(b) or 5(b)). Figures4(c) and 5(c) display the corresponding surge, sway and yaw angular velocities. The two controllers efficiently correct the undesirable deviations due to the persistent perturbations. From these results, one can conclude that the chattering phenomenon is lower using the proposed strategy. Indeed, the control law[START_REF] Levant | Universal SISO sliding-mode controllers with finite time convergence[END_REF] has only a discontinuous part. Here, the proposed controller is divided into a homogeneous part and a discontinuous one. Furthermore, contrary to[START_REF] Levant | Universal SISO sliding-mode controllers with finite time convergence[END_REF], one can easily hasten the finite-time transient process (see the convergence rate of σ 2 for both algorithms). These results show the robustness and the advantages of the proposed finite time convergent controller.

Fig. 4 .

 4 Fig. 4. Circular path tracking performance under unmodeled perturbations using the proposed algorithm.

  ∆w βu sin ψ + 2∆v βw + ∆w βv -2 ∆v w -∆v ∆w cos ψ ∆w τu + ∆v β 2 -∆v β -∆v w 2 + ∆v + ∆w βu cos ψ + 2∆v βw -∆v ∆w -2 ∆v w + ∆w βv sin ψ

Acknowledgements

The authors wish to thank the anonymous reviewers for useful comments and suggestions.

This work was partially supported by the JSPS Postdoctoral Fellowship and the FEDER (European Funds of Regional Development) under the ARCIR Robocoop and the AUTORIS-TAT T31 projects.

most appealing strategy is given by the following equation [START_REF] Levant | Universal SISO sliding-mode controllers with finite time convergence[END_REF]:

where the gain γ is taken as γ = 100.