Corinne Sarthou
email: sarthou@mnhn.fr

Denis Larpin
email: larpin@mnhn.fr

Émile Fonty
email: emile.fonty@free.fr

Sandrine Pavoine
email: pavoine@mnhn.fr

Jean-François Ponge
email: ponge@mnhn.fr

Stability of plant communities along a tropical inselberg ecotone in French Guiana (South America)

Keywords: Inselberg, French Guiana, Ecotone, Plant communities, Low forest, Open-rock vegetation, Diachronic change, Ecological barrier

We questioned whether and how plant communities vary in space and time along an inselbergrainforest ecotone in relation to present-day warming and whether biotic and non-biotic factors could explain the observed patterns. The study took place on a granitic inselberg in the French Guianan (South America) rainforest (Nouragues Natural Reserve: 4°5"N, 52°41"W). In a diachronic study (1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005) embracing a severe El-Niño event in 1997, we analysed vegetation structure and composition along three transects subsuming whole environmental and topographical variations in the transition zone from shrub vegetation at the fringe of open-rock vegetation to tall-tree rainforest. Data were analysed by PCA. Major variations in species and trait distribution were described in the low forest, with two floristic types evidenced by first PCA component and verified by cluster analysis: one with floristic composition reminiscent of open-rock vegetation but with higher and continuous canopy, the other typical of the low forest. There is no clear-cut

boundary between typical open-rock and low forest vegetation. Variation in species composition of typical low forest was evidenced by second PCA component, which displayed differences according to slope and altitude. Small (~1.5 m), although significant, shifts in the spatial distribution of plant species pointed to possible slow encroachment of typical low forest vegetation in the absence of disturbance. However, the stability of species and trait distribution was remarkable within the 10-yr interval considered, despite an otherwise recorded decrease in species richness and recruitment. The boundary between typical low forest and open-rock-like vegetation coincided with the spatial limit of the mineral soil above granite. Despite demographic accidents due to severe El Niño events, plant communities at the fringe of a tropical inselberg are stable at short-time both in composition and spatial distribution. In the absence of strong disturbances such as wildfire and further erosion, soil availability for roots could be interpreted as an environmental constraint to the successional development of forest vegetation. Soil development might thus act as an ecological barrier to forest encroachment, which could only be alleviated by erosion recovery, as otherwise demonstrated.

Introduction

Ecotones, as "zones of transition between adjacent ecological systems", have long time fascinated ecologists (synthesis in [START_REF] Hansen | Landscape boundaries: consequences for biotic diversity and ecological flows[END_REF] and challenged them to find causes and resulting patterns. Such discontinuities in vegetation composition may offer insights into the factors controlling the assembly of plant communities and thus may have significance for management and nature conservation.

Main effects of environmental change are likely to be seen first at ecotones due to physiological characteristics of species occurring there [START_REF] Neilson | Climatic constraints and issues of scale controlling regional biome[END_REF][START_REF] Kupfer | The suitability of montane ecotones as indicators of global climatic change[END_REF][START_REF] Allen | Drought-induced shift of a forest-woodland ecotone: rapid landscape response to climate variation[END_REF].

Many studies reported discrete vegetation boundaries on tropical mountains (e.g. [START_REF] Woldu | Partitioning an elevation gradient of vegetation from southeastern Ethiopia by probabilistic methods[END_REF][START_REF] Kitayama | An altitudinal transect study of the vegetation of Mount Kinabalu[END_REF][START_REF] Kitayama | An altitudinal transect analysis of the windward vegetation on Haleakala, a Hawaiian island mountain. II. Vegetation zonation[END_REF][START_REF] Fernandez-Palacios | Altitudinal pattern of vegetation variation on Tenerife[END_REF][START_REF] Kitayama | Biophysical conditions of the montane cloud forest of Mount Kinabalu, Sabah, Malaysia[END_REF][START_REF] Ashton | Floristic zonation of tree communities on wet tropical mountains revisited[END_REF][START_REF] Hemp | Climate change-driven forest fires marginalize the impact of ice cap wasting on Kilimanjaro[END_REF][START_REF] Martin | Tropical montane forest ecotones: climate gradients, natural disturbance, and vegetation zonation in the Cordillera Central, Dominican Republic[END_REF][START_REF] Sherman | Fire and vegetation dynamics in high elevation neotropical montane forests of Dominican Republic[END_REF]. Robust quantitative studies of compositional distribution that could address global change require highly replicated, fine-scale sampling (e.g. Vazquez and Givnish, 1998), but such studies are rare in the tropics, primarily because of the large sampling area needed for species census in species-rich tropical plant communities.

Inselbergs, from German "insel" for island and "berg" for mountain [START_REF] Bornhardt | Zur Oberflächengestaltung und Geologie Deutsch-Ostafrikas[END_REF], shaping isolated hills or groups of hills, constitute a peculiar model of mountain environment. In tropical areas, these frequently huge, Gyr-old monoliths mainly consist of Precambrian granites or gneisses and are defined as remnants of erosion processes within a plain landscape [START_REF] Cooke | Desert geomorphology[END_REF]. They form "xeric islands" within a rain forest matrix and harbour unique open-rock vegetation enduring harsh edaphic and microclimatic conditions. Floristic studies carried out on tropical inselbergs have been synthesised by [START_REF] Porembski | Inselbergs: biotic diversity of isolated rock outcrops in tropical and temperate regions[END_REF]. Since this synthesis, other surveys dealing with inselberg vegetation have been mainly concerned with description of plant communities [START_REF] Parmentier | La végétation des inselbergs du monument naturel de Piedra Nzas (Guinée Equatoriale continentale)[END_REF][START_REF] Oumorou | Aperçu de la végétation de quelques inselbergs du Bénin[END_REF][START_REF] Sarthou | Shrub vegetation on tropical granitic inselbergs in French Guiana[END_REF][START_REF] Parmentier | The inselberg flora of Atlantic Central Africa. I. Determinants of species assemblages[END_REF][START_REF] Parmentier | Grasslands and herbaceous fringes on inselbergs in Atlantic central Africa[END_REF]. Monocotyledonous mats, small grasslands and shrub thickets characterise the outcrop vegetation of inselbergs, but ecotones, i.e. transition areas between xerophilous outcrop vegetation and rainforest, have been rarely studied [START_REF] Parmentier | The inselberg flora of Atlantic Central Africa. I. Determinants of species assemblages[END_REF] and diachronic studies have never been conducted on tropical inselbergs. However, boundaries between such contrasted environments might allow study cases of directional changes in species and species trait distribution in the perspective of global warming, as this has been done at the upper forest limit in tropical mountains [START_REF] Vázquez | Altitudinal gradients in tropical forest composition, structure, and diversity in the Sierra de Manantlán[END_REF][START_REF] Ashton | Floristic zonation of tree communities on wet tropical mountains revisited[END_REF][START_REF] Hemp | Climate change-driven forest fires marginalize the impact of ice cap wasting on Kilimanjaro[END_REF][START_REF] Hemp | Continuum or zonation? Altitudinal gradients in the forest vegetation of Mt. Kilimandjaro[END_REF][START_REF] Martin | Tropical montane forest ecotones: climate gradients, natural disturbance, and vegetation zonation in the Cordillera Central, Dominican Republic[END_REF] and in forestgrassland or forest-scrub tropical and sub-tropical ecotones [START_REF] Allen | Drought-induced shift of a forest-woodland ecotone: rapid landscape response to climate variation[END_REF][START_REF] Grau | Rainfall variability, fire and vegetation dynamics in neotropical montane ecosystems in north-western Argentina[END_REF][START_REF] Favier | Modelling forest-savanna mosaic dynamics in man-influenced environments: effects of fire, climate and soil heterogeneity[END_REF].

In French Guiana, a granitic inselberg located in the Nouragues Biological Reserve has been particularly studied since two decades. Three ecosystems with genuine plant communities have been studied: open-rock vegetation (locally called "savane-roche"), low forest and tall-tree rainforest, the two latter being closed habitats. The two former vegetation types and specially epilithic plant communities have been described extensively [START_REF] Sarthou | Dynamique de la végétation pionnière sur un inselberg de la Guyane Française[END_REF][START_REF] Larpin | Les formations ligneuses sur un inselberg de Guyane Française: étude floristique, structurale et dynamique[END_REF][START_REF] Sarthou | Epilithic plant communities on inselbergs in French Guiana[END_REF][START_REF] Sarthou | Plant communities on a granitic outcrop[END_REF][START_REF] Larpin | The low forest (Nouragues inselberg)[END_REF][START_REF] Sarthou | Shrub vegetation on tropical granitic inselbergs in French Guiana[END_REF]. They are characterized by highly contrasted ecological conditions. In open-rock vegetation, exposed rocks are subject to high levels of insolation in combination with high evaporation rates.

Shallow soils and steep slopes cause a rapid run-off of water. Organic matter and roots accumulate locally under shrub thickets [START_REF] Kounda-Kiki | Humus profiles under main vegetation types in a rock savanna (Nouragues inselberg, French Guiana)[END_REF][START_REF] Dray | On the number of principal components: a test of dimensionality based on measurements of similarity between matrices[END_REF]. In the low forest, the temperature is more buffered and soils are deeper and mostly made of mineral matter overlaid by a thin litter layer. Both ecosystems were submitted to occasional fires during past dry periods as demonstrated by charcoal layers in the soil [START_REF] Charles-Dominique | Palaeoclimates and their consequences on forest composition[END_REF][START_REF] Kounda-Kiki | Humus profiles and successional development in a rock savanna (Nouragues inselberg, French Guiana): a micromorphological approach infers fire as a disturbance event[END_REF] and past erosive sequences [START_REF] Rosique | Évolution morphométrique holocène d"un bassin versant de la forêt guyanaise: la Nourague occidentale (Guyane française)[END_REF]. Whether of natural or human origin, fires dated from 10,000 to 100 yr BP are a driving force of vegetation dynamics. It is currently admitted that Guianan inselbergs, between severe and widespread fire events associated with colder and drier periods, are invaded by forest communities [START_REF] De Granville | Rain forest and xeric flora refuges in French Guiana[END_REF][START_REF] Ledru | Reconstitution palynologique de la forêt guyanaise au cours des 3000 dernières années[END_REF][START_REF] Charles-Dominique | Forest perturbations and biodiversity during the last ten thousand years in French Guiana[END_REF][START_REF] Larpin | Dynamique de la végétation sur l"inselberg des Nouragues (Guyane française) à différentes échelles de temps (annuelle à plurimillénaire)[END_REF][START_REF] Charles-Dominique | Incidents climatiques et modifications de l"écosystème forestier depuis 3000 ans en Guyane française[END_REF], 2001).

However, the continuum from the fringe of open rock vegetation to the rainforest remains rather unknown. In a previous paper, we reported a marked decrease in plant species richness and tree recruitment in a 10-yr period (1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005) embracing a severe El-Niño event in 1997 [START_REF] Fonty | A 10-year decrease in plant species richness on a neotropical inselberg: detrimental effects of global warming[END_REF]. In the present study, we explored the community scale and questioned the temporal change of the vegetation on the ecotone gradient over the same period. This is the first diachronic study of plant communities along a tropical inselberg ecotone in South America and we address the following questions (1) do present-day environmental conditions explain the spatial distribution of plant species in the contact zone between low forest and shrub vegetation fringes on rock outcrops? (2) did this distribution change during the recorded time lag?

Methods

Study site

Our study was conducted in French Guiana on a granitic inselberg 410m high in the Nouragues Natural Reserve (4°5"N, 52°41"W) in South America. The climate in the study area is seasonal, with a long rainy season from December to June and a short drier season from August to November. Annual precipitation averages 4000 mm. Mean annual air temperature lies between 25 and 27°C at the Nouragues station [START_REF] Grimaldi | Nouragues: dynamics and plant-animal interactions in a neotropical rain forest[END_REF]. Climatic measurements (synthesis in [START_REF] Fonty | A 10-year decrease in plant species richness on a neotropical inselberg: detrimental effects of global warming[END_REF] indicate a marked El Niño event in 1997. This granitic whaleback dome protruding ~300m from the rain forest matrix (Fig. 1) belongs to continental sets of rocks that constitute the base of the Guiana Shield extending north to Amazon River from Venezuela to Colombia [START_REF] Théveniaut | Connaissance géologique de la Guyane[END_REF]. Whenever the slope becomes too steep, differential erosion tends to increase more and more the slope, locally impeding forest settlement. This is the origin of present-day inselbergs. The topography is rugged, with steep slopes.

Open-rock vegetation is characterized by a discontinuous plant cover, separated by exposed rock, at the top and on southern and eastern sides of the inselberg (Figs. 1 and2). This xerophilous vegetation is composed of numerous small scattered patches of epilithic wind-and bird-disseminated herb species and shrubs. Two widespread plant communities give a typical aspect to the landscape [START_REF] Sarthou | Plant communities on a granitic outcrop[END_REF]: a herbaceous community dominated by the bromeliad Pitcairnia geyskesii established directly on granite and a shrub community dominated by the Clusiaceae Clusia minor forming thickets 2-8 m tall, with an organic soil [START_REF] Kounda-Kiki | Humus profiles under main vegetation types in a rock savanna (Nouragues inselberg, French Guiana)[END_REF]. Thickets in the contact zone with the low forest are likely denser and more diversified [START_REF] Sarthou | Dynamique de la végétation pionnière sur un inselberg de la Guyane Française[END_REF][START_REF] Sarthou | Plant communities on a granitic outcrop[END_REF]. Measurements of air temperature and relative air humidity show high diurnal variations [START_REF] Sarthou | Dynamique de la végétation pionnière sur un inselberg de la Guyane Française[END_REF][START_REF] Sarthou | Plant communities on a granitic outcrop[END_REF][START_REF] Rascher | Photosynthetic field capacity of cyanobacteria of a tropical inselberg of the Guiana Highlands[END_REF]. Due to shallow soils with a low water-holding capacity, moisture conditions in this ecosystem are harsher than everywhere on the inselberg [START_REF] Sarthou | Mécanismes de colonisation par la végétation d"un inselberg granitique en Guyane française[END_REF].

The low forest surrounds open-rock vegetation, forming a transition zone with the tall-tree rainforest, and is also established on the summit of the inselberg (Figs. 1 and2). In contrast with open-rock vegetation, the amplitude of temperature and moisture variations is much lesser, even during the dry season [START_REF] Larpin | Les formations ligneuses sur un inselberg de Guyane Française: étude floristique, structurale et dynamique[END_REF][START_REF] Larpin | The low forest (Nouragues inselberg)[END_REF]. Enrichment in water and nutrients by inflow from above located open-rock vegetation [START_REF] Sarthou | Mécanismes de colonisation par la végétation d"un inselberg granitique en Guyane française[END_REF][START_REF] Dojani | Nitrogen input by cyanobacterial biofilms of an inselberg into a tropical rain forest in French Guiana[END_REF] gives to the vegetation of the low forest a lush, species-rich aspect, with many epiphytes and terricolous herbs in the understory along with a diversity of shrubs and small trees [START_REF] Larpin | Les formations ligneuses sur un inselberg de Guyane Française: étude floristique, structurale et dynamique[END_REF][START_REF] Larpin | The low forest (Nouragues inselberg)[END_REF]. This vegetation has been described as a specific community characterized by Myrtaceae. Prominent features of the low forest are: low canopy (< 10 m), multi-stemming and vertical stratification of the vegetation (Fig. 2). Woody species are usually zoochorous and mainly dispersed by small birds [START_REF] Larpin | The low forest (Nouragues inselberg)[END_REF]. Soils are of varying depth (0-180 cm), increasing in depth from open-rock vegetation to the tall-tree rainforest (Fig. 2).

The tall-tree rainforest hosts a great diversity of trees, the height of which averages 30-35 m with emergent trees reaching 50 m in height [START_REF] Poncy | The lowland high rainforest: structure and tree species diversity[END_REF]. Soils are acid (pH < 5) clay-sandy Ferralsols (FAO, 2006) with a micro-aggregate organo-mineral texture of biological origin [START_REF] Pouvelle | Topsoil as affected by dung deposition under resting places of red howler monkey (Alouatta seniculus)[END_REF] and a sparsely distributed litter cover [START_REF] Grimaldi | Nouragues: dynamics and plant-animal interactions in a neotropical rain forest[END_REF]. Treefall gaps and erosive processes take a prominent part in the dynamics of vegetation [START_REF] Riéra | Rôle des perturbations actuelles et passées dans la dynamique et la mosaïque forestière[END_REF][START_REF] Sabatier | The influence of soil cover organization on the floristic and structural heterogeneity of a Guianan rain forest[END_REF].

Fires, whether of human or natural origin, affect the fate of vegetation and open cues to strong erosive processes [START_REF] Sabatier | The influence of soil cover organization on the floristic and structural heterogeneity of a Guianan rain forest[END_REF][START_REF] Rosique | Évolution morphométrique holocène d"un bassin versant de la forêt guyanaise: la Nourague occidentale (Guyane française)[END_REF]. Open as well as forested sites were submitted to occasional fires during past dry periods as demonstrated by charcoal layers in the soil [START_REF] Charles-Dominique | Palaeoclimates and their consequences on forest composition[END_REF][START_REF] Kounda-Kiki | Humus profiles and successional development in a rock savanna (Nouragues inselberg, French Guiana): a micromorphological approach infers fire as a disturbance event[END_REF]. The general disappearance of the original Amerindians at the end of the 18th century let this place devoid of permanent human occupation since about two centuries [START_REF] Hurault | Français et Indiens en Guyane: 1604-1972[END_REF].

Sampling

Three transects T4, T5 and T6 were established on southern, eastern and summit positions, respectively (Fig. 1). Transect length reached 89 m, 65 m and 52 m, respectively. All transects started on bare rock at the boundary of open-rock vegetation, going into the forest and stopped when the tall trees that characterize the rain forest first appeared (Fig. 2). They overwhelmed the whole area of what was described as the low forest by [START_REF] Larpin | Les formations ligneuses sur un inselberg de Guyane Française: étude floristique, structurale et dynamique[END_REF][START_REF] Larpin | The low forest (Nouragues inselberg)[END_REF]. These transects were chosen in order to describe the whole range of topographical conditions prevailing on the inselberg. Transect T4 showed an average 38% slope and was at 130 m mid elevation. Transect T5 showed an average 24% slope and was at 100 m mid elevation. Transect T6 showed an average 2% slope and was at 380 m mid elevation.

Vascular plants were identified to species or genus at the "Herbier de Guyane" in Cayenne and updated with a checklist by [START_REF] Funk | Checklist of the plants of the Guiana Shield (Venezuela: Amazonas, Bolivar, Delta Amacuro; Guyana; Surinam; French Guiana)[END_REF]. Census was taken each metre on adjacent quadrats (1x2 m). In each plot, the diameter of each stem was measured to the next cm, the canopy height was estimated visually and the number of individuals per quadrat was counted for every species. In case of multi-stemming, stems were pooled at the individual level for the calculation of species abundance per quadrat. The cover percentage of herbs and suffrutex species was estimated visually in each quadrat to the next 5%. The three transects were sampled again in April 2005 (same month and same rainy conditions as in 1995), after thorough examination of previous maps and retrieval of keystone trees which were used for replacing transects at the exact position they occupied 10 years sooner. The same well-trained botanists (D. Larpin and C. Sarthou) were present on the field in 1995 and 2005, thus avoiding biases in cover percentage estimates underlined by [START_REF] Leps | How reliable are our vegetation analyses?[END_REF]. The soil depth (organic and mineral) was measured in 2005 only, by forcing a 1-m steel post in the soil with a hammer, until the granite substrate was attained when not deeper than 70 cm.

Data processing

Vegetation data (number of individuals for woody species, percent cover for herbs and suffrutex species, in each quadrat and each transect) were analysed by Principal Components Analysis (PCA), using only species which were present in at least 10 quadrats over all transects and both years. Elevation, slope, depth of organic layer and mineral soil and year were added as passive variables in PCA and were projected on factorial scatter plots without having any influence on the calculation of factorial axes, also called principal components (indirect gradient analysis). All variables were standardized (mean 0, standard deviation 1), thus allowing data of different kinds and scales to be included in the same multivariate analysis, using correlation coefficient values in place of Euclidean distances [START_REF] Chae | Effect of using principal coordinates and principal components on retrieval of clusters[END_REF]. The choice of indirect rather than direct (RDA, DCA) gradient analysis was justified by the need to allow patterns merge from the data without any a priori hypotheses about the distribution of plant species and changes which could occur in the course of time. The choice of PCA over CoA (correspondence analysis) was justified by the nature of the data, which varied from occurrence data (for woody species) to cover percentages (for herbs and suffrutex species). The distribution of plant traits (life forms, fruit and dispersal types, functional types, seed sizes) was graphically studied in F1-F2 species scatter plots.

Cluster analysis [START_REF] Legendre | Numerical ecology[END_REF] was performed on plant species using hierarchical aggregative average-linkage clustering with Spearman rank correlation coefficients of raw data (used for PCA) as between-species similarity measurement.

Given that sampling was done along transect lines across variable environments, autocorrelation was expected [START_REF] Legendre | Numerical ecology[END_REF]. Correlations between species richness, canopy height (a surrogate of forest development), soil depth and Principal Component 1 of PCA (a surrogate of shift from open-rock to forest vegetation, as explained below) were analysed by simple or partial Mantel tests, the latter using distance along transects as co-variate [START_REF] Legendre | Numerical ecology[END_REF]. We used signed Mantel tests, in which the distance between two samples (quadrats) was estimated by the signed difference between corresponding values of the studied factor [START_REF] Oberrath | The signed Mantel test to cope with autocorrelation in comparative analyses[END_REF]. The significance of Mantel tests was tested by Monte-Carlo simulation procedure (5,000 runs).

To each plant species was assigned a mean distance from transect start, which was calculated by weighing all distances (from meter to meter) by abundance (for trees and shrubs) or cover percentage (for herbs and suffrutex species) and averaging these values over a whole transect, in 1995 and in 2005. Species were classified in groups (all species, trees and shrubs, herbs and suffrutex species) the mean values of which were compared in 1995 and 2005 by paired t-test, using species as independent replicates. All abovementioned calculations were done using XLSTAT statistical software [START_REF] Addinsoft | XLSTAT (data analysis software system)[END_REF].

Variation in the distribution of plant species between 1995 and 2005 was tested by co-inertia analysis [START_REF] Dolédec | Co-inertia analysis: an alternative method for studying species-environment relationships[END_REF] followed by Monte-Carlo test. The meaningfulness of species groups depicted by PCA was verified by two-way bottom-up hierarchical cluster analysis using Euclidean distances between coordinates of plant species along the first five factorial axes of PCA, with Bonferroni adjustment and Ward"s criterion (variance increase) for aggregation [START_REF] Dray | On the number of principal components: a test of dimensionality based on measurements of similarity between matrices[END_REF]. These analyses are implemented in the R programme [START_REF] Ihaka | R: a language for data analysis and graphics[END_REF].

Results

A total of 150 plant species were found in the three studied transects over both years 1995 and 2005.

They were comprised of 109 tree and shrub species (Appendix 1) and 41 herb and suffrutex species (Appendix 2), most of them being determined at the species (69%), genus (81%) or family level (95%).

Among woody species, Myrtaceae were the dominant family (20 species), followed by Rubiaceae (10 species). Among herbs and suffrutex species, Bromelieaceae were the dominant family (8 species), followed by Poaceae (8 species) and Orchidaceae (6 species).

The state of change of the plant community

A concomitant increase in local species richness (number of plant species per quadrat) and canopy height (height of tallest woody stems per quadrat) was observed along the three transects (Table 1). Whatever transects were considered, species richness and canopy height exhibited a highly significant positive correlation, which remained positive and highly significant when correcting for spatial influence. At the start of each transect, at the boundary of open-rock vegetation, the species richness was very low (1 to 3 species per quadrat) while it reached 25 to 30 species per quadrat at the boundary of the tall-tree rain forest (transect end). The canopy height rose from 0 at transect start to 10 to 30 m (according to transects), in the forest.

Despite a general increasing trend of canopy height, a great variation was observed in Transects 4 and 5.

Transect 4 (established on steep slope) exhibited several collapses in species richness, which can be explained by local wind-throws (treefall gaps) between 1995 and 2005.

The projection of 410 samples and five passive variables (elevation, slope, year, depth organic, depth mineral) in the F1-F2 bi-plot of PCA (Fig. 3a) showed that the year effect was negligible compared to that of depth of mineral soil (highly correlated with the first component F1, which extracted 7.2% of the total variation), and that of elevation and slope (both highly correlated with the second component F2, which extracted 5.5% of the total variation). In T4, T5 and T6 and in both years, quadrats were projected on both positive and negative sides of F1, with a clear discontinuity between two groups of quadrats in T4 and T6 (Fig. 3b). The first component was correlated with the depth of mineral soil above the granitic substrate, as indicated by a highly significant Mantel correlation coefficient (Table 1), which remained significant when spatial effects were discarded. The distribution of F1 values along the three transects showed a strong spatial dependence (Fig. 4): at start of transects, F1 values were negative and increased towards positive values with distance. Positive values of F1 were clearly associated with the presence of a mineral soil. It can be seen on T4 that each "accident" in the depth of the mineral soil (granite apparent or only covered with an organic layer) was associated with a reversal towards negative values of F1. The passage from negative to positive values of F1 was progressive in T5 but rather abrupt in T4 and T6.

The second principal component F2 separated T4 and T5 (lower elevation and higher slope) from T6 (higher elevation and lower slope). This separation concerned mainly samples far from transect start, i.e. those where mineral soil was present and with positive F1 coordinates (Fig. 4). Differences between years were not very pronounced (Figs. 3 and4), except that positive F1 coordinates were for the main higher in 1995 than in 2005, the contrary being shown on the negative side. As a result, the variance of F1 values was higher (9.0) in 1995 than in 2005 (5.3).

The projection of plant species in the F1-F2 scatter plot showed three groups of plant species, corresponding to three branches in the cloud of data (Fig. 5a). A species group was projected on the negative side of F1, while species with positive F1 values were separated in two groups by F2. Primary cluster analysis (Fig. 6) displayed a clear dichotomy between two main groups of species which were nearly identical to the two groups separated along the first component of PCA. A total of 28 among the 30 species classified in the first cluster (Community I) had negative values for F1, the other two species (Bromelia sp. 1 and Inga umbellifera) having positive values for F1 but not far from zero (Fig. 5). Conversely, a total of 49 among the 51 species classified in the second cluster (Community II) had positive values for F1, the other two species (Syagrus stratincola and Poacaeae sp. 4) having negative values for F1 but not far from zero (Fig. 5). The two-branch dichotomy which was observed along the second principal component of PCA was not retrieved by cluster analysis (Fig. 6).

Herbs and suffrutex species represented 50% of the species in Community I (Fig. 5b), belonging mainly to Bromeliaceae and Cyclanthaceae and with a high cover for Stelestylis surinamensis. Herbs and suffrutex species were poorly represented in Community II, comprising less than 2% of the species (Fig. 5b), and were mainly composed of Poaceae. The dominant families of woody plants in Community I were Myrtaceae (Myrcia saxatilis, M. guianensis, Eugenia florida, Myrciaria sp2), Clusiaceae (Clusia minor, C. nemorosa) and Melastomataceae (Ernestia granvillei, Miconia ciliata). Community II was characterized by 40 tree and shrub species belonging mainly to Myrtaceae (22.5%), Rubiaceae (15%), Mimosaceae (7.5%), Erythroxylaceae (5%) and Apocynaceae (5%) (Appendices 1 and 2). The floristic composition exhibited an abrupt change from Community I to Community II along T4 and T6 and a continuum in T5 (Fig. 4).

The second principal component of PCA expressed changes in the species composition of Community II which occurred according to the position of transects on the inselberg (Figs. 3 and5). This result points to the existence of sub-communities within Community II, which are separated by their position around the inselberg. However, cluster analysis did not display the existence of clear sub-groups of plant species within the second cluster (corresponding to positive F1 values), the stemming of branches being continuous within each of the two clusters corresponding to Community I and Community II, respectively.

Year (1995 vs 2005) did not influence to a great extent the distribution of plant species in the F1-F2 scatter plot. The absence of any significant variation in the distribution of plant species between 1995 and 2005 was verified by co-inertia analysis on T4, T5 and T6: both matrices (1995 and 2005) were positively correlated (r = 0.79, P = 0.001). However, it can be observed that positive F1 coordinates were in the main higher in 1995 than in 2005, the contrary being shown on the negative side (Figs. 3 and4). As a result, the variance of F1 values was higher (9.0) in 1995 than in 2005 (5.3). It must be noted, too, that the alternation of humps and mounds depicted by F1 values along each of the three transect lines (Fig. 4) showed a good correspondence between 1995 and 2005, which validated transect positioning ten years after the first installation.

Species traits were not equally distributed on both sides of F1, corresponding to the two abovementioned plant communities. In addition to the already reported habitat preference of herbs and suffrutex species for lower parts of transects (corresponding to the first cluster of cluster analysis or Community I), lianas (5 species kept for PCA) were all projected on the positive side of F1 (Fig. 5). This was also the case for trees, to the exception of one species (Humiria balsamifera). Apart from this exception, all woody species belonging to Community II were shrubs. Most plants with berries (16 among 24) and drupes (10 among 13) were projected on the positive side of F1, whereas most species having capsules (14 among 20) were projected on its negative side. However, plants with achenes (4 species) were distributed equally on both sides of F1, indicating the absence of any preference for the two communities. Other fruit types were negligible. In accordance with fruit types, dispersal types showed a discrepancy between both communities, anemochorous species belonging mostly to Community I (15 among 24) while zoochorous species belonged mostly to Community II (38 among 50), the other dispersal types being poorly represented (Fig. 5). With exception of climbing plants (lianas), which were only represented in Community II, other functional types did not show any pronounced preference for one or the other community. Small seeds (< 0.5cm) were mostly present in Community I (18 among 30) but most discrepancy occurred with large seeds (> 1cm) which, to the exception of Tabebuia capitata, belonged to Community II, while medium-sized seeds were present in similar proportions in both communities. Thus the first component of PCA depicted also a gradient of species traits, from a herb/suffrutex/shrub community characterized by dry fruits with small to medium seeds disseminated by wind to a tree community (with lianas) characterized by fleshy fruits with medium to large seeds disseminated by animals.

Further axes of PCA were neglected, since they displayed only very minor changes in floristic composition, not explained by environmental factors, and which were not validated by cluster analysis.

Community boundaries and their shifts from 1995 to 2005

New sampling done in 2005 did not reveal any pronounced displacement in the passage from one community to another, which occurred roughly at the same position on the transect lines. On both years, Community I was replaced by Community II once the mineral soil appeared (Fig. 4). Mantel tests showed that the first principal component of PCA for samples (F1) was positively correlated with depth of the mineral soil, and that this correlation remained positive and highly significant (P < 0.0002) when the effect of distance was eliminated (Table 1). However, the mean distance of plant species from the start of Transects 4, 5 and 6 varied to a weak (~1.5 m) but significant extent from 1995 to 2005 (Table 2): in average, there was a weak displacement of plant species along transects in the direction of the open environment, whatever their growth habit.

Discussion

Ecological determinants of plant assemblages

The ordination of vegetation samples by indirect gradient analysis (PCA) displayed the existence of two distinct communities within the low forest. Community I resembled, by its floristic composition although not by its physiognomony, the open-rock vegetation ("savane-roche") described by [START_REF] Sarthou | Dynamique de la végétation pionnière sur un inselberg de la Guyane Française[END_REF][START_REF] Sarthou | Plant communities on a granitic outcrop[END_REF], [START_REF] Sarthou | Epilithic plant communities on inselbergs in French Guiana[END_REF] and [START_REF] Sarthou | Shrub vegetation on tropical granitic inselbergs in French Guiana[END_REF] as a mosaic of epilithic herbaceous and shrub communities involved in a successional process [START_REF] Sarthou | Dynamique de la végétation pionnière sur un inselberg de la Guyane Française[END_REF][START_REF] Sarthou | Successional patterns on tropical inselbergs: a case study on the Nouragues inselberg (French Guiana)[END_REF]. According to observations by [START_REF] Sarthou | Dynamique de la végétation pionnière sur un inselberg de la Guyane Française[END_REF] shrub thickets dominated by Clusia minor and several Myrtaceae increase in height and density ("border thickets") as far as open-rock vegetation stems in the continuous canopy cover typical of the low forest. Therefore it seems that there is no clear-cut boundary between typical open-rock vegetation and the part of the low forest which we called Community I. As in open-rock vegetation this shrub community is established on accumulated organic matter (5 to 20-30 cm) without mineral soil. Humus forms resemble those described by [START_REF] Kounda-Kiki | Humus profiles under main vegetation types in a rock savanna (Nouragues inselberg, French Guiana)[END_REF][START_REF] Dray | On the number of principal components: a test of dimensionality based on measurements of similarity between matrices[END_REF] under open-rock vegetation. Our transect analysis suggests that species typical of the low forest according to [START_REF] Larpin | Les formations ligneuses sur un inselberg de Guyane Française: étude floristique, structurale et dynamique[END_REF][START_REF] Larpin | The low forest (Nouragues inselberg)[END_REF] pertain to Community II, with a continuum (T5) or a discontinuity (T4, T6) in the variation of the floristic composition, according to the distribution of the mineral soil. This points to the presence of plant species representative of open-rock vegetation within what is commonly viewed on the field as the low forest, as this has been already noted by [START_REF] Larpin | Les formations ligneuses sur un inselberg de Guyane Française: étude floristique, structurale et dynamique[END_REF][START_REF] Larpin | The low forest (Nouragues inselberg)[END_REF]. As ascertained from Figure 4, and according to [START_REF] Larpin | Les formations ligneuses sur un inselberg de Guyane Française: étude floristique, structurale et dynamique[END_REF], this phenomenon is currently occurring each time granite boulders outcrop and mineral soil is absent. Separate analyses, performed on each transect, did not reveal any further subdivision within the low forest community.

Soil depth and soil carbon were also considered to be the main determinants of community shifts on other tropical outcrops in Africa [START_REF] Parmentier | The inselberg flora of Atlantic Central Africa. I. Determinants of species assemblages[END_REF][START_REF] Parmentier | Grasslands and herbaceous fringes on inselbergs in Atlantic central Africa[END_REF] and Australia [START_REF] Withers | Overview of granite outcrops in Western Australia[END_REF]. Most detailed study on tropical inselberg ecotones was that of [START_REF] Parmentier | The inselberg flora of Atlantic Central Africa. I. Determinants of species assemblages[END_REF].

Using multivariate methods they showed on 27 inselbergs from Central Africa that at the local scale most pronounced changes in species and trait distribution occurred along the forest-open rock gradient, with a similar shift from prostrated to erected vegetation and corresponding life forms. However, their study, being not based on transects but rather on a wide array of dispersed samples, did not allow checking abrupt or continuous changes in floristic composition along ecotone gradients. Here we showed that contrasted situations may be observed on the same inselberg (Fig. 4): two transects (T4 and T6) showed an abrupt shift from Community I to Community II, while transect T5 showed a more continuous change in its floristic composition. This discrepancy cannot be explained by any variation in gradient length, given that in all cases soil conditions varied abruptly along transect lines (Fig. 4) and thus our results cannot exclude individualistic hypotheses of community structure [START_REF] Connell | Diversity in tropical rain forests and coral reefs[END_REF][START_REF] Vázquez | Altitudinal gradients in tropical forest composition, structure, and diversity in the Sierra de Manantlán[END_REF].

The development of a tall and species-rich community on deeper mineral soils can be explained, not only by more space for the vertical extension of tree root systems [START_REF] Poszwa | Do deep tree roots provide nutrients to the tropical rain forest?[END_REF], but also and more surely by the mineral soil acting as a sink for water flow and nutrients provided by mineral weathering and biological fixation [START_REF] Dojani | Nitrogen input by cyanobacterial biofilms of an inselberg into a tropical rain forest in French Guiana[END_REF]. This may explain, among other factors, the luxury of the typical low forest (in species as well as in individuals) compared to adjoining environments such as open-rock vegetation and tall-tree rainforest. Community II is the best representative of the species-rich typical low forest community which has been described by [START_REF] Larpin | Les formations ligneuses sur un inselberg de Guyane Française: étude floristique, structurale et dynamique[END_REF][START_REF] Larpin | The low forest (Nouragues inselberg)[END_REF] on the Nouragues inselberg and by other authors in other parts of the world (synthesis in [START_REF] Porembski | Inselbergs: biotic diversity of isolated rock outcrops in tropical and temperate regions[END_REF]. However, it seems that the distinction of a tree/shrub community, with strong floristic affinities with open-rock vegetation, has been overlooked in the absence of exhaustive sampling along large transect lines. Although epilithic, Community I, with its dense and closed canopy, seems to act as a buffer against destruction of the typical low forest (Community II) by storms and erosive events. This is reminiscent of the protection against hurricanes and wildfires which is offered by coniferous trees at tropical mountain tree-lines [START_REF] Martin | Tropical montane forest ecotones: climate gradients, natural disturbance, and vegetation zonation in the Cordillera Central, Dominican Republic[END_REF].

Stability of plant assemblages

The community composition did not change conspicuously in 10 years, pointing to stability of the forest fringe. However, the distance to the start of transects, always located in typical open-rock vegetation on granite, decreased to some extent for woody and to a lesser extent for non-woody plant species (Table 2).

Given the importance of soil for forest development, which has been registered in successional studies on terraces [START_REF] Viereck | An overview of the vegetation and soils of the floodplain ecosystems of the Tanana River, interior Alaska[END_REF], we suspect that the time scale chosen for our new census of vegetation was too short for discerning neatly any common trend of soil and vegetation development, which should rest on centuries if not millenaries [START_REF] Bremer | Inselbergs: geomorphology and geoecology[END_REF]. As a corollary, the absence of a mineral soil in parts of the inselberg covered by open-rock vegetation and in the distal part of the low forest can be considered as an environmental barrier for the forest community, thus explaining the contour of the forest ecosystem without any biological [START_REF] Houle | Seed availability and biotic interactions in granite outcrop plant communities[END_REF] nor climatic [START_REF] Sanz-Elorza | Changes in the high-mountain vegetation of the Central Iberian Peninsula as a probable sign of global warming[END_REF] interference. It can be reasonably thought that the passage from xeric inselberg vegetation to rain forest vegetation is a time-consuming successional process, including progressive and slow succession, as this has been shown to occur on tropical rocky outcrops [START_REF] Ibisch | Floristic, biogeographical, and vegetational aspects of Pre-Cambrian rock outcrops (inselbergs) in eastern Bolivia[END_REF]. Moderate disturbances such as wind-throws (single treefall gaps), which are often observed in the low forest [START_REF] Larpin | Les formations ligneuses sur un inselberg de Guyane Française: étude floristique, structurale et dynamique[END_REF][START_REF] Larpin | The low forest (Nouragues inselberg)[END_REF] and in the "savane roche" [START_REF] Sarthou | Dynamique de la végétation pionnière sur un inselberg de la Guyane Française[END_REF][START_REF] Sarthou | Plant communities on a granitic outcrop[END_REF] probably also contribute to explain local/temporary shifts in species richness, canopy height and community composition, such as those observed in our study (see Transect T4 in Figure 4). They may also contribute to explain the high species richness of the low forest according to the intermediate disturbance hypothesis [START_REF] Sheil | Disturbing hypotheses in tropical forests[END_REF]. The rate at which forest vegetation is encroaching on the studied inselberg (1.5 m over 10-yr) does not offer any opportunity that forest vegetation would totally replace openrock vegetation between two fire periods according to data by [START_REF] Charles-Dominique | Forest perturbations and biodiversity during the last ten thousand years in French Guiana[END_REF]. It may even be thought that present-day climate warming, and accompanying increased frequency of severe El Niño events in the Southern Oscillation Zone [START_REF] Malhi | Spatial patterns and recent trends in the climate of tropical rain forest regions[END_REF]) cause a reversion of the successional progress. Results of a parallel study showed that the recruitment of juveniles of woody species between 1995 and 2005 was severely affected by the severe El-Niño event which occurred in 1997 [START_REF] Fonty | A 10-year decrease in plant species richness on a neotropical inselberg: detrimental effects of global warming[END_REF]. In this study we observed a decrease of 24% for individuals and 22% for species over the 205 quadrats of the three transects (totalling 410 m 2 ).

About fire

The role of fire cannot be neglected, but only if we consider it at the scale of centuries rather than of years. Lightnings are frequently observed to occur, more especially in open-rock vegetation, where they have been suspected to play a significant role in the renewal of vegetation thickets [START_REF] Kounda-Kiki | Humus profiles and successional development in a rock savanna (Nouragues inselberg, French Guiana): a micromorphological approach infers fire as a disturbance event[END_REF][START_REF] Sarthou | Successional patterns on tropical inselbergs: a case study on the Nouragues inselberg (French Guiana)[END_REF], but their impact rarely extend over a few meters. This is in marked contrast with studies done on other inselbergs and tropical mountain ecotones where human influence (associated or not with pastoralism) and present-day climate warming increase the risk of uncontrolled severe wildfires which destroy the ecotone and make it more abrupt and far below its natural level [START_REF] Grau | Rainfall variability, fire and vegetation dynamics in neotropical montane ecosystems in north-western Argentina[END_REF][START_REF] Parmentier | The inselberg flora of Atlantic Central Africa. I. Determinants of species assemblages[END_REF][START_REF] Hemp | Climate change-driven forest fires marginalize the impact of ice cap wasting on Kilimanjaro[END_REF][START_REF] Martin | Tropical montane forest ecotones: climate gradients, natural disturbance, and vegetation zonation in the Cordillera Central, Dominican Republic[END_REF][START_REF] Sherman | Fire and vegetation dynamics in high elevation neotropical montane forests of Dominican Republic[END_REF]. When tropical ecotones are protected from fire, then the often-reported sharp transition from forest to open environments is replaced by a gradual change quite similar to that observed in our study site [START_REF] Allen | Drought-induced shift of a forest-woodland ecotone: rapid landscape response to climate variation[END_REF].

Conclusion

On the Nouragues inselberg, two plant communities, called Community I (wind-dispersed shrubs, herbs and suffrutex species) and Community II (animal-dispersed shrubs and trees, with numerous Myrtaceae) were shown to compose what is currently known as the "low forest", an ecotone between two strongly contrasted environments, open-rock vegetation ("savane roche") and tall-tree rainforest. The passage from one to the other community may be abrupt or progressive, according to cases, but is always related to soil development, Community I, which exhibit strong floristic affinities with open-rock vegetation, is established on a thick humus layer directly overlying the granitic substrate, while Community II displays a shallow mineral soil with a thin litter layer at the ground surface. Despite demographic accidents due to severe El Niño events, plant communities at the fringe of the Nouragues tropical inselberg are stable at shorttime both in composition and spatial distribution. In the absence of strong disturbances such as wildfire and further erosion, soil availability for roots could be interpreted as an environmental constraint to the successional development of forest vegetation. Soil development might thus act as an ecological barrier to forest encroachment, which could only be alleviated by erosion recovery, as otherwise demonstrated.

Figure legends Figure 1 .Figure 2 .Figure 4 .Figure 5 .Figure 6 .

 legends12456 Figure legends

  Fig. 1

  Fig. 6

Table 1 .

 1 Mantel correlation tests at quadrat scale with and without distance as co-variate. *** = P<0.0002

	Transect 4 Transect 5	Transect 6	Observed t value (2005 vs 1995)

Table 2 .

 2 Changes in the mean distance (m) of plant species from start of transect between 1995 and 2005. Between-year differences were tested by paired Student's t test. * = P<0.05; ** = P<0.01 List of species names and codes of trees and shrubs found in the three studied transects. Stars indicate taxa which were included in PCA and cluster analysis (present in at least 10 quadrats)

	Appendix 1.
	Species name	Code	Family	Raunkiaer's life forms	Fruit types	Dispersion modes	Seed size	Community
	Alibertia myrciifolia	Amy*	Rubiaceae	microphanerophyte	berry	zoochory	0.5-1 cm	II
	Antonia ovata	Aov	Loganiaceae	mesophanerophyte	capsule	anemochory	unknown (winged)
	Apocynaceae sp. 1	As1	Apocynaceae	unknown	unknown	unknown	unknown
	Asclepiadaceae sp.	Ass*	Asclepiadaceae	liana	follicle	anemochory	unknown	II
	Aspidosperma cruentum	Acr	Apocynaceae	megaphanerophyte	follicle	anemochory	>2 cm (winged)
	Aspidosperma marcgravianum Ama*	Apocynaceae	megaphanerophyte	follicle	anemochory	>2 cm (winged)	II
	Aspidosperma sp.	Asp*	Apocynaceae	mesophanerophyte	follicle	anemochory	>2 cm (winged)	II
	Bignoniaceae sp. 1	Bi1	Bignoniaceae	liana	capsule	anemochory	unknown
	Brosimum guianense	Bgu	Moraceae	megaphanerophyte	fleshy	endozoochory	0.5-1 cm
	Burseraceae sp. 1	Bu1	Burseraceae	mesophanerophyte	drupe	endozoochory	unknown
	Burseraceae sp. 2	Bu2	Burseraceae	mesophanerophyte	drupe	endozoochory	unknown
	Calyptranthes lepida	Cle*	Myrtaceae	mesophanerophyte	berry	zoochory	<0.5 cm 0.5-1 cm	II
	Casearia sp.	Cas	Flacourtiaceae	mesophanerophyte	capsule	zoochory	unknown
	Cassipourea guianensis	Cag*	Rhizophoraceae	mesophanerophyte	capsule	zoochory	0.5-1 cm	II
	Chrysobalanaceae sp. 1	Ch1	Chrysobalanaceae phanerophyte	drupe	endo/synzoochory	unknown
	Clusia minor	Cmi*	Clusiaceae	microphanerophyte	capsule	zoochory	<0.5 cm	I
	Clusia nemorosa	Cne*	Clusiaceae	microphanerophyte	capsule	zoochory	0.5-1 cm	I
	Coccoloba sp.	Csp*	Polygonaceae	liana	fleshy	zoochory/hydrochory	0.5-1 cm	II
	Croton tafelbergicus	Cta*	Euphorbiaceae	microphanerophyte	capsule	auto/barochory	<0.5 cm	I
	Croton sp. 2	Cs2	Euphorbiaceae	microphanerophyte	capsule	auto/barochory	<0.5 cm
	Cupania diphylla	Cdi	Sapindaceae	mesophanerophyte	capsule	endozoochory	0.5-1 cm
	Cybianthus guianensis	Cgu*	Myrsinaceae	microphanerophyte	drupe	zoochory	0.5-1 cm	II
	Daphnopsis granitica	Dgr*	Thymeleaceae	microphanerophyte	drupe	zoochory	0.5-1 cm	II
	Dileniaceae sp. 1	Di1	Dileniaceae	liana	unknown	zoochory	unknown (arilled)
	Duroia sp.	Dsp	Rubiaceae	phanerophyte	berry	zoochory	0.5-1 cm
	Eriotheca surinamensis	Esu*	Bombacaceae	microphanerophyte	capsule	anemochory	0.5-1 cm	II
	Ernestia granvillei	Egr*	Melastomataceae	nanophanerophyte	capsule-like	barochory or anemochory	<0.5 cm	I
	Erythroxylum citrifolium	Eci*	Erythroxylaceae	microphanerophyte	drupe	zoochory	0.5-1 cm	I
	Erythroxylum ligustrinum	Eli*	Erythroxylaceae	mesophanerophyte	drupe	zoochory	0.5-1 cm	II
	Erythroxylum squamatum	Esq*	Erythroxylaceae	mesophanerophyte	drupe	zoochory	0.5-1 cm	II
	Eugenia albicans	Eal	Myrtaceae	microphanerophyte	berry	zoochory	0.5-1 cm
	Eugenia florida	Efl*	Myrtaceae	microphanerophyte	berry	zoochory	0.5-1 cm 1-2 cm	I
	Eugenia marowynensis	Ema*	Myrtaceae	mesophanerophyte	berry	zoochory	1-2 cm	II
	Eugenia ramiflora	Era*	Myrtaceae	microphanerophyte	berry	zoochory	0.5-1 cm	II
	Eugenia sp. 1	Es1	Myrtaceae	mesophanerophyte	berry	zoochory	0.5-1 cm
	Eugenia sp. 2	Es2	Myrtaceae	microphanerophyte	berry	zoochory	0.5-1 cm
	Euplassa pinata	Epi	Proteaceae	mesophanerophyte	drupe	zoochory	1-2 cm
	Guapira eggersiana	Geg*	Nyctaginaceae	mesophanerophyte	fleshy	zoochory	0.5-1 cm	II
	Hebepetalum humiriifolium	Heb	Linaceae	mesophanerophyte	drupe	zoochory	0.5-1 cm
	Henriettea sp.	Hen	Melastomataceae	mesophanerophyte	berry	zoochory	<0.5 cm
	Heteropteris sp.	Hsp*	Malpighiaceae	liana	samara	anemochory	0.5-1 cm (winged)	II
	Himatanthus bracteatus	Hbr	Apocynaceae	mesophanerophyte	capsule	anemochory	>2 cm
	Hippocrateaceae sp. 1	Hi1	Hippocrateaceae	liana or microphanerophyte unknown	zoochory or anemochory	unknown
	Hirtella racemosa	Hra	Chrysobalanaceae mesophanerophyte	drupe	zoochory	0.5-1 cm 1-2 cm
	Humiria balsamifera	Hba*	Humiriaceae	mesophanerophyte	drupe	zoochory	0.5-1 cm 1-2 cm	I
	Inga lateriflora	Ila	Mimosaceae	mesophanerophyte	pod	endozoochory	1-2 cm
	Inga stipularis	Isi*	Mimosaceae	mesophanerophyte	pod	endozoochory	1-2 cm	II
	Inga umbellifera	Ium*	Mimosaceae	microphanerophyte	pod	endozoochory	1-2 cm	II
	Inga virgultosa	Ivi*	Mimosaceae	mesophanerophyte	pod	endozoochory	0.5-1 cm 1-2 cm	II
	Inga sp.	Isp	Mimosaceae	mesophanerophyte	pod	endozoochory	>2 cm
	Manilk ara bidentata	Mbi*	Sapotaceae	megaphanerophyte	berry	zoochory	>2 cm	II
	Maytenus myrsinoides	Mmy*	Celastraceae	mesophanerophyte	capsule	zoochory	1-2 cm	II
	Melastomataceae sp. 1	Me1*	Melastomataceae	phanerophyte	unknown	autochory or zoochory	<0.5 cm	II
	Melastomataceae sp. 2	Me2	Melastomataceae	phanerophyte	unknown	autochory or zoochory	<0.5 cm
	Miconia ciliata	Mci*	Melastomataceae	nanophanerophyte	berry	zoochory	<0.5 cm	I
	Miconia holosericea	Mho*	Melastomataceae	mesophanerophyte	berry	zoochory	0.5-1 cm	II
	Morinda sp.	Msp*	Rubiaceae	microphanerophyte	fleshy	zoochory	<0.5 cm 0.5-1 cm	II
	Myrcia citrifolia	Mct*	Myrtaceae	mesophanerophyte	berry	zoochory	0.5-1 cm	II
	Myrcia fallax	Mfa*	Myrtaceae	mesophanerophyte	berry	zoochory	0.5-1 cm	II
	Myrcia guianensis	Mgu*	Myrtaceae	microphanerophyte	berry	zoochory	0.5-1 cm	I
	Myrcia quitarensis	Mqu*	Myrtaceae	mesophanerophyte	berry	zoochory	0.5-1 cm	II
	Myrcia saxatilis	Msa*	Myrtaceae	microphanerophyte	berry	zoochory	0.5-1 cm	I
	Myrcia sylvatica	Msy*	Myrtaceae	microphanerophyte	berry	zoochory	0.5-1 cm	II
	Myrciaria floribunda	Mfl	Myrtaceae	mesophanerophyte	berry	zoochory	0.5-1 cm 1-2 cm
	Myrciaria sp. 1	Ms1*	Myrtaceae	phanerophyte	berry	zoochory	0.5-1 cm	II
	Myrciaria sp. 2	Ms2*	Myrtaceae	phanerophyte	berry	zoochory	0.5-1 cm	I
	Myrtaceae sp. 1	My1	Myrtaceae	phanerophyte	fleshy	zoochory	0.5-1 cm or 1-2 cm
	Myrtaceae sp. 3	My3	Myrtaceae	phanerophyte	fleshy	zoochory	0.5-1 cm or 1-2 cm
	Myrtaceae 28	M28*	Myrtaceae	phanerophyte	fleshy	zoochory	0.5-1 cm or 1-2 cm II
	Myrtaceae 40	M40	Myrtaceae	phanerophyte	fleshy	zoochory	0.5-1 cm or 1-2 cm
	Neea ovalifolia	Nov	Nyctaginaceae	mesophanerophyte	drupe-like	zoochory	unknown
	Ocotea sp.	Osp*	Lauraceae	microphanerophyte	berry	zoochory	0.5-1 cm 1-2 cm	II
	Ouratea candollei	Oca	Ochnaceae	mesophanerophyte	drupelet	zoochory	0.5-1 cm
	Ouratea leblondii	Ole*	Ochnaceae	microphanerophyte	drupelet	zoochory	0.5-1 cm	II
	Oxandra asbeck ii	Oas	Annonaceae	mesophanerophyte	fleshy	zoochory	1-2 cm
	Parinaria excelsa	Pex	Chrysobalanaceae megaphanerophyte	drupe	zoochory	>2 cm
	Park ia sp.	Psp	Mimosaceae	megaphanerophyte	pod	zoochory	1-2 cm
	Peltogyne paniculata	Ppa*	Caesalpiniaceae	megaphanerophyte	pod	zoochory	>2 cm	II
	Petrea volubilis	Pvo	Verbenaceae	liana	wing-like calyx lobes anemochory	0.5-1 cm
	Phyllanthus attenuatus	Pat	Euphorbiaceae	microphanerophyte	capsule	probable autochory	<0.5 cm
	Picramnia guianensis	Pgu	Simaroubaceae	microphanerophyte	berry	zoochory	0.5-1 cm 1-2 cm
	Piptocoma schomburgk ii	Psc*	Asteraceae	microphanerophyte	achene	anemochory	<0.5 cm	I
	Polygala spectabilis	Pos*	Polygalaceae	nanophanerophyte	capsule	anemochory or myrmechochory 0.5-1 cm	I
	Pourouma sp.	Pou	Cecropiaceae	mesophanerophyte	drupe-like	zoochory	0.5-1 cm
	Protium heptaphyllum	Phe*	Burseraceae	mesophanerophyte	drupe	zoochory	0.5-1 cm 1-2 cm	II
	Psychotria ctenophora	Pct*	Rubiaceae	microphanerophyte	berry	zoochory	0.5-1 cm	II
	Psychotria cupularis	Pcu*	Rubiaceae	microphanerophyte	drupe	zoochory	<0.5 cm 0.5-1 cm	II
	Psychotria hoffmannseggiana Pho*	Rubiaceae	nanophanerophyte	drupe	zoochory	<0.5 cm	II
	Psychotria moroidea	Pmo	Rubiaceae	microphanerophyte	drupe	zoochory	0.5-1 cm
	Rubiaceae sp. 1	Ru1	Rubiaceae	phanerophyte	unknown	zoochory or anemochory	unknown
	Rubiaceae sp. 2	Ru2	Rubiaceae	phanerophyte	unknown	zoochory or anemochory	unknown
	Rudgea crassiloba	Rcr*	Rubiaceae	microphanerophyte	drupe	zoochory	0.5-1 cm	II
	Sagotia racemosa	Sra*	Euphorbiaceae	mesophanerophyte	capsule	autochory	0.5-1 cm	II
	Sapium montanum	Smo	Euphorbiaceae	microphanerophyte	capsule	zoochory	<0.5 cm 0.5-1 cm
	Schefflera decaphylla	Sde*	Araliaceae	megaphanerophyte	drupe	zoochory	>2 cm (winged)	II
	Securidaca uniflora	Sun	Polygalaceae	liana	samara	anemochory	0.5-1 cm (winged)
	Smilax sp.	Ssp	Smilacaceae	liana	berry	zoochory	0.5-1 cm
	Souroubea guianensis	Sgu	Marcgraviaceae	liana	berry	zoochory	<0.5 cm
	Tabebuia capitata	Tca*	Bignoniaceae	microphanerophyte	capsule	anemochory	>2 cm	I
	Tapirira guianensis	Tgu*	Anacardiaceae	mesophanerophyte	drupe	zoochory	0.5-1 cm	II
	Terminalia amazonia	Tam*	Combretaceae	mesophanerophyte	drupe	anemochory	0.5-1 cm (winged)	II
	Ternstroemia dentata	Tde*	Theaceae	mesophanerophyte	berry	zoochory	1-2 cm	II
	Thyrsodium guianense	Thg	Anacardiaceae	mesophanerophyte	drupe	endozoochory	1-2 cm
	Zygia tetragona	Zte*	Mimosaceae	mesophanerophyte	pod	endozoochory	1-2 cm	II
	Undetermined 1	Un1	unknown	phanerophyte	unknown	unknown	unknown
	Undetermined 2	Un2	unknown	phanerophyte	unknown	unknown	unknown
	Undetermined 3	Un3	unknown	phanerophyte	unknown	unknown	unknown
	Undetermined 4	Un4	unknown	phanerophyte	unknown	unknown	unknown
	Undetermined 5	Un5	unknown	unknown	unknown	unknown	unknown

life forms Fruit types Dispersion modes Seed size Community

	Aechmea melinonii	Ame	Bromeliaceae	hemicryptophyte	berry	zoochory	<0.5 cm 0.5-1 cm
	Aganisia pulchella	Apu*	Orchidaceae	hemicryptophyte	capsule	anemochory	<0.5 cm	I
	Anthurium jenmanii	Aje*	Araceae	hemicryptophyte	berry	zoochory	<0.5 cm	I
	Bromelia sp. 1	Bs1*	Bromeliaceae	hemicryptophyte	berry	zoochory	<0.5 cm 0.5-1 cm	II
	Calathea squarrosa	Csq*	Marantaceae	geophyte	berry	zoochory or myrmechory 0.5-1 cm 1-2 cm	I
	Chamaecrista desvauxii	Cde	Fabaceae	chamaephyte	pod	anemochory	<0.5 cm
	Chelonanthus purpurascens	Cpu*	Gentianaceae	hemicryptophyte	capsule	anemochory	<0.5 cm	I
	Cleistes rosea	Cro	Orchidaceae	therophyte	capsule	anemochory	<0.5 cm
	Cuphea black ii	Cbl*	Lythraceae	chamaephyte	capsule	anemochory	<0.5 cm	I
	Cyperaceae sp. 1	Cy1	Cyperaceae	hemicryptophyte	achene	autochory or anemochory <0.5 cm
	Disteganthus lateralis	Dla*	Bromeliaceae	hemicryptophyte	berry	zoochory	0.5-1 cm	II
	Elleanthus brasiliensis	Ebr	Orchidaceae	hemicryptophyte	capsule	anemochory	<0.5 cm
	Encyclia ionosma	Eio	Orchidaceae	hemicryptophyte	capsule	anemochory	<0.5 cm
	Episcia sphalera	Esp	Gesneriaceae	hemicryptophyte	capsule	autochory	<0.5 cm
	Guzmania lingulata	Gli*	Bromeliaceae	hemicryptophyte	capsule	anemochory	<0.5 cm	I
	Ichnanthus nemoralis	Ine	Poaceae	hemicryptophyte	caryopsis	anemochory	<0.5 cm
	Jessenia bataua	Jba	Arecaceae	microphanerophyte	drupe	barochory or zoochory	>2 cm
	Lindsaea sp.	Lsp	Dennstaedtiaceae hemicryptophyte	sporangium	anemochory	<0.5 cm
	Ludovia lancifolia	Lla*	Cyclanthaceae	hemicryptophyte	berry	zoochory	<0.5 cm	I
	Macrocentrum cristatum	Mcr*	Melastomataceae chamaephyte	capsule	anemochory	<0.5 cm	I
	Olyra obliquifolia	Oob*	Poaceae	hemicryptophyte	caryopsis	anemochory	<0.5 cm	II
	Paradrymonia campostyla	Pac	Gesneriaceae	liana	capsule	autochory	<0.5 cm
	Pariana campestris	Pca*	Poaceae	hemicryptophyte	caryopsis	anemochory	<0.5 cm	II
	Phramipedium lindleyanum	Phl	Orchidaceae	hemicryptophyte	capsule	anemochory	<0.5 cm
	Pitcairnia geysk esii	Pge*	Bromeliaceae	hemicryptophyte	capsule	anemochory	<0.5 cm (winged)	I
	Poaceae sp. 1	Po1*	Poaceae	hemicryptophyte	caryopsis	anemochory	<0.5 cm	II
	Poaceae sp. 2	Po2	Poaceae	hemicryptophyte	caryopsis	anemochory	<0.5 cm
	Poaceae sp. 3	Po3	Poaceae	hemicryptophyte	caryopsis	anemochory	<0.5 cm
	Poaceae sp. 4	Po4*	Poaceae	hemicryptophyte	caryopsis	anemochory	<0.5 cm	I
	Poaceae sp. 5	Po5	Poaceae	hemicryptophyte	caryopsis	anemochory	<0.5 cm
	Sauvagesia aliciae	Sal*	Ochnaceae	chamaephyte	capsule	anemochory	<0.5 cm	I
	Scleria cyperina	Scy*	Cyperaceae	hemicryptophyte	achene	anemochory	<0.5 cm	I
	Scleria secans	Sse*	Cyperaceae	liana	achene	anemochory	<0.5 cm	II
	Selaginella sp.	Ssp*	Selaginellaceae	hemipcryptophyte	sporangium	anemochory	<0.5 cm	I
	Stelestylis surinamensis	Ssu*	Cyclanthaceae	hemicryptophyte	berry	zoochory	<0.5 cm	I
	Stylosanthes guianensis	Sgu	Fabaceae	chamaephyte	pod	anemochory	<0.5 cm
	Syagrus stratincola	Sst*	Arecaceae	micro-mesophanerophyte drupe	zoochory	>2 cm	I
	Vanilla ovata	Vov	Orchidaceae	liana	capsule	anemochory	<0.5 cm
	Vriesea gladioliflora	Vgl*	Bromeliaceae	hemicryptophyte	capsule	anemochory	<0.5 cm 0.5-1 cm (plumose) I
	Vriesea pleiostica	Vpl	Bromeliaceae	hemicryptophyte	capsule	anemochory	<0.5 cm 0.5-1 cm (plumose)
	Vriesea splendens	Vsp*	Bromeliaceae	hemicryptophyte	capsule	anemochory	<0.5 cm 0.5-1 cm (plumose) II

components of PCA

Acknowledgements

We want to acknowledge the staff of the Nouragues Research Station (CNRS UPS 656, dir. P.

Charles-Dominique) for accommodation and technical help.


Srinath Govindaswamy

Thierry Floquet
email: thierry.floquet@ec-lille.fr

Sarah K Spurgeon
email: s.k.spurgeon@kent.ac.uk

Linear Matrix Inequality based static output feedback sliding mode control for discrete time systems

This paper discusses the design of a static output feedback based sliding mode controller for MIMO nonminimum phase discrete time systems. In previous work, it was shown that by identifying a minimal set of current and past outputs to determine an extended output signal, an augmented system can be obtained which permits the design of a sliding manifold based upon output information only, which renders the sliding mode dynamics stable. In this paper, it is shown that if the extended outputs chosen span the state zero directions of an invariant zero of the system, then the invariant zero disappears from the augmented system. Linear matrix inequalities are then used for sliding surface design. Such controllers have great potential within reconfigurable control schemes for application in the presence of multiple sensor failures. The methodology is applied to an aircraft system to show the efficacy of the design philosophy.

I. INTRODUCTION

In continuous time, a sliding mode is generated by means of discontinuities in the control signals about a surface in the state space [START_REF] Utkin | Sliding Modes in Control Optimization[END_REF]. It is required that the discontinuity surface, usually called the sliding surface, is attained from any initial condition in a finite time interval. For an appropriately selected controller, the motion on the surface, or sliding mode, is completely insensitive to any matched uncertainty in the system [START_REF] Utkin | Sliding Modes in Control Optimization[END_REF], [START_REF] Edwards | Sliding Mode Control: theory and applications[END_REF]. In a discrete control implementation, the control signal is held constant during the sample period and hence it is not possible, in general, to attain a sliding mode which would require the control to switch at infinite frequency. As a result, the invariance properties of continuous time sliding-mode control can be lost. The obvious solution of sampling at high frequency, which will closely approximate continuous time, may not be possible for given hardware specifications. This has led to interest in the area of discrete time sliding-mode control (DSMC). For the case of uncertain discrete systems, it is not possible to ensure the states remain on a surface within the state space and for this reason much of the early DSMC literature focused on establishing a discrete time counterpart to the (continuous time) reachability condition [START_REF] Bartoszewicz | Comments on discrete-time variable structure control systems[END_REF], [START_REF] Gao | Discrete-time variable structure control systems[END_REF], [START_REF] Sapturk | On the stability of discrete-time sliding mode control systems[END_REF]. A comprehensive overview of these early developments is given in [START_REF] Milosavljevic | Discrete-time VSS[END_REF]. One distinctive feature is that DSMC does not necessarily require the use of a discontinuous control strategy [START_REF] Spurgeon | Hyperplane design techniques for discrete-time variable structure control systems[END_REF]. The results presented in [START_REF] Hui | On discrete-time variable structure sliding mode control[END_REF], [START_REF] Spurgeon | Hyperplane design techniques for discrete-time variable structure control systems[END_REF] show that an appropriate choice of sliding surface, used with the equivalent control required to ensure sliding, can guarantee a bounded motion about the surface in the presence of bounded matched uncertainty and that the use of a relay/switch in the control law is detrimental to performance.

Early contributions in sliding mode control were developed in a framework in which all the system states are available. This may not be realistic for practical engineering problems and has motivated the need for output feedback controllers. A number of algorithms have been developed for robust stabilization of uncertain systems which are based on sliding surfaces and output feedback control schemes [START_REF] Zak | On variable structure output feedback controllers for uncertain dynamic systems[END_REF], [START_REF] Edwards | Sliding mode stabilisation of uncertain systems using only output information[END_REF]. In [START_REF] Zak | On variable structure output feedback controllers for uncertain dynamic systems[END_REF], a geometric condition is developed to guarantee the existence of the sliding surface and the stability of the reduced order sliding motion. Edwards and Spurgeon derived an algorithm [START_REF] Edwards | Sliding mode stabilisation of uncertain systems using only output information[END_REF], [START_REF] Edwards | Sliding Mode Control: theory and applications[END_REF] which is convenient for practical use. Based on the work in [START_REF] Zak | On variable structure output feedback controllers for uncertain dynamic systems[END_REF], some dynamic feedback sliding mode controllers have been proposed [START_REF] Kwan | Further results on variable output feedback controllers[END_REF], [START_REF] Shyu | A dynamic output feedback controller for mismatched uncertain variable systems[END_REF]. In all the above output feedback sliding mode control schemes, it is an a priori requirement that the system under consideration is minimum phase and relative degree one and that a particular sub-system must be output feedback stabilisable [START_REF] Edwards | Sliding Mode Control: theory and applications[END_REF].

Compared with continuous time sliding-mode strategies, the design problem in discrete time is much less mature. Other than early work in [START_REF] Sira-Ramirez | Nonlinear discrete variable structure systems in quasi-sliding mode[END_REF], much of the literature assumes all states are available [START_REF] Furuta | Variable structure control with sliding sector[END_REF], [START_REF] Hui | On discrete-time variable structure sliding mode control[END_REF], [START_REF] Tang | Sliding surface design for discrete VSS using LQR technique with a preset eigenvalue[END_REF]. Discrete sliding mode control schemes which have restricted themselves to output measurements alone have often been observer based schemes with or without disturbance estimation [START_REF] Lee | A discrete-time sliding mode controller and observer with computation delay[END_REF], [START_REF] Tang | Discrete variable structure control for linear multivariable systems[END_REF]. Recent exceptions have been the work in [START_REF] Monsees | Discrete-time sliding mode control[END_REF] which considers both static and dynamic output feedback problems, and the discrete time versions of certain higher-order sliding-mode control schemes [START_REF] Bartolini | Digital second-order sliding mode control for uncertain nonlinear systems[END_REF], [START_REF] Bartolini | Digital sliding mode control with O(T 3 ) accuracy[END_REF].

For continuous time systems, it was shown in [START_REF] Floquet | An output feedback sliding mode control strategy for MIMO systems of arbitrary relative degree[END_REF] that the relative degree condition associated with the solution of the existence problem can be weakened if a classical sliding mode observer is combined with sliding mode exact differentiators to generate additional independent output signals from the available measurements. In [START_REF] Govindaswamy | On output sampling based sliding mode control for discrete time systems[END_REF], it was shown that by using the output signal at the current time instant together with a limited amount of information from previous sample instants, the class of systems for which an output feedback based sliding mode controller can be developed is significantly broadened. It is shown here that if the extended output spans the state zero directions of an invariant zero of the original plant, the invariant zero disappears from the augmented system. The design of the sliding surface is formulated as an LMI problem. It is shown that by extending the available outputs, the methodology can be applied to cases where systems experience multiple sensor failures and require reconfigurable control using the remaining available outputs.

The paper is structured as follows. Section 2 presents the problem motivation. The existence problem is considered in Section 3 and a control law to ensure the sliding mode is attained is given in Section 4. A motivational reconfigurable control example, from the class of discrete systems which could not previously be stabilised by output feedback sliding mode control, is presented to illustrate the approach.

II. MOTIVATION

Consider the discrete, linear, time invariant state space system representation given below:

x k+1 = Ax k + Bu k ( 1
)
y k = [(y 1 ) k ...(y p ) k ] T = Cx k , (y i ) k = C i x k (2)
where x k ∈ R n is the state vector, y k ∈ R p is the output vector and u k ∈ R m is the control input. It is assumed that m ≤ p, the pair (A, B) is controllable and without loss of generality, that rank(C) = p and that column rank(B) = m.

Consider the development of a control law based on output measurements only which will induce an ideal sliding motion on the surface

s = {x ∈ R n : FCx k = 0} (3)
for some selected matrix F ∈ R m×p . It is well known that for a unique equivalent control to exist, the matrix FCB ∈ R m×m must have full rank. As rank(FCB) ≤ min{rank(F), rank(CB)} (4) it follows that both F and CB must have full rank. As F is a design parameter, it can be chosen to be full rank.

A necessary condition for FCB to be full rank, and thus for solvability of the output feedback sliding mode design problem, thus becomes that CB must have rank m. If this rank condition holds and any invariant zeros of the triple {A, B,C} lie in the unit disk, then the existence of a matrix F defining the surface (3), which provides a stable sliding motion with a unique equivalent control is determined from the stabilizability by output feedback of a specific, welldefined subsystem of the plant [START_REF] Edwards | Sliding Mode Control: theory and applications[END_REF].

III. THE EXISTENCE PROBLEM

It will be assumed in this work that the matrix A is invertible. Such a property often occurs in discrete-time linear systems. For instance, consider the continuous time linear time invariant system ż(t) = Fz(t) + Gu(t)

(5)

If the system (1)-( 2) is the discretized form of the continuous time system (5) under sampling i.e,

A = e (Ft) , B = T 0 e Ft dτG ( 6
)
then it is shown in [START_REF] Rugh | Linear Systems Theory[END_REF] (page 386) that the state space matrix A is invertible. Thus, the system (2) can be rewritten as:

x k = A -1 (x k+1 -Bu k ) (7) y k = Cx k (8)
A. Generation of the extended output Consider system (1)-( 2) without any a priori assumptions relating to either the stability of the invariant zeros or the fulfillment of the matching condition. The main idea here is to construct a matrix C:

C =              C 1 . . . C 1 A -µ 1 +1 . . . C p . . . C p A -µ p +1              (9)
such that C is full rank, rank( CB) = rank(B), and any invariant zeros of the triple {A, B, C} lie inside the unit disk. Also, the µ i are chosen such that p = ∑ p i=1 µ i is minimal. Note that C = C means that the original system is output feedback stabilizable using existing methods (see e.g. [START_REF] Edwards | Sliding Mode Control: theory and applications[END_REF]).

B. Zeros and Zero directions

Consider the system in ( 1)-( 2

) with x k ∈ R n , u k ∈ R m , y k ∈ R p ,
where A, B,C are all real matrices with appropriate dimensions. Then, the number z is defined as the invariant zero of (1)-( 2) iff there exists vectors 0 = x z ∈ C n and u z ∈ C m , such that the triple z, x z , u z satisfies zI -A B C 0

x z u z = 0 (10)
where, I is the identity matrix of dimension n. Here, x z is called the state zero direction and u z is called the input zero direction. As seen in [START_REF] Govindaswamy | On output sampling based sliding mode control for discrete time systems[END_REF], it can be shown that any invariant zeros of the triple {A, B, C} are amongst the invariant zeros of the triple {A, B,C}. It can also be shown that if an appropriate choice of extended outputs is available, the invariant zeros from the original triple (A, B,C) may disappear from the augmented system. This is particularly useful if any of the invariant zeros of (A, B,C) are unstable. The choice of augmented outputs to remove transmission zeros is explained in the lemma below.

Lemma 1

Let z be the invariant zero of the original triple (A, B,C) with zero direction x z . Assume the zero direction x T z is spanned by the rows of the augmented output matrix C. Then the invariant zero z is not present in the augmented triple (A, B, C). If z is an invariant zero of the original triple with zero direction x z , it follows that Cx z = 0 and thus x z is in the null space of C.

By construction of the augmented output, x z belongs to the range space of C. The additional rows of C are constructed by post multiplying particular rows of C by a full rank matrix of the form A -l . It follows that the augmented rows of C lie in the range of C and thus Cx z cannot be identically zero as x z lies in the null space of C by definition. Thus, z is not an invariant zero of (A, B, C).

Hence by suitably selecting the augmented outputs such that they span the state zero direction of an unstable invariant zero, it is possible to remove unstable invariant zeros from the augmented system.

C. Design of the sliding manifold and analysis of the equivalent dynamics

Assume that it is possible to construct a matrix C as defined in [START_REF] Floquet | An output feedback sliding mode control strategy for MIMO systems of arbitrary relative degree[END_REF]. Then, the requirements for solvability of the output feedback sliding mode design problem are determined by the triple {A, B, C}. It remains to find a suitable sliding variable that depends on the available measurements only. Extend the original outputs as shown below:

ỹk =              (y 1 ) k . . . (y 1 ) k-µ 1 +1 . . . (y p ) k . . . (y p ) k-µ p +1             
From the system (7)-( 8), it can be computed that:

y k = Cx k y k-1 = CA -1 (x k -Bu k-1 ) y k-2 = CA -2 x k -CA -2 Bu k-1 -CA -1 Bu k-2
. . .

y k-j = CA -j x k -CA -j Bu k-1 -• • • -CA -1 Bu k-j = CA -j x k - j ∑ l=1 CA -l Bu k-j+l-1
Thus, the extended output matrix ỹ can be expressed as follows:

ỹk = Cx k -M     ū1 ū2 . . . ūp     (11)
with, for i = 1, ..., p:

ūi =      u k-1 u k-2 . . . u k-µ i +1      M = diag M 1 , ..., M p M i =            0 0 . . . . . . 0 C i A -1 B 0 . . . . . . 0 C i A -2 B C i A -1 B . . . . . . . . . . . . 0 . . . . . . C i A -1 B 0 CA -µ i +1 B . . . . . . C i A -2 B C i A -1 B           
where 1) . If the outputs are extended such that rank( C) is n (i.e when C is invertible), it can be seen from ( 11) that it is possible to write the state as a function of ỹ (the current and the past outputs) and u i (the past inputs). In this case the transmission zeros can always be removed from the augmented system and the controller design is performed as a state feedback design using the estimates of the states. The following class of sliding manifolds, that only depend on known variables, can then be defined

M i ∈ R µ i ×(µ i -
s k = F ỹk + FM     ū1 ū2 . . . ūp     (12)
= F Cx k [START_REF] Hui | On discrete-time variable structure sliding mode control[END_REF] where F ∈ m× p is a design parameter.

To analyse the stability of the resulting sliding motion, it is now convenient to introduce a coordinate transformation to the usual regular form, making the final p states of the system depend directly on the extended outputs [START_REF] Edwards | Sliding Mode Control: theory and applications[END_REF]:

A = A 11 A 12 A 21 A 22 B = 0 B 2 C = 0 T (14) x = x1 x2
where T ∈ R p× p is an orthogonal matrix, A 11 ∈ R (n-m)×(n-m) and the remaining sub-blocks in the system matrix are partitioned accordingly. The corresponding switching surface parameter is given by

p-m ↔ m ↔ F 1 F 2 = FT ( 15
)
where T is the matrix from equation ( 14). As a result

F C = F 1 C f F 2 (16)
where

C f = 0 ( p-m)×(n-p) I ( p-m) (17)
Therefore F CB = F 2 B 2 and the square matrix F 2 is nonsingular. The canonical form in ( 14) can be viewed as a special case of the regular form normally used in sliding mode controller design, and thus the reduced-order sliding motion is governed by a free motion with system matrix

A s 11 = A 11 -A 12 F -1 2 F 1 C f ( 18
)
which must therefore be stable.

If K ∈ R m×( p-m) is defined as K = F -1 2 F 1 then A s 11 = A 11 -A 12 KC f ( 19
)
and the problem of hyperplane design is equivalent to a static output feedback problem for the system (A 11 , A 12 ,C f ). In order to utilize the existing literature it is necessary that the pair (A 11 , A 12 ) is controllable and (A 11 ,C f ) is observable. The former is ensured as (A, B) is controllable. The observability of (A 11 ,C f ), is not so straightforward, but can be investigated by considering the canonical form below.

Lemma 2

Let (A, B, C) be a linear system with p > m and rank ( CB) = m. Then a change of coordinates exists so that the system triple with respect to the new coordinates has the following structure:

• The system matrix can be written as

A = A 11 A 12 A 21 A 22
where A 11 ∈ R (n-m)×(n-m) and the sub-block A 11 when partitioned has the structure r) for some r ≥ 0 and the pair (A o 22 , A o 21 ) is completely observable.

A 11 =   A o 11 A o 12 0 A o 22 A m 12 0 A o 21 A m 22   where A o 11 ∈ R r×r , A o 22 ∈ R (n-p-r)×(n-p-r) and A o 21 ∈ R ( p-m)×(n-p-
• The input distribution matrix B and the output distribution matrix C have the structure in [START_REF] Janardhanan | Multirate Output Feedback based Robust Quasi-Sliding Mode Control of Discrete time systems[END_REF].

For a proof and a constructive algorithm to obtain this canonical form see [START_REF] Edwards | Sliding mode stabilisation of uncertain systems using only output information[END_REF].

In the case where r > 0, the intention is to construct a new system ( Ã11 , B1 , Cf ) which is both controllable and observable with the property that

λ (A s 11 ) = λ (A o 11 ) ∪ λ ( Ã11 -B1 K Cf ).
To this end, as in [START_REF] Edwards | Sliding mode stabilisation of uncertain systems using only output information[END_REF], partition the matrices A 12 and A m 12 as

A 12 = A 121 A 122 and A m 12 = A m 121 A m 122
where A 122 ∈ R (n-m-r)×m and A m 122 ∈ R (n-p-r)×( p-m) and form a new sub-system represented by the triple ( Ã11 , A 122 , Cf ) where

Ã11 = A o 22 A m 122 A o 21 A m 22 Cf = 0 ( p-m)×(n-p-r) I ( p-m) ( 20
)
It follows that the spectrum of A s 11 decomposes as

λ (A 11 -A 12 KC f ) = λ (A o 11 ) ∪ λ ( Ã11 -A 122 K Cf )
Lemma 3 [START_REF] Edwards | Sliding mode stabilisation of uncertain systems using only output information[END_REF] The spectrum of A o 11 represents the invariant zeros of (A, B, C).

It follows directly that for a stable sliding motion, the invariant zeros of the system (A, B, C) must lie inside the unit disk and the triple ( Ã11 , A 122 , Cf ) must be stabilisable with respect to output feedback.

The matrix A 122 is not necessarily full rank. Suppose rank(A 122 ) = m then, as in [START_REF] Edwards | Sliding mode stabilisation of uncertain systems using only output information[END_REF], it is possible to construct a matrix of elementary column operations T m ∈ R m×m such that

A 122 T m = B1 0 (21)
where B1 ∈ R (n-m-r)×m and is of full rank. If K m = T -1 m K and K m is partitioned compatibly as To design the sliding surface consider the method for design of a static output feedback gain as given in [START_REF] Mehdi | Static output feedback design for uncertain linear systems[END_REF]. The technique in [START_REF] Mehdi | Static output feedback design for uncertain linear systems[END_REF] introduces slack variables to decouple the Lyapunov matrix and the static output feedback gain. With the additional slack variables and a chosen state space variable, an LMI problem is solved to obtain the static output feedback controller. The following theorem is required to formulate the design of the sliding surface using this method.

K m = K 1 K 2 m m-m then Ã11 -A 122 K Cf = Ã11 -B1 0 K m Cf = Ã11 -B1 K 1 Cf

Theorem 1

A static output feedback gain K is stabilizing if and only if there exists a positive definite matrix P = P T > 0 in R n×n , non singular matrices G ∈ R m×m and E 4 ∈ R n×n , a non-null matrices E 1 ∈ R n×n and L ∈ R m×p and arbitrary matrices E 2 ∈ R n×n , E 3 ∈ R m×n such that the following LMI:

    E 1 A o + A T o E T 1 -P * E 2 A o -P E 3 A o + A T 122 E T 1 + (L Cf -GK o ) A T 122 E T 2 E 4 A o -E T 1 P -E T 2 * * * * E 3 A 122 + A T 122 E T 3 -(G + G T ) * E 4 A 122 -E T 3 -E 4 -E T 4    < 0 (22)
is feasible for a given state feedback gain K o that stabilizes the pair ( Ã11 , A 122 ) with the static output feedback gain given by

K = G -1 L (23)
The matrix A o here is defined as

A o = Ã11 + A 122 K o .
For a conclusive proof for the above theorem refer to [START_REF] Mehdi | Static output feedback design for uncertain linear systems[END_REF].

IV. REACHABILITY PROBLEM It is necessary to force s(k) to zero to ensure the sliding dynamics are reached. Here the control selected has two components and is of the form:

u k = K y ỹk -(Φ -A 22 )s k (24)
Consider

T s = I n-m 0 F 1 C f F 2 (25)
with the sub matrices of T s conformably partitioned. The matrix T s is nonsingular since by construction F 2 is nonsingular. Let

(x 1 ) k s k = T s (x 1 ) k (x 2 ) k ( 26
)
Then with the new coordinates the nominal system can be written as:

(x 1 ) k+1 = Ā11 (x 1 ) k + Ā12 s k (27) s k+1 = F 2 Ā21 (x 1 ) k + F 2 Ā22 F -1 2 s k + Λu k with Ā11 = A 11 -A 12 M , Ā12 = A 12 F -1 2 , Ā21 = M Ā11 + A 21 -A 22 M and Ā22 = M Ā12 + A 22 , with M = F -1 2 F 1 C f and Λ = F 2 B 2 .
The first component of ( 24) is developed for stabilising the system [START_REF] Tang | Discrete variable structure control for linear multivariable systems[END_REF] and is calculated by using the algorithm given in [START_REF] Mehdi | Static output feedback design for uncertain linear systems[END_REF]. The second component of the control law is used for shaping the reaching response of the plant and is a design choice. The control law chosen is similar to the linear control law given in [START_REF] Monsees | Discrete-time sliding mode control[END_REF]. The control law here does not use any dynamics in the sense that there is no explicit estimation of states or an explicit compensator to achieve control. Thus the design can be considered as a static output feedback control law achieved with the available signals.

V. MOTIVATIONAL EXAMPLE Consider the example of an aircraft from [START_REF] Edwards | Sliding Mode Control: theory and applications[END_REF] given below. The plant model represents the fifth-order lateral dynamics of an L-1011 fixed wing aircraft, with the actuator dynamics removed. The plant system triple is given by:

A =      0 0 1 0 0 0 -.154 -.0042 1.54 0 0
.249 -1.000 -5.20 0 0.0386 -0.996 -0.0003 -.117 0 0 0.500 0 0 -0.5

     B =      0 0 -0.744 -0.032 0.337 -1.1200 0.02 0 0 0      C =    0 1 0 0 -1 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0   
The plant states are: The sampling rate chosen for discretizing the plant was t = .1s. Assume that the aircraft has lost the side slip angle and roll rate.The invariant zeros for the resulting plant triple (A, B,C) are at [1.00 .9929 -.9671] and thus a stable sliding surface cannot be designed using the remaining two outputs. By augmenting the bank angle and the washed out yaw rate once, it is found that the invariant zeros are removed from the system with the augmented outputs. The augmented matrix C in this case is given as Here Φ is chosen as .95I m , where I m is the identity matrix of dimension m × m. The simulations were performed with the control law given in [START_REF] Shyu | A dynamic output feedback controller for mismatched uncertain variable systems[END_REF] and with the initial conditions [0 .01 0 0 0]. The simulation results are shown below. Recall that this control law is performing system reconfiguration following the loss of two sensors. The key design requirement is that the system response remain stable given only measurements from the remaining sensors. In such circumstances one would not expect to be able to match the performance achievable from the complete set of measurements.

x =      φ bank
C =    0

VI. CONCLUSIONS

A static output feedback sliding mode-type control has been developed for discrete systems which employs information from both the previous and current output measurements. It has been shown that the invariant zeros of the original system can be removed from an augmented system if the minimal set of current and past outputs used to determine the extended signal span the state input direction associated with the invariant zero. The sliding surface design was performed using Linear Matrix Inequalities. It has been shown that the theory is applicable for systems in the presence of sensor failure by considering a particular aircraft problem. It is hypothesized that the methodology presented in this paper provides a useful core for developing reconfigurable control schemes for discrete time implementation.

Fig. 1 .

 1 Fig. 1. Response of the statesx k , Control Input u k , sliding surface s k , and augmented plant output ỹk

  The initial state feedback gain K o is obtained by pole placement and the poles are placed at [ .85 .975 .9]. Using the algorithm given in[START_REF] Mehdi | Static output feedback design for uncertain linear systems[END_REF], the gain K for the reduced order subsystem ( Ã11 , A 122 , Cf ) is calculated as

					1.0000	0	0	-1.0000	
	0.0003 1.0593 0.0004 1.0000 0 0	-0.1595 -1.0513 0 0	 
	1.0000 0.0004 -0.1052 -0.0270	0
	Here the orthogonal transformation matrix T is
			-0.5345 0.4990	-0.0132 -0.6820	
	T =	 	0.4980 0.4973	-0.4661 -0.0185 -0.7310 0.5349 -0.6829 0.0149	 
			0.4680	0.4975	0.7302	-0.0169
	and the subsystem ( Ã11 , A 122 , Cf ) obtained for the sliding
	surface design is
		Ã11 =	 	0.9933 0.0056	-0.0328 0.0307 1.1861 -0.1149	 
					-0.0053 0.2090	0.8684
	A 122 =	 	0.0050 -1.2966 0.0211 -0.0414 -1.3871 0.0344	  Cf =	0 1 0 0 0 1
					K =	0.2838 -0.3432 -6.8063 0.2682
	with the λ ( Ã11 -A 122 K Cf ) at [0.9849 + 0.0281i 0.9849 -
	0.0281i 0.0969]. The output feedback gain K y is obtained by
	first obtaining a state feedback gain by placing the poles at
	[.98 .975 .96 .985 .95]. Using the algorithm in [18] modified
	for the system given in (27), the static output feedback gain
	is calculated as:
	K y =	-5.2236 7.4514	10.0550 -15.1651 2.1197 1.2640 -1.7262 0.0532