Xiaoxia Deng

Lianfu Chen

Enwei Tian

Dayong Zhang

Tanming Wattana

Hui Yu

Finn Kjellberg

Simon T Segar

Xiaoxia Deng

Chen Lianfu

Tian Enwei

Zhang Dayong

Y U Hui

Kjellberg Finn

Low host specificity and broad geographical ranges in a community of parasitic non-pollinating fig wasps (Sycoryctinae; Chalcidoidea)

Keywords: Beta-diversity, community assembly, parasitoid, population genetics, mutualism

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

Introduction

Herbivorous insects and their parasitoids make up over 50% of all described insect species [START_REF] Price | Resource-driven terrestrial interaction webs[END_REF]. In addition to being of high economic relevance both trophic levels contribute 50 substantially to biodiversity and ecosystem function (B. A. [START_REF] Hawkins | Predators, parasitoids, and pathogens as mortality agents in phytophagous insect populations[END_REF]. Herbivores and parasitoids are also model systems for studying community assembly and coevolution. One of the most fundamental questions in ecology is how multitrophic communities and networks assemble across time and space. Under the expectation of classical cospeciation, entire communities might codiversify such that interactions are 55 inherited across speciation events resulting in predictable and replicated community structure.

Widespread evidence suggests that such a process is rare and that coevolution in most networks is diffuse and context dependent [START_REF] Thompson | The Coevolutionary Process[END_REF], but still a key driver of network structure [START_REF] Segar | The role of evolution in shaping ecological networks[END_REF]. It is, therefore, necessary to study species across their entire range to fully understand the process behind community patterns. More generally, 60 geographical variation in the specialisation of host-parasitoid networks can be driven by contrasting patterns of beta-diversity among trophic levels, making it important to quantify both local and regional network structure [START_REF] Galiana | The geographical variation of network structure is scale dependent: Understanding the biotic specialization of host-parasitoid networks[END_REF].

A major determinant of parasitoid network structure is host specificity because this will determine the number of trophic links formed. Indeed, [START_REF] Hawkins | Pattern and process in host-parasitoid interactions[END_REF] points to host 65 concealment as a major predictor of parasitoid specificity: concealed hosts are predicted to host generalist ectoparasitic idiobionts and exposed hosts specialist endoparasitic koinobionts. In natural communities of herbivores, parasitoids have been found to be hostspecific (an average of 1.8 hosts in a New Guinean caterpillar community [START_REF] Hrcek | Parasitism rate, parasitoid community composition and host specificity on exposed and semi-concealed caterpillars from a tropical rainforest[END_REF]), but this can vary in contrast to expectations between 70 host guilds with parasitoids of semi-concealed hosts being the most host specific. Parasitoids of the miner guild can also be host specific (an average of 2.8 hosts in one Belizean web) but vary in this degree [START_REF] Leppänen | Ecological versus phylogenetic determinants of trophic associations in a plant-leafminer-parasitoid food web[END_REF][START_REF] Lewis | Structure of a diverse tropical forest insect-parasitoid community[END_REF].

Evidence for 'Host Associated Differentiation' (the evolution of specialists across host genotypes) suggests that parasitoids of highly concealed galling insects [START_REF] Nicholls | Partitioning of herbivore hosts across time and food plants promotes diversification in the Megastigmus dorsalis oak gall parasitoid complex[END_REF][START_REF] Stireman | Cascading host-associated genetic differentiation in parasitoids of phytophagous insects[END_REF] can even be amongst the most host specific of insects. However, realised host breadth of parasitoids can be determined by multiple additional factors including evolutionary history (of both host and parasitoid), host abundance and geographical range (B. A. [START_REF] Hawkins | Pattern and process in host-parasitoid interactions[END_REF].

Prevailing evidence points to low levels of ecologically equivalent species in herbivores [START_REF] Butterill | Gall-forming insects in a lowland tropical rainforest: Low species diversity in an extremely specialised guild[END_REF][START_REF] Hrcek | Parasitism rate, parasitoid community composition and host specificity on exposed and semi-concealed caterpillars from a tropical rainforest[END_REF][START_REF] Novotny | Insects on plants: Explaining the paradox of low diversity within specialist herbivore guilds[END_REF] but a high frequency of morphologically highly similar (cryptic) species among their parasitoids [START_REF] Li | COI and ITS2 sequences delimit species, reveal cryptic taxa and host specificity of fig-associated Sycophila (Hymenoptera, Eurytomidae)[END_REF][START_REF] Smith | Extreme diversity of tropical parasitoid wasps exposed by iterative integration of natural history, DNA barcoding, morphology, and collections[END_REF][START_REF] Smith | DNA barcodes reveal cryptic host-specificity within the presumed polyphagous members of a genus of parasitoid flies (Diptera: Tachinidae)[END_REF] which only in some cases stems from host associated differentiation [START_REF] Hernández-López | Host tracking or cryptic adaptation? Phylogeography of Pediobius saulius (Hymenoptera, Eulophidae), a parasitoid of the highly invasive horse-chestnut leafminer: Parasitoid host races[END_REF]. Indeed, cryptic species are common in parasitoid networks and their inclusion is crucial to inform our understanding of network structure and turnover [START_REF] Hrcek | What do molecular methods bring to host-parasitoid food webs[END_REF][START_REF] Van Veen | Food web structure of three guilds of natural enemies: Predators, parasitoids and pathogens of aphids[END_REF]. Meso-diverse insect communities in enclosed microcosms, such as those associated with oak (Quercus) galls, have become model systems for studying host specificity [START_REF] Nicholls | Partitioning of herbivore hosts across time and food plants promotes diversification in the Megastigmus dorsalis oak gall parasitoid complex[END_REF] and community assembly [START_REF] Bunnefeld | Whole-genome data reveal the complex history of a diverse ecological community[END_REF] due to their replicated nature and intercontinental distributions. The current population genetic structure among Quercus species and their associated insect fauna has been highly dependent on glacial cycles [START_REF] Stone | Reconstructing community assembly in time and space reveals enemy escape in a western palearctic insect community[END_REF]. The multi-trophic wasp communities associated with [START_REF] Cook | Mutualists with attitude: Coevolving fig wasps and figs[END_REF]. Being largely tropical in distribution they offer a tractable counterpoint to the temperate oak galling communities. For example, functional community structure has been shown to be retained across the paleotropics, despite turnover of every species in the community of the canopy hemi-epiphyte F. benjamina [START_REF] Darwell | Conserved community structure and simultaneous divergence events in the fig wasps associated with Ficus benjamina in Australia and China and[END_REF].

While figs and their pollinators provide one of the strongest examples of codiversification on a global scale [START_REF] Cruaud | An extreme case of plant-insect codiversification: Figs and fig-pollinating wasps[END_REF], regional and local dynamics can demonstrate multiple trajectories. Accumulating evidence for multiple pollinators per host fig and even pollinator sharing is emerging (J. [START_REF] Cook | Speciation in fig wasps[END_REF] with recent examples demonstrating both continuity [START_REF] Bain | Geographic structuring into vicariant species-pairs in a wide-ranging, high-dispersal plant-insect mutualism: The case of Ficus racemosa and its pollinating wasps and[END_REF] and turnover in pollinator species over islands [START_REF] Rodriguez | Diversification and spatial structuring in the mutualism between Ficus septica and its pollinating wasps in insular South East Asia[END_REF] continents [START_REF] Yu | 6 Sycoscapter-H7 Sycoscapter-H3 Sycoscapter-H4 Sycoscapter-H19 Sycoscapter-H8 Sycoscapter-H20 Sycoscapter-H5 Sycoscapter-H2 Sycoscapter-H1 Sycoscapter-H9 Sycoscapter sp1[END_REF] and elevational gradients [START_REF] Souto-Vilarós | Faster speciation of fig-wasps than their host figs leads to decoupled speciation dynamics: Snapshots across the speciation continuum[END_REF]. Our increased understanding of speciation dynamics in pollinators has also shed more light on their diversification. For instance, the pollinators associated with small dioecious hosts are thought to be particularly prone to genetic isolation due to their limited dispersal capabilities when compared to the above canopy dispersers associated with monoecious fig species [START_REF] Ahmed | Wind-borne insects mediate directional pollen transfer between desert fig trees 160 kilometers apart[END_REF][START_REF] Compton | The flight heights of chalcid wasps (Hymenoptera, Chalcidoidea) in a lowland Bornean rain forest: Fig wasps are the high fliers[END_REF][START_REF] Harrison | Dispersal of fig pollinators in Asian tropical rain forests[END_REF][START_REF] Yu | 6 Sycoscapter-H7 Sycoscapter-H3 Sycoscapter-H4 Sycoscapter-H19 Sycoscapter-H8 Sycoscapter-H20 Sycoscapter-H5 Sycoscapter-H2 Sycoscapter-H1 Sycoscapter-H9 Sycoscapter sp1[END_REF]. The concept of high pollinator host specificity, however, has survived scrutiny in most cases.

The discovery of multiple pollinators per host has often been driven by molecular evidence [START_REF] Darwell | Molecular species delimitation of a symbiotic figpollinating wasp species complex reveals extreme deviation from reciprocal partner specificity[END_REF][START_REF] Molbo | Cryptic species of fig-pollinating wasps: Implications for the evolution of the figwasp mutualism, sex allocation, and precision of adaptation[END_REF]e.g. Yu et al., 2019) and has gradually become reconciled with a 'split-and-sort' mechanisms for diversification (J. [START_REF] Cook | Speciation in fig wasps[END_REF]. Diversification dynamics of parasitoids are often linked to those of their hosts [START_REF] Forister | Phylogenetic cascades and the origins of tropical diversity[END_REF]. Broad scale studies (at the level of the Ficus section) demonstrate congruence between host, pollinator and non-pollinator phylogenies for African fig species [START_REF] Jousselin | One fig to bind them all: Host conservatism in a fig wasp community unraveled by cospeciation analyses among pollinating and nonpollinating fig wasps[END_REF]. Putative pollinator cleptoparasites in the genus Philotrypesis [START_REF] Joseph | Recherches sur les chalcidiens, Blastophaga psenes (L.) et Phylotrypesis caricae (L.) du figuier (Ficus caricaL.)[END_REF] and two lineages of Otitesella ("uluzi" and "sesquinianellata"), one a galler the other a cleptoparasite of the galler (Segar, Pereira, Compton, & Cook, 2013, Figure 4) all showed high levels of host specificity. Both codiversification and host tracking are likely to have occurred throughout the diversification of the parasitoid fig wasps [START_REF] Segar | The global phylogeny of the subfamily Sycoryctinae (Pteromalidae): Parasites of an obligate mutualism[END_REF]. Within this general picture of host tree specialists, several species of neotropical Idarnes wasps, including gallers and cleptoparasites [START_REF] Farache | Insights into the structure of plant-insect communities: Specialism and generalism in a regional set of non-pollinating fig wasp communities[END_REF] [START_REF] Darwell | Cryptic diversity in a fig wasp community-morphologically differentiated species are sympatric but cryptic species are parapatric[END_REF]. Ficus rubiginosa hosts two parapatric species of Sycoscapter 'short' and Philotrypesis 'black'. Population genetic data from the whole range [START_REF] Sutton | One step ahead: A parasitoid disperses farther and forms a wider geographic population than its fig wasp host[END_REF] reveals that another species, Sycoscapter 'long', demonstrates higher levels of population connectivity and/or higher levels of dispersal than in one pollinating wasp, which has a disjunct distribution (but see [START_REF] Kjellberg | Tracking the elusive history of diversification in plant-herbivorous insect-parasitoid food webs: Insights from figs and fig wasps[END_REF].

Expectations and predictions.

Here we conduct extensive sampling of the wasp community associated with the small shrub, F. hirta. Previous genetic data [START_REF] Yu | 6 Sycoscapter-H7 Sycoscapter-H3 Sycoscapter-H4 Sycoscapter-H19 Sycoscapter-H8 Sycoscapter-H20 Sycoscapter-H5 Sycoscapter-H2 Sycoscapter-H1 Sycoscapter-H9 Sycoscapter sp1[END_REF] have shown that with nine parapatric species it has the largest set of pollinators reported to date for any Ficus species. We connect this large number with the small size of the plant which may lead to localized wasp dispersal in comparison to the large strangling Ficus species, and with the seemingly elevated rate of speciation in pollinating fig wasps in comparison to fig trees [START_REF] Moe | Rapid evolution of pollinator-mediated plant reproductive isolation[END_REF]. We use a range of genetic markers to establish the taxonomic equivalence of this spatially structured genetic differentiation and test the hypothesis of high connectivity between populations and/or high levels of dispersal in nonpollinating fig wasps. According to existing evidence from a single other gall/parasitoid wasp community [START_REF] Sutton | One step ahead: A parasitoid disperses farther and forms a wider geographic population than its fig wasp host[END_REF], we expect lower species diversity and turnover within parasitic fig wasps than for the pollinators, with parasites capable of utilizing most members of the pollinating wasp complex as hosts. Due to high dispersal ability and a resilience to founder effects (conferred by the limited necessity for outbred males) we expect that species ranges of parasites will not be shaped by historical contingencies but rather by inter-specific competition and species' ecological traits. Following on from range disparities and reduced parasite specificity we predict that pair-wise comparisons across sites will reveal multiple examples of parasites spread across several pollinator hosts.

Further, in systems presenting a linear distribution (as is the case for F. rubiginosa) different species may abut on the same ecological barrier, incidentally leading to statistical association between parasites and between parasites and hosts. Such incidental correlations are expected to be less frequent in systems presenting two-dimensional distributions (as is the case for F. hirta), as ecological obstacles and ecological gradients are more diversified and different species groups may respond differently to these factors. At the community scale we expect conserved ecological function throughout the species' range (as appears to be the case in other fig-wasp systems), with the same niches existing and being filled throughout the range of F. hirta. This is driven by an underlying expectation that fig wasp communities should be generally saturated and structured largely by competition for limited resources. Indeed, published data suggests that while not saturated at the individual tree-crop level, fig-wasp communities are saturated when several crops are sampled more widely (Compton & Hawkins, 1992;B. A. Hawkins & Compton, 1992), in other words there is saturation at the regional level.

Methods (a) Study species

Ficus hirta Vahl is a shrub of secondary vegetation (Fig. S1). Its distribution extends from the island of Java in the south to China in the north and westwards into northeast India and Nepal [START_REF] Yu | 6 Sycoscapter-H7 Sycoscapter-H3 Sycoscapter-H4 Sycoscapter-H19 Sycoscapter-H8 Sycoscapter-H20 Sycoscapter-H5 Sycoscapter-H2 Sycoscapter-H1 Sycoscapter-H9 Sycoscapter sp1[END_REF]. Ficus hirta is pollinated by the host specific species complex of Valisia javana Mayr (Agaonidae, Chalcidoidea, Hymenoptera) [START_REF] Yu | 6 Sycoscapter-H7 Sycoscapter-H3 Sycoscapter-H4 Sycoscapter-H19 Sycoscapter-H8 Sycoscapter-H20 Sycoscapter-H5 Sycoscapter-H2 Sycoscapter-H1 Sycoscapter-H9 Sycoscapter sp1[END_REF]. Pollinating wasps enter the fig (a closed urn-shaped receptacle) pollinate and oviposit in flowers.

Three non-pollinating chalcid fig-wasp species have been reported from F. hirta [START_REF] Yu | Plant geographic phenotypic variation drives diversification in its associated community of a phytophagous insect and its parasitoids[END_REF]. Philotrypesis josephi and Sycoscapter hirticola in northeast India [START_REF] Nair | Two new Torymidae (Hymenoptera, Chalcidoidea) from Ficus hirta[END_REF] and Sycoryctes simplex in Java [START_REF] Mayr | Fig insecten[END_REF].

Philotrypesis (tribe Philotrypesini), Sycoscapter and Sycoryctes (tribe Sycoryctini) are monophyletic genera belonging to the subfamily Sycoryctinae (Pteromalidae) [START_REF] Segar | The global phylogeny of the subfamily Sycoryctinae (Pteromalidae): Parasites of an obligate mutualism[END_REF]. They oviposit into ovaries containing Valisia larvae by inserting their long ovipositor through the fig wall. Philotrypesis are cleptoparasites while Sycoryctini are parasitoids [START_REF] Conchou | The non-pollinating fig wasps associated with Ficus guianensis: Community structure and impact of the large species on the fig/pollinator mutualism[END_REF]. Philotrypesis, Sycoryctini, and pollinating Valisia are morphologically remarkably different (Fig. S1). Offspring of fig wasps 190 mate in the fig, often between siblings, before dispersal from their natal fig, a feature that facilitates the initial establishment of rare colonizers [START_REF] Yu | 6 Sycoscapter-H7 Sycoscapter-H3 Sycoscapter-H4 Sycoscapter-H19 Sycoscapter-H8 Sycoscapter-H20 Sycoscapter-H5 Sycoscapter-H2 Sycoscapter-H1 Sycoscapter-H9 Sycoscapter sp1[END_REF].

(b) Sampling

Samples were collected from South China to Java (Supporting Information Table S1). Wasp collection protocols follow [START_REF] Yu | 6 Sycoscapter-H7 Sycoscapter-H3 Sycoscapter-H4 Sycoscapter-H19 Sycoscapter-H8 Sycoscapter-H20 Sycoscapter-H5 Sycoscapter-H2 Sycoscapter-H1 Sycoscapter-H9 Sycoscapter sp1[END_REF]. Spatial structuring has been investigated for the 195 pollinators of F. hirta, Valisia spp. [START_REF] Yu | 6 Sycoscapter-H7 Sycoscapter-H3 Sycoscapter-H4 Sycoscapter-H19 Sycoscapter-H8 Sycoscapter-H20 Sycoscapter-H5 Sycoscapter-H2 Sycoscapter-H1 Sycoscapter-H9 Sycoscapter sp1[END_REF] and microsatellite data and cytochrome c oxidase I (COI) data have been obtained for Philotrypesis and Sycoscapter for samples from South-East China [START_REF] Yu | Plant geographic phenotypic variation drives diversification in its associated community of a phytophagous insect and its parasitoids[END_REF] S1). We sequenced the ITS2 gene for a total of 133 individuals from 15 sites for Philotrypesis and 54 205 individuals from 11 sites for Sycoryctini (Table S1). Molecular procedures were the same as in Yu et al. [START_REF] Yu | Plant geographic phenotypic variation drives diversification in its associated community of a phytophagous insect and its parasitoids[END_REF], except for Cytb amplification (see Table S2). Sequences were aligned as in Yu et al. [START_REF] Yu | Plant geographic phenotypic variation drives diversification in its associated community of a phytophagous insect and its parasitoids[END_REF]. A 628 fragment of the Cytb gene and a 180 bp fragment of the ITS2 gene were sequenced for Philotrypesis, while for the Sycoryctini the fragment lengths were 652 and 234 bp respectively.

210

The within genus phylogenetic positions of Philotrypesis, Sycoscapter and Sycoryctes species associated with Ficus section Eriosycea has not been ascertained. Therefore, we used our Sycoryctini samples as an outgroup of our Philotrypesis and conversely. Maximum likelihood trees were constructed using MEGA 6.0 [START_REF] Tamura | MEGA6: Molecular evolutionary genetics analysis version 6.0[END_REF] for Cytb and ITS2 separately, and node supports were assessed based on 2000 bootstrap replicates. We calculated Kimura-2-parameter (K2P) distances for Cytb haplotypes and ITS2 between all individuals and within and between clades evidenced by the Maximum Likelihood tree, using MEGA 6.0. Classical indices of genetic diversity were estimated using GenALEx 6.1 [START_REF] Peakall | GENALEX 6: Genetic analysis in Excel. Population genetic software for teaching and research[END_REF]. FIS values were calculated separately for each location. To provide a global representation of the data, we performed a factorial correspondence analysis as implemented in GENETIX [START_REF] Belkhir | GENETIX 4.05, logiciel sous Windows TM pour la génétique des populations[END_REF]. We used Bayesian clustering to assign multilocus microsatellite genotypes to clusters using STRUCTURE 2.2 [START_REF] Pritchard | Inference of population structure using multilocus genotype data[END_REF]. The admixture ancestry and correlated allele frequencies model was used with three independent runs each of 500,000 MCMC iterations and 500,000 burn-in steps. We ran STRUCTURE varying K (the number of clusters) from 2 to 13 for Philotrypesis and for Sycoscapter-Sycoryctes.

(e) Co-occurrence analysis

We analysed the co-occurrence i) of species across sites, ii) between Philotrypesis and Sycoscapter and iii) between hosts and parasites. First, we plotted the occurrences of the different species on a map and made visual comparisons of the differences in the range limits between the species groups. Second, we constructed co-occurrence matrices for each species of each genus for all sites from which they were recorded (e.g. a 9 x 32 matrix for Valisia) and tested for aggregation or segregation by comparing the observed V-ratio (Variance ratio; mean pairwise covariance in association) to the distribution obtained from a set of 1,000 randomised matrices created by shuffling matrix fill (retaining row sums and allowing 240 column totals to vary randomly and equiprobably). We discarded 500 randomisations as 'burnin'. We implemented this analysis using co-occurrence null models in the R package 'EcoSimR' [START_REF] Gotelli | EcoSimR: Null model analysis for ecological data[END_REF] by setting the metric to "V-ratio" and the algorithm to "Sim 2". We repeated this analysis for a matrix of coocurrence between Philotrypesis species and Sycoscapter species.

245 Third, we calculated the observed Czekanowski niche overlap index for Philotrypesis and Valisia species (a 4 x 9 matrix) and all Sycoscapter and Valisia species (a 3 x 9 matrix). The observed niche overlap index was compared to a distribution obtained from 1,000 randomised matrices created by shuffling matrix fill generated by reshuffling row values obtaining the observed number of interactions per species. We use only binary and not 250 abundance-based data. This analysis was also implemented in R using the package 'EcosimR' by running niche overlap null models, we used the Czekanowski and RA3 algorithm.

Results

Sequence data. The geographic distribution of the different haplotypes and associated S6-S7). The cumulative distribution of K2P distances for Cytb presents a marked barcoding gap between clades for both Philotrypesis and Sycoryctini (Fig. 2).

Therefore, in the following, we will accept each clade as a separate species. We made the conservative choice of considering Philotrypesis samples from Sand as belonging to sp2 pending deeper sequencing. Genetic distances were smaller between Philotrypesis species than between Sycoryctini species (Fig. 1, S6-S7). Among all the data we have only one case

where Philotrypesis species co-occurred in a single location while there was no cooccurrence of Sycoryctini species (Tables S2-S5). The Sycoryctini sequences from Java are highly divergent for both Cytb and ITS2 (Tables S6-S7). Morphological inspection showed that samples from Java belonged to genus Sycoryctes while samples from the continent belonged to genus Sycoscapter as currently circumscribed [START_REF] Segar | The global phylogeny of the subfamily Sycoryctinae (Pteromalidae): Parasites of an obligate mutualism[END_REF]. Critical inspection of the description of Sycoscapter hirticola [START_REF] Nair | Two new Torymidae (Hymenoptera, Chalcidoidea) from Ficus hirta[END_REF] confirmed assignment to genus. S8-S9. For Philotrypesis there was too much variation within species and too little variation among species to allow clear species separation (Fig. S3), while factorial correspondence analysis confirmed the differentiation between species in Sycoryctini (Fig. S4). Four genotypes from location Xi grouped with sp1 genotypes while 3 genotypes grouped with sp2 genotypes (Fig. S4). One genotype from Java came out with sp2 genotypes. Bayesian assignment to cluster gave the same insights as the correspondence analysis (Fig. S5-S6). For Sycoscapter, the presence in location Xi of both sp1 and sp2, and the presence of an individual of sp2 on Java was supported by the Bayesian assignment.

Microsatellite data. Diversity indices are given in Table

Pollinators presented higher FIS values than Philotrypesis and Sycoryctini demonstrating more frequent brother-sister mating (Fig. S7). Sample sizes and distances among locations 285 allowed investigating genetic isolation by distance (IBD) for Valisia spp 1 and 2, Philotrypesis spp. 1, 2, 4, and Sycoscapter sp1. In Valisia there was no IBD, but differentiation within sp1 between Hainan island, southeast China and Vietnam [START_REF] Tian | Lack of genetic isolation by distance, similar genetic structuring but different demographic histories in a figpollinating wasp mutualism[END_REF]. IBD was present in Philotrypesis and Sycoscapter, except for Philotrypesis sp4 (Fig. S8). As well as Valisia, both Philotrypesis and Sycoscapter segregated across sites (Table 1), such that the entire geographic range was divided into largely non-overlapping blocks occupied by 305 a single species for each genus. Our co-occurrence analysis provides statistical support for this striking pattern presented in Figure 3 and outlined above. It is clear that the geographic limits of the species belonging to the two tribes of parasites do not correspond (the mean and simulated mean are not statistically different, Table 1). In other words, there is no broad congruence between species ranges among members of different genera. Furthermore, there is no niche partitioning between either Sycoscapter or Philotrypesis across their Valisia hosts (Czekanowski niche overlap is no different from random in either case, Table 1). Neither parasite genus shows greater levels of specialisation than expected by chance.

Discussion

While at least eight species of Valisia are associated with Ficus hirta throughout its range, we found four species of Philotrypesis, three species of Sycoscapter and one species of Within most parasitic species we found evidence for genetic isolation by distance. Such a pattern of well delimited sister taxa displaying within species isolation by distance suggests an older colonisation of the range by extant parasite species or more limited dispersal than in the sole other non-pollinating fig wasp for which genetic data is available, a Sycoscapter associated with F. rubiginosa [START_REF] Sutton | One step ahead: A parasitoid disperses farther and forms a wider geographic population than its fig wasp host[END_REF]. An alternative explanation is lower population density in the parasites than in the pollinators of Ficus hirta. However, there is only a limited difference in density between the two groups, which is probably not sufficient to explain the lack of IBD among pollinator populations coupled with strong IBD among parasite populations. Indeed, the slope of genetic differentiation is predicted to be simply inversely proportional to density, all other factors remaining constant [START_REF] Rousset | Genetic differentiation and estimation of gene flow from F-statistics under isolation by distance[END_REF]. The difference may relate to differences in fruiting phenology and distribution between Ficus hirta, a small dioecious shrub (continuous fruiting, growing in patches) and F. rubiginosa, a monoecious hemiepiphytic figs (synchronised crop on a tree, more dispersed individuals).

Several of the Sycoryctine species studied here had larger geographical ranges than pollinators. For example, Philotrypesis sp 1 occurs from southern China to southern Thailand. Competitive exclusion could be a major driver in conjunction with some ecological differentiation of the parapatric distribution across sites within Philotrypesis and Sycoscapter.

Indeed, the geographic distributions of diversity do not appear to be the same for each trophic group, with little correspondence seen between the ranges of Philotrypesis and Sycoscapter, and between either and Valisia. Climatic modelling would outline the role of environmental gradients versus competition, but the evidence presented here suggests that they are not the sole drivers of species distributions. While host associated differentiation is largely absent at higher trophic levels some parasites are restricted to single pollinator species, possibly because of their restricted range. The northern populations of F. hirta that host the more restricted Philotrypesis sp3 are phenotypically distinct and produce larger figs with thick walls [START_REF] Yu | Plant geographic phenotypic variation drives diversification in its associated community of a phytophagous insect and its parasitoids[END_REF]. Perhaps the observed pattern is contingent on encounter frequency and asymmetric coevolution across trophic levels [START_REF] Lapchin | Host-parasitoid association and diffuse coevolution: When to be a generalist?[END_REF]; diffuse coevolution). [START_REF] Segar | How to be a fig wasp down under: The diversity and structure of an Australian fig wasp community[END_REF][START_REF] Weiblen | Speciation in fig pollinators and parasites[END_REF]. This divergence is highly suggestive of divergent selection driven by competition on a shared host. In our study, sister species are largely allopatric reflecting a situation of competitive exclusion. Certainly, the congeneric individuals studied here are largely similar with respect to ovipositor length (apart from Philotrypesis sp3.) and hard to distinguish morphologically. More detailed morphological appraisal of each molecular entity would certainly be highly valuable for determining potential niche overlap.

In a similar vein formal tests of phylogenetic congruence between these various taxonomic groups would be an important next step.

Despite being absent in high canopy sampling efforts [START_REF] Harrison | Fig wasp dispersal and the stability of a keystone plant resource in Borneo[END_REF] the single species of non-pollinating fig wasp studied to date appears to be a reasonable disperser [START_REF] Sutton | One step ahead: A parasitoid disperses farther and forms a wider geographic population than its fig wasp host[END_REF]. Sutton et al. [START_REF] Sutton | One step ahead: A parasitoid disperses farther and forms a wider geographic population than its fig wasp host[END_REF] report no or almost no IBD in a wide ranging Sycoscapter wasp. A general point might be the longevity of these wasps, known to live for as long as 35 days when fed sugar water [START_REF] Compton | African fig wasp parasitoid communities[END_REF][START_REF] Joseph | Recherches sur les chalcidiens, Blastophaga psenes (L.) et Phylotrypesis caricae (L.) du figuier (Ficus caricaL.)[END_REF].

In contrast to non-feeding pollinators and other wasps that enter monoecious figs and which disperse by wind, externally ovipositing wasps can wait for suitable clutches to become locally available. Gradual, but regular, inter-generational dispersal of these generalist parasitoids is likely given their strategy of laying small clutches of offspring across multiple figs (James M. [START_REF] Cook | Molecular markers reveal reproductive strategies of non-pollinating fig wasps: Molecular revelation of fig wasp behaviour[END_REF]. We might expect broad geographic ranges across parasitoid and cleptoparasitic fig wasps in general. Among the species studied here, some Philotrypesis species have remarkable ranges, but display higher levels of IBD than Valisia pollinators. These lines of evidence suggest a long-term host association between non-pollinating fig wasps and F. hirta and moderate levels of dispersal in these 'slow' wasps [START_REF] Venkateswaran | Life-history strategy, resource dispersion and phylogenetic associations shape dispersal of a fig wasp community[END_REF], or lower densities. Furthermore, Ficus hirta is a dioecious species with a clumped distribution in the landscape, a population structure that is likely to offset some long distance dispersal in wasps [START_REF] Kjellberg | Tracking the elusive history of diversification in plant-herbivorous insect-parasitoid food webs: Insights from figs and fig wasps[END_REF]. We suggest that strong gene flow in pollinating wasps, offset by a degree of local dispersal and in combination with limited phenotypic variation in F. hirta, will result in multiple phenotypically homogeneous pollinator species. Each species will be easily excluded by another species at its margins. In contrast, parasites display clinal genetic variation, likely reflecting local adaptation, and are hence more resistant to competition at the limits of their range.

A generalized pattern of geographic species turnover in fig-wasp community

The geographic variation of associated communities of fig wasps specialised on a single Ficus host have now been investigated using molecular markers for two Ficus species [START_REF] Darwell | Molecular species delimitation of a symbiotic figpollinating wasp species complex reveals extreme deviation from reciprocal partner specificity[END_REF][START_REF] Darwell | Cryptic diversity in a fig wasp community-morphologically differentiated species are sympatric but cryptic species are parapatric[END_REF][START_REF] Yu | Plant geographic phenotypic variation drives diversification in its associated community of a phytophagous insect and its parasitoids[END_REF]this study, 2019). In both cases, the wasps are mainly structured into groups of parapatric, ecologically equivalent species. In both communities, pollinating wasp species diversity is twice that within parasite group, and the different parasite functional groups on a fig host present similar numbers of parapatric species (4 on Ficus hirta, 1-2 on F. rubiginosa). A direct correlate is that these parasitoids and cleptorasites are not host specialists, as they parasitize several host species. Such heterogeneous diversification patterns may be general. Indeed, in communities of oak galling wasps, galling species present more divergent populations through their range than their parasitoids [START_REF] Stone | Reconstructing community assembly in time and space reveals enemy escape in a western palearctic insect community[END_REF]. The pattern observed here could fit within the general scenario where parasitoids of endophytic herbivores (e.g. gallers and leaf miners) are less specialized than parasitoids of exophytic herbivores [START_REF] Askew | Parasitoid communities: Their size, structure and development[END_REF]. Endophytic parasitoids are generally idiobionts and more subject to interspecific competition (B. A. [START_REF] Hawkins | Influences of host feeding-niche and foodplant type on generalist and specialist parasitoids[END_REF], a process which underpins our hypothesis. Another more general example is that of leaf mining insects, a group of endophytes attacked in some cases by rather generalist parasitoids whose host use better correlates to plant than host phylogeny [START_REF] Ives | Phylogenetic analysis of trophic associations[END_REF][START_REF] Leppänen | Ecological versus phylogenetic determinants of trophic associations in a plant-leafminer-parasitoid food web[END_REF][START_REF] Lopez-Vaamonde | Evolutionary dynamics of host-plant use in a genus of leaf-mining moths[END_REF].

The age of divergence among Valisia host species is estimated at over 10 Ma with some subgroups splitting as recently as 2.6 Ma [START_REF] Yu | 6 Sycoscapter-H7 Sycoscapter-H3 Sycoscapter-H4 Sycoscapter-H19 Sycoscapter-H8 Sycoscapter-H20 Sycoscapter-H5 Sycoscapter-H2 Sycoscapter-H1 Sycoscapter-H9 Sycoscapter sp1[END_REF] the age of Western Palearctic cynipid wasp communities is around 3.5 Ma. The progenitor oak gall wasp community likely assembled before the expansion of oaks into from Asia around 5-7 Ma [START_REF] Stone | Reconstructing community assembly in time and space reveals enemy escape in a western palearctic insect community[END_REF].

The accumulation of species in these last two systems has operated over similar timescales.

While climactic history clearly differs, there are strong parallels in the ecology of the parasites in both systems. These studies highlight the need for range-wide studies for understanding host specificity.

The diversity within group and among locations (beta diversity) was more than twice as high in Ficus hirta than in F. rubiginosa while the number of groups (alpha diversity) was much higher in F. rubiginosa resulting in a total of at least 17 non-pollinating wasp species [START_REF] Darwell | Cryptic diversity in a fig wasp community-morphologically differentiated species are sympatric but cryptic species are parapatric[END_REF][START_REF] Segar | How to be a fig wasp down under: The diversity and structure of an Australian fig wasp community[END_REF]. A simple explanation could be that a shared factor, more limited wasp dispersal, reduces the number of species that can survive locally while facilitating geographic speciation. In agreement, alpha-diversity of fig-wasp communities has been shown to correlate with tree height and breeding system (Compton & Hawkins, 1992). Comparative studies will allow unravelling the causal factors involved.

A generalized pattern of geographic stability in fig-wasp community structure

The geographic stability of community structure reported here is frequent in fig-wasp communities. Indeed, there was no, or almost no, latitudinal turnover in parasite species number for 26 Ficus species in southern Africa (B. A. Hawkins & Compton, 1992). The structure of the community associated with Ficus rubiginosa in Australia is constant 435 throughout its range despite geographic turnover in species composition [START_REF] Darwell | Cryptic diversity in a fig wasp community-morphologically differentiated species are sympatric but cryptic species are parapatric[END_REF][START_REF] Segar | How to be a fig wasp down under: The diversity and structure of an Australian fig wasp community[END_REF]. Similarly, the communities associated with F. benjamina in Hainan

Island and in Australia, while sharing no species, presented highly similar structures [START_REF] Darwell | Conserved community structure and simultaneous divergence events in the fig wasps associated with Ficus benjamina in Australia and China and[END_REF]. Such geographic stability suggests a role for deterministic processes in which host-plant traits determine the structure of their associated fig-wasp communities. Indeed, fig- 440 wasp communities present independently evolved convergent structure across continents [START_REF] Segar | Convergent evolution of 665 multitrophic community structure over three continents[END_REF], and the complexity of fig-wasp communities in southern Africa is determined by ecological factors such as tree height rather than by phylogenetic history (Compton & Hawkins, 1992).

A generalized pattern of limited spatial turn over in community structure of plant445 associated insects

As for fig wasps, the structure of the communities constituted by deciduous-oak galling wasps and their parasitoids across the Western Palearctic is remarkably stable. Indeed, species turnover between refugia in Iran, the Balkan and Iberia is extremely limited [START_REF] Stone | Reconstructing community assembly in time and space reveals enemy escape in a western palearctic insect community[END_REF], and this is achieved despite contrasting biogeographic patterns for the different 450 species. However, if we accept long distance dispersal to be frequent, as documented for the oak associated communities and for fig wasps, then we may expect current genetic structure to reflect the history of the individual species, with occasional regional extinctions of some taxa followed by re-colonization, but also range-wide invasions with species replacements.

Homogeneous spatial genetic structure and age of colonization within a community is not a prediction [START_REF] Alvarez | Phylogeography and historical biogeography of obligate specific mutualisms[END_REF]. Nevertheless, stability of community structure without geographic species turnover was observed in the most extensive survey of geographic variation in tropical herbivorous insects, in the New Guinea lowlands, over distances of 500-1000 km (V. [START_REF] Novotny | Low beta diversity of herbivorous insects in tropical forests[END_REF]. Species turnover should probably be scaled relative to geographical distances, to variation in ecological conditions and to age of the examined communities.

Conclusions

  plants of the genus Ficus have great potential to serve as a complimentary system to gall wasps. Pollinating fig wasps (Agaonidae, Chalcidoidea) co-exist with members of several other chalcid (sub) families. Some fig wasps, including the pollinating wasps, gall fig ovules, other fig wasps parasitise pollinators and/or other gall formers (as parasitoids or cleptoparasites) while a third trophic level of hyper-parasitoids can also be present (J. M

  and a few African parasitoid species of genus Arachonia (McLeish, Beukman, van Noort, & Wossler, 2012) are generalists, developing in the figs of several local Ficus species. Our knowledge at finer scales is more limited, and to date only two studies have addressed the phylogeography and population genetics of non-pollinating fig wasps. Wide ranging fig species, for example the Australian Ficus rubiginosa, are associated with distinct species of non-pollinators throughout their range

  . We sampled representatives of each fig wasp species emerging from individual figs at each location, however we did not record complete fig contents or individual fig level occurrence data at all sites.200 (c) Gene sequence data Cytoplasmic Cytochrome b (Cytb) was sequenced as preliminary results gave inconsistent results for COI. Cytb sequences were obtained, generally for a single wasp per fig, for a total of 124 Philotrypesis from 18 sites and 76 Sycoryctini from 13 sites (Table

  For Philotrypesis, 262 individuals, each from a different fig, were genotyped at 6 unlinked microsatellite loci. For Sycoryctini, 203 individuals, each from a different fig, were genotyped at 7 unlinked microsatellite loci. Molecular techniques follow Yu et al. (2018).

255

  diversity parameters are given in Tables S2-S5. A Cytb maximum likelihood tree separated the non-pollinating wasps into 4 clades of Philotrypesis and 4 clades of Sycoryctini (Fig 1). Individuals originating from locations monomorphic for Philotrypesis or Sycoryctini Cytb clades were sequenced for ITS2. Each of the 7 sequenced Cytb clades was monomorphic for ITS2, presenting a distinctive haplotype (Fig. S2). The distribution of K2P distances within and among clades shows low within clade and large between clade distances for both Cytb and ITS2 (Tables

290

  Host-specificity and co-occurrence. The geographic distributions of the different species are shown in Fig. 3. The limits of the species distributions for the three types of wasps are visually different. For instance, the distribution of Valisia sp1 was established by sequencing 176 individuals for COI or for ITS2. In all locations where Valisia sp 1 was found, not a single sequence from any other Valisia species was found. Reciprocally, locations where 295 Valisia sp1 was not found were established by sequencing 358 individuals either for COI or for ITS2. The distribution of Valisia sp1 was totally cohesive, and formed one block. Philotrypesis sp 1 was found in locations alongside Valisia sp1 (118 individuals of Philotrypesis sp1 sequenced either for CytB or for ITS2) and in locations where Valisia sp1 was not present (44 individuals of Philotrypesis sp1 sequenced either for CytB or for ITS2). 300 In Fig. 3, it can be seen that Philotrypesis sp4 uses Valisia sp6 in DAL and Valisia sp5 in SNP while Sycoscapter sp2 uses Valisia sp2 in QMS, Tai and likely CS vs. Valisia sp6/7 in Wu and Valisia sp7 in CH.

Sycoryctes.

  Our expectation of lower parasite diversity and limited host-specificity was confirmed. Philotrypesis sp1 parasitizes at least four species of the pollinating Valisia and Sycoscapter sp2 at least two species of Valisia. These parasitic wasp species are not specialists of host insect species. Ultimately, direct confirmation of host associations will only be obtained by sequencing gall content to determine the host-parasite association present in individual galls (e.g.[START_REF] Sow | 680 Deciphering host-parasitoid interactions and parasitism rates of crop pests using DNA metabarcoding[END_REF]. Do non-pollinators exist locally but fail to develop, for example? Nevertheless, the non-overlapping distributions of pollinator species gives strength to our correlative conclusion. It is also important to note here that our sampling design did not allow for detailed (e.g. at the level of the individual fig) pair-wise comparisons of co-occurrence across fig wasp species.

Finally, while the

  community of fig wasps associated with Ficus hirta is not diverse, the community structure recorded here (one pollinator, one cleptoparasite and one parasitoid) is largely conserved across the host range. This is in line with our expectations derived from other fig-wasp systems. Indeed, while Hawkins and Compton (1992) stressed a lack of saturation in fig wasp communities reared from individual crops, their extensive data set showed that both community structure and species richness remain constant for several Ficus sampled at a regional scale. Sympatric, congeneric fig wasps in the same trophic guild tend to diverge in ovipositor length which is a proxy for fig development stage at oviposition

  Comprehensive data on geographic variation in fig wasp community structure, spatial genetic structure, and phylogenetic history is providing new insights into the ecological and evolutionary determinants of community structure and composition. In the case of F. hirta fewer cleptoparasite and parasitoid species are found than obligate pollinator species. These non-pollinating fig wasps are generalist, in the sense that they can utilise several pollinator species as hosts. Furthermore, while range size varies among similar non-pollinating fig wasp species, it is generally large and non-overlapping. As the two types of parasitic wasps are non-specialists and present different geographic patterns, co-diversification is not a process shaping local and regional communities. Isolation by distance is lower in pollinators than non-pollinators. Finally, most populations of F. hirta are utilised by a single pollinator species along with one cleptoparasite and one parasitoid. Our results highlight the links between ecology, life history and evolutionary history in shaping community structure, patterns of genetic diversity and host specificity. Fig wasp communities are becoming a major biological model to understand the diversification of insect communities feeding on plants in the tropics, with real hopes of establishing a set of general rules. Data Accessibility Microsatellite genotype data for all individuals of Philotrypesis and Syocryctini are archived in the Dryad Digital Repository at http://datadryad.org, doi: 10.5061/dryad.5x69p8d0x. YH designed research, collected samples, analyzed data and co-wrote the manuscript. CLF 720 analyzed data. STS co-wrote the manuscript. FK contributed to data analyses and co-wrote the manuscript. DXX performed laboratory work and analyzed data. EWT performed laboratory work. DYZ designed research. WT collected samples in Thailand.

Figure legends 725 Fig. 1 .

 7251 Figure legends

Fig. 2 .

 2 Fig. 2. Cumulative distribution of Kimura pairwise genetic distances (K2P) for Cytb for

Fig. 3 .

 3 Fig. 3. Geographic distribution of the three sets of fig wasps associated with Ficus hirta and 735 genetic variation of the plant. Locations indicated with coloured lettering are those for which species assignment was based on SSR genotypes only. Philotrypesis are assumed to be cleptoparasites of Valisia, Sycoscapter and Sycoryctes are assumed to be parasitoids of Valisia. Valisia are the pollinators of Ficus hirta and feed on galled plant tissue. For the plant, genetic variation is clinal and is illustrated by percentage of assignment to two extreme 740 gene pools.

Table 1 .

 1 Results of coocuurence analysis between i) species within each wasp genus at each site and ii) the two main parasite wasp genera and each Valisia pollinator species. SES: standardised effect size.

		Mean of simulated
	V -ratio	data		SES	P -value
	Cooccurence
	Philotrypesis at each site	0.06	1.005	-3.534 p<0.001
	Sycoscapter at each site	0.11	1.003	-3.121 p<0.001
	Valisia at each site	0.07	1.001	-3.964 p<0.001
	Philotrypesis and

Acknowledgements

We thank Prof. Xia Nianhe, Peng Yanqiong, Vu Quang Nam, Tien Tran, Nong Van Duy, Zhang Xianchun (collecting permit number in Indonesia: No.1799IPH.1.02/KS.01/XI/2014), Harry Wiriadinata, Arief Hidayat, Wei Ran, Chen Wenli, Mashhor Mansor, Nora Sikin, Serena Lee (provided Singapore samples), Khin Me Aung, Chuck Cannon, Thitima Tharawoot, and Liang Dan for sample collection and help in the field. This work was supported by the National Natural Science Foundation of China (31630008; 31670395), Key Special Project for Introduced Talents Team of Southern Marine Science and Engineering Guangdong Laboratory (Guangzhou) (GML2019ZD0408), the National Natural Science Foundation of China (31971568), Province Natural Science Foundation of Guangdong (c20140500001306), and the CAS President's International Fellowship Initiative (2018VBA0040). Simon Segar is grateful for departmental support from Harper Adams University.

Supporting information

Additional Supporting Information may be found in the online version of this article:


Brigitte Bidegaray-Fesquet

Nacim Meslem
email: nacim.meslem@gipsa-lab.grenoble-inp.fr

Fairouz Zobiri
email: fairouz.zobiri@kuleuven.be

Event-Based Set-point Tracking Based on a State Observer

We propose an event-based framework for set-point tracking involving three linear time invariant systems, a real, a reference and an observer system. To be actually event-based using only the knowledge of the output of the real system, we show that the two systems (reference and observer) involved in the event generator have to take into account the output observation error. The resulting setup is proved to be able to follow a Lipschitz predefined set-point trajectory with no Zeno phenomenon.

I. INTRODUCTION

Event-based control is an alternative to continuous or periodic control, that enables to address the control of embedded systems with low resources [START_REF] Johan | Comparison of periodic and event based sampling for first-order stochastic systems[END_REF], [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF], [START_REF] Peng | A survey on recent advances in eventtriggered communication and control[END_REF], [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]. In this paradigm, the infrequent control updates are suitable for low power systems that spare the energy used in actuators, or to relax the communication requirements in embedded and networked systems.

The stabilization problem of continuous-time systems with event-triggered controllers has been widely studied in the literature. For example, the case of linear systems is considered in [START_REF] Forni | Eventtriggered transmission for linear control over communication channels[END_REF], [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF], [START_REF] Meslem | Event-triggered algorithms for continuous-time systems based on reachability analysis[END_REF], [START_REF] Zobiri | Selftriggered stabilizing controllers for linear continuous-time systems[END_REF], the case of nonlinear systems is addressed in [START_REF] Anta | To sample or not to sample: selftriggered control for nonlinear systems[END_REF], [START_REF] Marchand | A general formula for event-based stabilization of nonlinear systems[END_REF], [START_REF] Meslem | Event-based controller synthesis by bounding methods[END_REF], [START_REF] Seuret | Stability of non-linear systems by means of event-triggered sampling algorithms[END_REF], [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF], and the case of switched linear systems is studied in [START_REF] Li | Stabilization of switched linear neutral systems: An event-triggered sampling control scheme[END_REF], [START_REF] Xiang | Event-triggered control for continuous-time switched linear systems[END_REF], [START_REF] Xiao | Eventtriggered control of continuous-time switched linear systems[END_REF], [START_REF] Zobiri | Eventtriggered stabilizing controllers for switched linear systems[END_REF]. Note that, the eventtriggering conditions of all the aforementioned approaches are based on the knowledge of the whole state vector of the system at all times. Thus, to overcome this constraint, event-triggered stabilizing controllers based on output feedback are proposed in [START_REF] Abdelrahim | Co-design of output feedback laws and event-triggering conditions for linear systems[END_REF], [START_REF] Donkers | Output-based event-triggered control with guaranteed L∞-gain and improved and decentralized event-triggering[END_REF] and on estimated state feedback are introduced in [START_REF] Forni | Eventtriggered transmission for linear control over communication channels[END_REF], [START_REF] Meslem | Event-based stabilizing controller using a state observer[END_REF]. In [START_REF] Forni | Eventtriggered transmission for linear control over communication channels[END_REF] the experiential stability problem of a specific closed-loop interconnection between a continuous-time linear system and a continuous-time linear controller via a digital channel is considered. The global asymptotic stability of the plant is guaranteed when an estimate of its state is provided by a continuous-time Luenberger observer. The control is updated when a Lyapunov function using an extended system involving both the estimated state and the control error is not sufficiently decreasing or when the error becomes large with respect to the estimated state. This type of triggering condition is also found in the pioneering work [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]. A non-zero dwelling time is also enforced with a timer. In [START_REF] Meslem | Event-based stabilizing controller using a state observer[END_REF] a combination of a stable scalar reference system and a classical state observer is used to design a new event-triggering condition. This condition is based on a comparison between the Lyapunov function of the reference system and a quadratic function of the estimated state.

On the other hand, the set-point tracking problem with event-triggered policy is tackled in [START_REF] Zobiri | Eventbased sampling algorithm for setpoint tracking using a state-feedback controller[END_REF]. It is shown that, for continuous-time linear systems, it is possible for the system outputs to track reference signals even if the control signals are not updated continuously. Indeed the control values are updated only when the behavior of the system is no longer satisfactory. To assess the behavior a condition involving the full state of the system is used. This is not very realistic for practical applications and for two reasons: first the full state of a system is usually not measurable; second, the event generator is supposed to have access to the state for all times, which would induce exorbitant communication costs. However, in the case when the system is observable, it is possible to estimate the state of a system using a state observer. Thus, in this present work, we show how to extend the event-triggered algorithm presented in [START_REF] Zobiri | Eventbased sampling algorithm for setpoint tracking using a state-feedback controller[END_REF] to the case of estimated state feedback.

II. A SIMPLE OBSERVER IMPLEMENTATION

The aim of this section is to show that a direct extension of our previous results, on the design of event-triggered controllers, to the case of estimated state feedback is not always possible. Thus, in the next section, we propose a new structure of the reference system to overcome this issue.

A. Event-triggering based on measured state feedback

Consider the LTI system

ẋ(t) = Ax(t) + Bu(t), y(t) = Cx(t), (1)
for which we want the output vector y(t) ∈ R m to follow a predefined reference vector r(t) ∈ R m . In this system, the state vector x(t) belongs to R n for all t ≥ 0, and the control signal u(t) ∈ R p . Therefore the state matrix A, input matrix B and observation matrix C respectively belong to M n×n (R), M n×p (R), and M m×n (R).

In [START_REF] Zobiri | Eventbased sampling algorithm for setpoint tracking using a state-feedback controller[END_REF] an event-based tracking strategy is investigated, by using a piecewise constant control signal ū(t) where for t ∈ [t k , t k+1 ) ū(t) = -Kx(t k ) + Gr(t k ) ≡ ūk , and the t k , k ∈ N, are the control update times. They correspond to time instants when an event-triggering condition is fulfilled. This condition involves the comparison of the state x(t), at each time instant, to the state of a reference system ẋr (t) = (A -BK)x r (t) + BGr(t), y r (t) = Cx r (t).

(

) 2
Under the assumption that the pair (A, B) is controllable and the pair (A, C) is observable, the feedback matrix K ∈ M p×n (R) is constructed for both systems such that A-BK is Hurwitz with prescribed eigenvalues. Defining the gain matrix

G ∈ M p×m (R) by G = -C(A -BK) -1 B -1 ,
we know that lim t→∞ y r (t) -r(t) = 0.

The strategy defined in [START_REF] Zobiri | Eventbased sampling algorithm for setpoint tracking using a state-feedback controller[END_REF] is proved to ensure that the tracking error x(t) -x r (t) remains small (i.e. x(t) -x r (t) ≤ ε) so that y(t) is also close to the reference r(t) at order ε as time goes to infinity.

However, the full state vector is not always available for measurement, which makes unworkable an event-triggering condition based on the full state. A possible solution to this problem is to estimate the value of the state through a state observer.

B. Construction of a state observer

We therefore first suggest to implement a state observer as

ẋ(t) = Ax(t) + B û(t) + L(y(t) -ŷ(t)), ŷ(t) = C x(t). (3)
where the observer gain L ∈ M n×m (R) is chosen such that A-LC is Hurwitz. In this system, the control is also piecewise constant and for t ∈ [t k , t k+1 ), û(t) = -K x(t k ) + Gr(t k ) is computed using the observed state. This control is sent and used by the real system. The tracking error which is used to define the control update instants t k is now taken as the difference between the observed state and the reference state

ê(t) = x(t) -x r (t).
The event-triggering condition is constructed from this error ê as it is done in [START_REF] Zobiri | Eventbased sampling algorithm for setpoint tracking using a state-feedback controller[END_REF] from x -x r . Namely a pseudo-Lyapunov function is constructed

V (ê(t)) = ê(t) T P ê(t),
where P is a positive definite matrix that satisfies the Lyapunov equation

(A -BK) T P + P (A -BK) = -Q,
and Q is a positive definite matrix. A new control is computed and applied when the following event occurs V (ê(t)) = δ. More precisely for a predefined δ > 0

t k = inf{t ≥ t k-1 such that V (ê(t)) ≥ δ}. ( 4
)
The construction of matrix P and functional V is quite classical. However we call V only a pseudo-Lyapunov function because the update condition in (4) only ensures that V remains below a threshold. This will be later seen in Figure 4 which shows a typical behavior of V for such an event-based update condition.

C. Event-triggering property

The previous tracking implementation is valid if it fulfills the following conditions

• it manages to track the reference trajectory,

• there is no Zeno phenomenon, i.e. the control updates are really discrete in time.

To ensure this, it is necessary that when V (ê) reaches the threshold δ and the control is updated, the function V (ê(t)) is pushed below the threshold, that is its time derivative is negative. Computing this derivative at time t k we obtain

dV (ê(t)) dt t=t k = -ê(t k ) T Qê(t k )+2ê(t k ) T P L(y(t k )-ŷ(t k )).
The output observation error y(t) -ŷ(t) is generally nonzero, and may be relatively large in transient time. Therefore we cannot ensure that the time derivative of the Lyapunov function is non positive at the update time t k . Let us show on an example that such a situation can occur.

D. Impact of the output observer error on the event-triggering strategy

To show that the above implementation is not valid as an event-triggered approach, we use a simple stabilization example from the literature (see e.g. [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]), i.e. we want to track the zero function. To this aim we consider the system

ẋ1 ẋ2 = 0 1 -2 3 x 1 x 2 + 0 1 u, (5)
Though the control manages to stabilize the system and drive the states to zero, it is updated at every sampling instant in transient time, contrary to the requirements of an eventbased scheme. Figure 2 shows the control signal and the Lyapunov function. We notice that there is first a transient part during which the Lyapunov function is much too large and therefore the update condition is always fulfilled, leading to a continuous control. This destroys the performance of the system even if it eventually shows event-based features at permanent regime.

III. A NEW APPROACH

A. Redefining the reference system

To get around the problem, we would like to simply have

dV (ê(t)) dt t=t k = -ê(t k ) T Qê(t k ). (6)
We can perform this without changing the event-triggering conditions, nor the observer system, but only the reference system. Indeed we will also feed the reference system with the output observation error, leading to the modified dynamics

ẋr (t) = (A -BK)x r (t) + BGr(t) + L(y(t) -ŷ(t)), y r (t) = Cx r (t). (7)
We now consider the systems (1), (3), and [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF] to track the reference trajectory r and show that is qualified as an event-based approach. The modified communication scheme is pictured in Figure 3.

B. Tracking property

The observation error e(t) = x(t) -x(t) has the following dynamics ė(t) = (A -LC)e(t), and since A -LC is Hurwitz, the observation error tends to zero as time goes to infinity:

lim t→∞ (x(t) -x(t)) = 0. (8)
Moreover, by construction of the control update events, V (ê(t)) ≤ δ. Indeed, when this condition is violated, the control is updated, and the derivative (6) is negative, pushing V (ê(t)) below the threshold again. This yields an upper bound for ê : if we denote λ min (P ) the lowest (positive) eigenvalue of P , for all times

x(t) -x r (t) ≤ δ λ min (P ) ≡ ε.
Equation ( 8) also implies that y(t) -ŷ(t) tends to zero as t goes to infinity. Therefore the dynamics of equation ( 7) tends to that of ẋr (t) = (A -BK)x r (t) + BGr(t) which is well known to ensure the tracking of r(t).

These three limits ensure that the output of the real system tracks r(t) with a precision of ε in the limit t → ∞. In the case of a stability problem, where r ≡ 0, this ensures that the real system is asymptotically stable with error ε.

C. Intersample delay

For the algorithm to be event-based, we have also to prove that there exists a minimum delay between two update instants, more precisely, there exists τ > 0 such that t k+1 -t k > τ for all k ∈ N.

To study this have to consider the derivative of ê for t ∈ [t k , t k+1 ):

ė(t) = (A -BK)ê(t) + B (v(t) -v(t k )) ,
where v(t) = -K x(t) + Gr(t). This leads to

dV (ê(t)) dt = -ê(t) T Qê(t) + 2ê(t) T P B(v(t) -v(t k )).
The proof follows exactly the same steps as in [START_REF] Zobiri | Eventbased sampling algorithm for setpoint tracking using a state-feedback controller[END_REF]. A minimum delay τ exists if r(t) is a Lipschitz function, which is the case in real systems for which a low pass filter is used on the reference input to avoid the saturation of actuators.

More precisely, we show that there exists a constant β that only depends on the threshold δ and on the eigenvalues of P and Q such that dV (ê(t))/dt ≤ -β over a time interval [t k , t k + τ 1 ] and τ 1 depends on the model parameters and the Lipschitz constant of the reference vector. Besides the minimum delay τ 1 is smaller than the actual dwelling time, since do not ask the pseudo-Lyapunov to decrease but allow it to slightly grow before a new update of the control.

D. Test on a stabilization problem

To test this approach, we first use it for the previous stabilization problem. Figure 4 shows the control signal and the time evolution of the Lyapunov function. Here V (ê(t)) remains below δ = 0.3 up to small numerical overshoots due to the fact that the numerical discretization does not catch exactly the event times. Although the controls of Figures 2 and4 seem very similar, on Figure 4 there are only 585 control updates for 300,000 sampling instants (we have set the simulation time step to ∆t = 10 -4 ).

We have chosen ∆t = 10 -4 to satisfy a trade-off between the preservation of dynamics properties of the continuous-time system in the performed numerical simulations and a reasonable computational time. Table I compares both approches for ∆t = 10 -4 , ∆t = 10 -3 , and ∆t = 10 -2 in terms of the number of control updates and the maximum of the pseudo-Lyapunov function. We analyze separately two parts in the time evolution of the systems: a transient part (roughly for t ∈ [0, 7.4]) where both strategies show different features, and a steady state part where they are essentially the same. In the transient part, the first strategy is very bad and the updates occur at each simulation time, which means that the control is continuous in time. As can be also seen in Figure 2 the pseudo-Lyapunov function V takes very large values during this part of the simulation. On the contrary, for the second approach using (7) for the reference system, the number of updates during the transient part is much lower and not directly connected to the number of simulation steps. Choosing a relatively small value of ∆t allows to capture the events more precisely. The events are always captured after they occur, and ∆t represents the maximum delay in the event detection. This explains that the maximum value for V in Table I is always greater that δ = 0.3. Detecting events in time leads to a better approximation of the dynamics and end up in a lower number of control updates in the steady state part.

V(e)

Fig. 4. The control signal (above) and the Lyapunov function for system (5) and control tracking implementation (1), (3), and [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF] Figure 5 shows the time evolution of the states of the real and observed systems. Both systems are stabilized to the precision induced by the choice of δ.

E. A tracking problem

To test tracking we use a control moment gyroscope [5] which is described by a 5-state system, with

A =       0 0 0 1 0 0 0 0 0 1 0 0 0 0 J D Ω I C +I D 0 0 0 0 0 0 0 -J D Ω I D +K A +K B +K C 0 0       , B =      0 0 0 0 0 1 I C +I D -1 J B +J C 0 0 0     
, C = 1 0 0 0 0 0 1 0 0 0 , and K A = 0.067, I B = 0.012, J B = 0.018, K B = 0.030, I C = 0.0092, J C = 0.023, K C = 0.022, I D = 0.015, J D = 0.027, Ω = 42. The gain matrix K is chosen in order to solve the linear quadratic regulator (LQR) problem which minimizes

∞ 0 (x T Q C x + u T Ru)dt with Q C = 100 C T C and R = 3 0 0 0.02 ,
For this system, the second actuator u 2 is more powerful than the first one u 1 . For this reason we have chosen unbalanced entries in matrix R in order to have a relatively smooth response y. Taking for example R = I would lead to large ripples on y 1 .

The observer gain matrix L is chosen allowing the online estimation of the state vector solving the problem

find x(t) minimizing E[(x(t) -x(t)) T (x(t) -x(t))],
where E(•) stands for the mathematical expectation of the square of the estimation error. To do that, a linear quadratic minimization problem design method is applied for which W = BB T and V = 0.1 I are the weighting matrices used to solve the associated Riccati equation.

Finally Q = 2 I is used to construct the Lyapunov matrix P and the control update threshold is δ = 10 -4 .

The target reference trajectory is a circle centered at the origin and we choose to let the real system depart from an initial point far off the target, namely from the origin. The time evolution of the two components of the output of the real system is plotted on Figure 6. There is a transient part during which the real system draws near the reference trajectory and then an almost periodic part during which the real system remains near the reference, as can be seen both on Figures 6 and7. The latter Figure displays the trajectories in the output phase plane (y 1 , y 2 ). The transient part itself can be split into two. Indeed, at the very beginning, the real system output remains constant. This is due to fact that matrix A does not lead to an unstable system. It has three zero and two conjugate pure imaginary eigenvalues, and therefore the free dynamics is a nutation. It takes time for the system to destabilize and for the pseudo Lyapunov function to reach δ. This is one of the reasons to choose a very small δ.

This computation, performed with a time step ∆t = 10 -3 , uses 325 control updates, which means updating the control 3.25% of the computation instants. This medium value for the time step fulfills our needs. A larger time step would mean missing the event times and therefore letting the system diverge too much from its prescribed trajectory. A smaller time step would render the numerical algorithm more stable and would lead to a longer first part of the transient behavior.

IV. CONCLUSION

We have been able to design an event-based tracking setup for linear time invariant systems able to follow a Lipschitz predefined reference trajectory. It also avoids the Zeno phenomenon. Its triggering mechanism only needs the knowledge of the measured output of the real system. In a further work we intend to extend this approach to the case when the real system is also subject to process and measurement noises. Our approach also makes use of the same matrices in the real system and the two other computed systems. An interesting question is the impact of model uncertainties on the control performances.

Figure 1 Fig. 1 .

 11 Figure1summarizes the communications between the systems (1)-(3), showing clearly that only the output of the real system is used by the two other systems. The dashed lines represent communications that only occur at control update times t k .

Fig. 2 .

 2 Fig.2. The control signal (above) and the Lyapunov function for system(5) and control tracking implementation (1)-[START_REF] Johan | Comparison of periodic and event based sampling for first-order stochastic systems[END_REF]

Fig. 3 .

 3 Fig.3. Communications between the real, reference and observer systems (1)-(7)-[START_REF] Johan | Comparison of periodic and event based sampling for first-order stochastic systems[END_REF]. Dashed lines are event-triggered communications. The new communication channel is thicker.

Fig. 5 .

 5 Fig. 5. Time evolution of the real (top) and observed (bottom) states

Fig. 6 .

 6 Fig. 6. Time evolution of the two outputs of the real and reference systems

Fig. 7 .

 7 Fig. 7. Trajectory of the outputs in the phase plane

TABLE I .

 I NUMBER OF CONTROL UPDATES AND MAXIMUM VALUES FOR V FOR BOTH APPROACHES.

ACKNOWLEDGMENT

This work has been partially supported by the LabEx PERSYVAL-Lab (ANR-11-61 LABX-0025-01).