Anne Houlès
email: houles.anne@gmail.com

François Joussemet
email: francois.joussemet@uclouvain.be

Bryan Vincent
email: bryan.vincent@cirad.fr

Laure Hannibal
email: laure.hannibal@ird.fr

Magali Patrois

Karine Gotty
email: karine.gotty@inrae.fr

Marc Ducousso
email: marc.ducousso@cirad.fr

Philippe Jourand
email: philippe.jourand@ird.fr

Michael Wink

Samuele Voyron

Erica Lumini

The Nurse Plant Acacia spirorbis Enriches Ectomycorrhizal Community Composition of a Target Species: Tristaniopsis calobuxus

Keywords: agroforestry, ectomycorrhizal share, forest nursery, fungal ITS, New Caledonia, tree legume, ultramafic soil

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

Introduction

Tropical ecosystems have been the subject of far fewer studies than temperate ecosystems [START_REF] Alexander | Mycorrhizas in Tropical Forests: A neglected research imperative[END_REF], thus our knowledge of the unique New Caledonian ecosystem is extremely new and fragmentary [START_REF] L'huillier | Mines et Environnement En Nouvelle-Calédonie: Les Milieux sur Substrats Ultramafiques et Leur Restauration[END_REF][START_REF] Pillon | Infertile landscapes on an old oceanic island: The biodiversity hotspot of New Caledonia[END_REF]. Recent work carried out in these ecosystems revealed the importance of ectomycorrhizal (ECM) symbiosis in New Caledonia (e.g., [START_REF] Amir | Les bactéries et les champignons du sol sur roches ultramafiques[END_REF][START_REF] Carriconde | Host effects in high ectomycorrhizal diversity tropical rainforests on ultramafic soils in New Caledonia[END_REF]). However, anthropogenic activities (bush fires, nickel mining and urbanization) are major threats to the preservation of these ecosystems in New Caledonia [START_REF] Jaffré | Threatened plants of New Caledonia: Is the system of protected areas adequate?[END_REF][START_REF] Jaffré | Threats to the conifer species found on New Caledonia's ultramafic massifs and proposals for urgently needed measures to improve their protection[END_REF][START_REF] Losfeld | Mining in New Caledonia: Environmental stakes and restoration opportunities[END_REF][START_REF] Ibanez | is the system of protected areas now adequate in New Caledonia?[END_REF], and so this tropical archipelago is now considered as a world biodiversity hot spot [START_REF] Myers | Biodiversity hotspots for conservation priorities[END_REF][START_REF] Isnard | How did the ultramafic soils shape the flora of the New Caledonian hotspot?[END_REF]. In this context, studies on the use of trees to rehabilitate mine sites began in the 1970s [START_REF] Jaffré | La végétalisation des anciens sites miniers[END_REF]. Since then, only two major endemic tree species, Acacia spirorbis Labill. and Casuarina collina J. Poiss. ex Pancher & Sebert, have been shown to be able to grow rapidly in such extreme soil conditions [START_REF] Jaffré | Sur la présence de deux Cupressaceae, Neocallitropsis pancheri (Carr.) Laubenf. et Libocedrus austrocaledonica Brongn. & Gris dans le Massif du Paéoua et localités nouvelles de Gymnospermes en Nouvelle-Calédonie[END_REF].

However, these two tree species are not naturally present in most original mine ecosystems nor do they enable the spontaneous return of the original native endemic species. Thus, even if these two species grow well in these harsh conditions, they do not provide satisfactory ecosystem services, notably, because natural plant successions do not occur spontaneously [START_REF] Mccoy | Fire and succession in the ultramafic maquis of New Caledonia[END_REF][START_REF] Wulff | Conservation priorities in a biodiversity hotspot: Analysis of narrow endemic plant species in New Caledonia[END_REF].

To test the setting of a controlled plant succession, we took the opportunity to use the remains of a 12-year-old field plantation of A. spirorbis, of which the ECM community was recently described [START_REF] Houlès | Ectomycorrhizal communities associated with the legume Acacia spirorbis growing on contrasted edaphic constraints in New Caledonia[END_REF]. In this plantation, the mortality after planting had left some scattered isolated trees on bare soil surrounded by the native endemic ecosystem locally dominated by the ECM Tristaniopsis guillainii Vieill. ex Brongn. & Gris. Considering the vicinity of sources of ECM fungal propagules and the dispersal abilities of ECM propagules [START_REF] Peay | Measuring ectomycorrhizal fungal dispersal: Macroecological patterns driven by microscopic propagules[END_REF], we assumed that natural diversified ECM inoculums would not be a limiting factor for nursery saplings being colonised by the variety of established ECM fungi. However, the conditions in nurseries, i.e., confined space, highly fertile soil, the occasional use of fungicides and abundant watering, prevent ectomycorrhizal interactions [START_REF] Vaario | The effect of nursery substrate and fertilization on the growth and ectomycorrhizal status of containerized and outplanted seedlings of Picea abies[END_REF][START_REF] Kazantseva | Effects of growth medium, nutrients, water, and aeration on mycorrhization and biomass allocation of greenhouse-grown interior Douglas-fir seedlings[END_REF], and mostly involve opportunistic species with high dispersal abilities [START_REF] El Karkouri | Diversity of ectomycorrhizal fungi naturally established on containerised Pinus seedlings in nursery conditions[END_REF][START_REF] Trocha | Genetic diversity of naturally established ectomycorrhizal fungi on Norway Spruce seedlings under nursery conditions[END_REF][START_REF] Fernández | Ectomycorrhizas naturally established in Nothofagus nervosa seedlings under different cultivation practices in a forest nursery[END_REF]. As result, ECM fungal communities associated with nursery saplings are markedly different from those encountered in the field [START_REF] Henry | Transfer to forest nurseries significantly affects mycorrhizal community composition of Asteropeia mcphersonii wildings[END_REF]. Transplanting from the nursery to the natural environment has been widely studied, especially in the framework of controlled inoculation trials using selected ECM fungal strains [START_REF] Selosse | Temporal persistence and spatial distribution of an American inoculant strain of the ectomycorrhizal basidiomycete Laccaria bicolor in a French forest plantation[END_REF][START_REF] Karkouri | Dominance of the mycorrhizal fungus Rhizopogon rubescens in a plantation of Pinus pinea seedlings inoculated with Suillus collinitus[END_REF]. Nevertheless, after planting, opportunistic nursery species must make way for indigenous species, to ensure the survival and development of the saplings.

In these conditions, following Walker and del Moral [START_REF] Walker | Primary Succession and Ecosystem Rehabilitation[END_REF], who defined facilitation as any positive influence of one species on another, we hypothesised that a species such as A. spirorbis could act as an ECM facilitator [START_REF] Nara | Ectomycorrhizal fungi on established shrubs facilitate subsequent seedling establishment of successional plant species[END_REF] for another species such as Tristaniopsis calobuxus Brongn. & Gris. ECM facilitation is effective only if most ECM fungi are not specific [START_REF] Richard | Diversity and specificity of ectomycorrhizal fungi retrieved from an old-growth mediterranean forest dominated by Quercus iIlex[END_REF][START_REF] Ryberg | Ectomycorrhizal diversity on Dryas octopetala and Salix reticulata in an alpine cliff ecosystem[END_REF][START_REF] Diédhiou | Multi-host ectomycorrhizal fungi are predominant in a Guinean tropical rainforest and shared between canopy trees and seedlings[END_REF][START_REF] Smith | Ectomycorrhizal fungal diversity and community structure on three co-occurring leguminous canopy tree species in a neotropical rainforest[END_REF][START_REF] Smith | The ectomycorrhizal fungal community in a neotropical forest dominated by the endemic dipterocarp Pakaraimaea dipterocarpacea[END_REF]. We thus needed to check the ability of ECM fungal species associated with A. spirorbis to associate with T. calobuxus before and after planting. The remains of a 12-year-old plantation of A. spirorbis established on bare soil on the top of the Koniambo Massif (altitude 851 m above sea level) was used to test the ability of A. spirorbis to act as a nurse plant.

Materials and Methods

Study Site and Experimental Design

The Koniambo Massif in New Caledonia is currently a nickel mine exploited by the company Koniambo Nickel SAS. The general features of the massif have been described in Perrier et al. [START_REF] Perrier | Occurence of mycorrhizal symbioses in the metal-rich lateritic soils of the Koniambo Massif, New Caledonia[END_REF]. The plantation site is a 60 × 30 m rectangle located on the Koniambo Massif, between the latitudes 21 • 00 20-21 S and longitudes 164 • 50 17-19 E at an altitude of 851 m. The main characteristics of the soil in the study site are listed in Table S1.

The experimental plantation was established in May 2015 and contained the remains of a former plantation of A. spirorbis dating from 2003 in the form of scattered 12-year-old trees. The site is surrounded by diversified maquis vegetation with the recurring presence of the ECM endemic shrub species T. guillainii. Such surrounding vegetation is a potential source of ECM inoculums for planted T. calobuxus saplings. We thus delimited three zones in the maquis in which T. guillainii was sampled to assess ECM fungal diversity (Figure 1).

A preliminary prospective study was designed to select the better species to test the ability of A. spirorbis to act as a nurse plant through the integration of the target species in a common ectomycorrhizal network, in respect to the project timeline.

Tristaniopsis calobuxus was growing quickly in nursery conditions and needed a shorter co-planting period compared with T. guillainii to obtain relevant results (data not shown). These two Tristaniopsis species share some similarities and have trouble growing in a post-mining ultramafic soil on their own. Therefore, we chose T. calobuxus rather than T. guillainii. 2a). The squares represent plots at least 10 m away from A. spirorbis (see Figure 2b). The circles outside planting areas are the exact positioning of harvested T. guillainii trees in the natural maquis surrounding the planting areas.

A preliminary prospective study was designed to select the better species to test the ability of A. spirorbis to act as a nurse plant through the integration of the target species in a common ectomycorrhizal network, in respect to the project timeline.

Tristaniopsis calobuxus was growing quickly in nursery conditions and needed a shorter co-planting period compared with T. guillainii to obtain relevant results (data not shown). These two Tristaniopsis species share some similarities and have trouble growing in a post-mining ultramafic soil on their own. Therefore, we chose T. calobuxus rather than T. guillainii.

In each zone, three T. guillainii were sampled, and the coordinates of each tree are given in Table S2. In these conditions, two types of plots were planted: one in the close vicinity of a 12-year-old A. spirorbis and the second outside the vicinity of an A. spirorbis. The first type of plantation was a circle with an A. spirorbis at the centre; T. calobuxus with three repetitions were planted on each radius, the first at 1 m, the second at 2 m and the third at 3 m from the limit of the A. spirorbis canopy; three A. spirorbis were not associated to T. calobuxus and were used as controls. The second type of plot was a standard 4 row/4 line plantation with 1 m space between the lines and the rows; 3 plots were planted with T. calobuxus, at least 10 m away from any 12-year-old A. spirorbis (Figure 2). The plant survival and growth (collar diameter and height) were measured quarterly after transplantation from May 2015 until July 2016 (Table S3).

Nursery Growth

All saplings came from the same nursery. Seeds used in this nursery are always sterilized as already described in Jourand et al. [START_REF] Jourand | Ectomycorrhizal Pisolithus albus inoculation of Acacias spirorbis and Eucalyptus globulus grown in ultramafic topsoil enhances plant growth and mineral nutrition while limits metal uptake[END_REF]. Briefly, seeds were pre-treated for 2 min in 70% ethanol supplemented with 0.1% (v/v) Tween 20, then surface sterilized for 5 min in H2O2 (30%) and washed 3 times in sterile water. In each zone, three T. guillainii were sampled, and the coordinates of each tree are given in Table S2. In these conditions, two types of plots were planted: one in the close vicinity of a 12-year-old A. spirorbis and the second outside the vicinity of an A. spirorbis. The first type of plantation was a circle with an A. spirorbis at the centre; T. calobuxus with three repetitions were planted on each radius, the first at 1 m, the second at 2 m and the third at 3 m from the limit of the A. spirorbis canopy; three A. spirorbis were not associated to T. calobuxus and were used as controls. The second type of plot was a standard 4 row/4 line plantation with 1 m space between the lines and the rows; 3 plots were planted with T. calobuxus, at least 10 m away from any 12-year-old A. spirorbis (Figure 2). The plant survival and growth (collar diameter and height) were measured quarterly after transplantation from May 2015 until July 2016 (Table S3).

Sampling Ectomycorrhizal Root Tips and Molecular Analyses

Nursery Growth

All saplings came from the same nursery. Seeds used in this nursery are always sterilized as already described in Jourand et al. [START_REF] Jourand | Ectomycorrhizal Pisolithus albus inoculation of Acacias spirorbis and Eucalyptus globulus grown in ultramafic topsoil enhances plant growth and mineral nutrition while limits metal uptake[END_REF]. Briefly, seeds were pre-treated for 2 min in 70% ethanol supplemented with 0.1% (v/v) Tween 20, then surface sterilized for 5 min in H 2 O 2 (30%) and washed 3 times in sterile water.

Sampling Ectomycorrhizal Root Tips and Molecular Analyses

In order to describe the original ECM fungal communities at planting time, in May 2015, 15 ECM root tips were sampled per plant for DNA analysis on three T. calobuxus saplings randomly selected in each plot (for a total of 3 × 6 × 15 = 270 root tips). Using the root tracking method described in Perrier et al. [START_REF] Perrier | Occurence of mycorrhizal symbioses in the metal-rich lateritic soils of the Koniambo Massif, New Caledonia[END_REF], 15 root tips from the nine A. spirorbis were used as nurse plant candidates and nine T. guillainii from the surrounding maquis (Figure 1 and Table S2) were harvested (total of 15 × (9 + 9) = 270 root tips).

To describe ECM fungal communities after a growth period of 13 months (July 2016), three randomly chosen T. calobuxus per plot were uprooted, and 15 root tips were harvested per plant (total of 3 × 6 × 15 = 270 root tips). A root tip was considered ECM when a fungal mantle surrounded it. The root tips were preserved in 2% cetyltrimethylammoniumbromide (CTAB), 100 mM Tris-HCl (pH 8.0), 1.4 M NaCl, 20 mM EDTA) at -20 • C until molecular analysis.

Using Sanger sequencing, each root tip was subjected to molecular analyses to characterise the ECM fungus putatively involved in the ECM symbiosis. For this purpose, the total DNA was extracted from each ECM root tip using the Wizard Genomic DNA Purification (Promega, Charbonnière-les-Bains, France) following the protocol described by Carriconde et al. [START_REF] Carriconde | Population evidence of cryptic species and geographical structure in the cosmopolitan ectomycorrhizal fungus, Tricholoma scalpturatum[END_REF]. We used the nuclear ribosomal internal transcribed spacer (ITS) as a universal fungal barcode [START_REF] Schoch | Nuclear ribosomal Internal Transcribed Spacer (ITS) region as a universal DNA barcode marker for fungi[END_REF]. A fragment of approximately 650 bp was amplified using primers ITS1F (5 -TTTCCGTAGGTGAACCT-3 ) [START_REF] Gardes | ITS primers with enhanced specificity for Basidiomycetes-Application to the identification of mycorrhizae and rusts[END_REF] and ITS4 (5 -TCCTCCGCTTATTGATATGC-3 ) [START_REF] White | Amplification and direct sequencing of fungal ribosomal RNA genes for phylogenetics[END_REF]. The PCR reaction used for this amplification was prepared in a total volume of 25 µL containing a 1 µL aliquot of genomic DNA diluted 1/20, 1 µM of each primer, 1.5 units of Taq DNA polymerase (Promega, Charbonnières, France), 5 µL of 5X Promega Taq polymerase buffer, 2 mM of MgCl 2 and 200 µM of dNTP Promega. Amplification was performed with a Eppendorf ® Mastercycler ® thermocycler (Eppendorf, Hamburg, Germany) programmed as follows: one cycle of 3 min at 94 • C, followed by 35 cycles of 45 s at 94 • C, 45 s at 55 • C, 1 min at 72 • C and a final extension of 10 min at 72 • C. PCR products were checked by electrophoresis on a 1% agarose gel (Sigma) in 0.5% TBE buffer with 2 µL ethidium bromide (BET, 10 µg/mL for 100 mL gel). A size marker 1000 pb (Promega) was used to estimate the weight of the different fragments. The DNA bands were then visualised by fluorescence under UV light and photographed. Only for samples with simple well-defined bands, bidirectional sequencing with primers ITS4 and ITS1F was carried out by an external service (Genoscreen, Lille, France) on an automated sequencer using genetic analysis based on capillary electrophoresis ABI3730XL.

The resulting sequences were analysed and assembled into contigs with Geneious ® 7.1.9 software (http://www.geneious.com, accessed on 6 July 2021) (Biomatters Ltd., Auckland, New Zealand). DNA sequences were assigned to operational taxonomic units (OTU) using a sequence identity cut-off of 97% of the ITS sequences, as reported in Smith et al. [START_REF] Smith | Ectomycorrhizal fungal diversity and community structure on three co-occurring leguminous canopy tree species in a neotropical rainforest[END_REF] using software CD-Hit Est ® [START_REF] Li | Cd-Hit: A fast program for clustering and comparing large sets of protein or nucleotide sequences[END_REF]. The taxonomic identification and lineage attribution of the generated OTUs were carried out using the UNITE database, in relation to the fungal nomenclature reported by Tedersoo et al. [START_REF] Tedersoo | Ectomycorrhizal lifestyle in fungi: Global diversity, distribution, and evolution of phylogenetic lineages[END_REF].

Data Analysis

Statistical analyses were performed in the R environment [START_REF] Core | R: A Language and Environment for Statistical Computing[END_REF], with the following packages: vegan [START_REF] Oksanen | Community Ecology Package[END_REF] and RVAidememoire [START_REF] Hervé | RVAideMemoire: Testing and Plotting Procedures for Biostatistics (0.9[END_REF]. To test the sampling method, the observed and estimated sample completeness curves for each plot were constructed using iNEXT online software (http://chao.stat.nthu.edu.tw (accessed on 6 July 2021; [START_REF] Chao | Statistical challenges of evaluating diversity patterns across environmental gradients in megadiverse communities[END_REF]). The species richness ( 0 D), Shannon ( 1 D) and Simpson ( 2 D) indexes were calculated using the iNEXT R package [START_REF] Hsieh | INEXT: An R package for rarefaction and extrapolation of species diversity (Hill numbers)[END_REF] in order to evaluate ECM fungal diversity among the treatments [START_REF] Zack | Fungal biodiversity and patterns[END_REF]. The estimated and asymptotic richness index 0 D counts species equally without considering their relative abundance, the Shannon diversity index 1 D counts species in proportion to their abundance, and the Simpson diversity index 2 D discounts all but the dominant species in the assemblage. Venn diagrams were created online, with the help of the Venny2 website [START_REF] Oliveros | An Interactive Tool for Comparing Lists with Venn Diagrams[END_REF]. The total species richness of each community was estimated using Specpool in the vegan package [START_REF] Oksanen | Community Ecology Package[END_REF] with the first-order jackknife estimator [START_REF] Smith | Nonparametric estimation of species richness[END_REF]. The global representation of the lineages identified in the sample is represented with an NMDS, based on a dissimilarity matrix of Bray-Curtis. Supplementary analyses such as Chi 2 tests, were carried out to test if several parameters, such as host specificity, the year of sampling, the distance of sampling from A. spirorbis, or the combination of all treatments could affect the structuration of ECM fungal communities. All OTUs belonging to the /non-ectomycorrhizal lineage were excluded from the analyses.

Results

Diversity and Comparison of ECM Fungal Communities Associated with the Different ECM Tree Species

From a total of 810 root tips, 312 ECM ITS DNA sequences were obtained. These sequences were merged into 97 ECM OTUs which were grouped into 11 fungal lineages. One representative sequence of each OTU was deposited in the DNA Data Bank of Japan (DDJB) with the accession numbers LC271284 to LC271380. The taxonomic identification of all OTUs is available in Table S4. Summarized data of the ITS sequences are presented in Table S5. The sampling effort is represented with completeness curves (Figure 3).

The distribution of lineages according to the OTUs is presented in Figure 4. At transplantation, we observed 26 OTUs grouped in seven lineages on A. spirorbis, 14 OTUs grouped in four lineages on T. guillainii, and five OTUs grouped in four lineages on T. calobuxus saplings. Thirteen months after transplanting, the number of OTUs associated with T. calobuxus was 21 (multiplied four times), and the number of shared OTUs between A. spirorbis and T. guillainii increased from one to five (Table 1; Figure 5). Concerning lineages, we observed four new lineages on the T. calobuxus 13 months after transplantation: /boletus, /pisolithus-scleroderma, /russulales and /coltricia. Except for /boletus, which was present only on T. guillainii, all other lineages were observed on A. spirorbis ECM (Figure 4). Table 1. Ecological diversity indexes of the ECM fungal OTUs according to all treatments. The names of the treatments include the host species (AS: A. spirorbis; TG: T. guillainii; TC: T. calobuxus), the year of sampling (2015 or 2016) and the distance of sampling from A. spirorbis, expressed in meters (where none correspond to TC grown in nursery). The estimated and asymptotic richness index 0 D counts species equally without considering their relative abundance, the Shannon diversity index 1 D counts species in proportion to their abundance, and the Simpson diversity index 2 D discounts all but the dominant species in the assemblage. As shown in Figure 5, the ECM OTUs of A. spirorbis (Figure 5A) and T. calobuxus (Figure 5B) shifted over time, with the emergence of 17 and 19 unique OTUs in 2016. Moreover, the number of ECM OTUs shared between A. spirorbis and T. calobuxus increased from one to five after one year of plantation (Figure 5C,D). Regarding the ECM OTUs of T. calobuxus in relation to the sampling distance from the nurse plant, 28.6% were shared between a 3 m radius and a sampling 10 m away (Figure 5E). A strong host effect was detected in 2015, where 43 of 44 OTUs were unique to their host species (Figure 5F).

Diversity Indexes

Treatment

A non-metric multidimensional scaling (NMDS) based on a dissimilarity matrix of Bray-Curtis revealed that ECM fungal communities differed in relation to the host species (A. spirorbis, T. calobuxus, or T. guillainii), and changed after a year of plantation (Figure 6). These results were further supported by Chi square tests (Table S6; must be read in parallel with the Venn diagrams (Figure 5)). As shown in Figure 5, the ECM OTUs of A. spirorbis (Figure 5A) and T. calobuxus (Figure 5B) shifted over time, with the emergence of 17 and 19 unique OTUs in 2016. Moreover, the number of ECM OTUs shared between A. spirorbis and T. calobuxus increased from one to five after one year of plantation (Figure 5C,D). Regarding the ECM OTUs of T. calobuxus in relation to the sampling distance from the nurse plant, 28.6% were shared between a 3 m radius and a sampling 10 m away (Figure 5E). A strong host effect was detected in 2015, where 43 of 44 OTUs were unique to their host species (Figure 5F).

A non-metric multidimensional scaling (NMDS) based on a dissimilarity matrix of Bray-Curtis revealed that ECM fungal communities differed in relation to the host species (A. spirorbis, T. calobuxus, or T. guillainii), and changed after a year of plantation (Figure 6). These results were further supported by Chi square tests (Table S6; must be read in parallel with the Venn diagrams (Figure 5)).

Discussion

First, in terms of ECM diversity, our results were consistent with those reported by Wasseem et al. [START_REF] Waseem | ectomycorrhizal fungal diversity associated with endemic Tristaniopsis spp. (Myrtaceae) in ultramafic and volcano-sedimentary soils in New Caledonia[END_REF] who recorded seven ECM lineages associated with T. guillainii, and with those of Houlès et al. [START_REF] Houlès | Ectomycorrhizal communities associated with the legume Acacia spirorbis growing on contrasted edaphic constraints in New Caledonia[END_REF] who observed 12 ECM lineages associated with A. spirorbis, data both collected partly from the Koniambo Massif where our field trial was located. In contrast, none of the ECM fungi were shared with Tristaniopsis in Bangka Island (Indonesia) [START_REF] Helbert; Turjaman | Ectomycorrhizal fungal communities of secondary tropical forests dominated by Tristaniopsis in Bangka Island, Indonesia[END_REF].

Discussion

First, in terms of ECM diversity, our results were consistent with those reported by Wasseem et al. [START_REF] Waseem | ectomycorrhizal fungal diversity associated with endemic Tristaniopsis spp. (Myrtaceae) in ultramafic and volcano-sedimentary soils in New Caledonia[END_REF] who recorded seven ECM lineages associated with T. guillainii, and with those of Houlès et al. [START_REF] Houlès | Ectomycorrhizal communities associated with the legume Acacia spirorbis growing on contrasted edaphic constraints in New Caledonia[END_REF] who observed 12 ECM lineages associated with A. spirorbis, data both collected partly from the Koniambo Massif where our field trial was located. In contrast, none of the ECM fungi were shared with Tristaniopsis in Bangka Island (Indonesia) [START_REF] Helbert; Turjaman | Ectomycorrhizal fungal communities of secondary tropical forests dominated by Tristaniopsis in Bangka Island, Indonesia[END_REF].

The spontaneous ECM communities of nursery saplings usually differ from those associated with plants growing in the natural environment [START_REF] El Karkouri | Diversity of ectomycorrhizal fungi naturally established on containerised Pinus seedlings in nursery conditions[END_REF][START_REF] Trocha | Genetic diversity of naturally established ectomycorrhizal fungi on Norway Spruce seedlings under nursery conditions[END_REF][START_REF] Fernández | Ectomycorrhizas naturally established in Nothofagus nervosa seedlings under different cultivation practices in a forest nursery[END_REF][START_REF] Henry | Transfer to forest nurseries significantly affects mycorrhizal community composition of Asteropeia mcphersonii wildings[END_REF][START_REF] Schmidt | Urbanization erodes ectomycorrhizal fungal diversity and may cause microbial communities to converge[END_REF]. Indeed, nursery conditions, i.e., confined space, high soil fertility and abundant watering, significantly influence mycorrhizal interactions resulting in the selection of opportunistic ECM species and contribute to low ECM fungal diversity [START_REF] Schmidt | Urbanization erodes ectomycorrhizal fungal diversity and may cause microbial communities to converge[END_REF][START_REF] Rudawska | Ectomycorrhizal Fungal Assemblages of nursery-grown Scots pine are influenced by age of the seedlings[END_REF]. In the absence of any inoculation practices, we can assume that, at the end of the nursery stage, the T. calobuxus ECM fungal community was composed of ruderal species, which are easily dispersed by spores [START_REF] Peay | Measuring ectomycorrhizal fungal dispersal: Macroecological patterns driven by microscopic propagules[END_REF][START_REF] Tedersoo | Ectomycorrhizal lifestyle in fungi: Global diversity, distribution, and evolution of phylogenetic lineages[END_REF] and mostly adapted to survive in nursery conditions. Our observations of ECM diversity associated with T. calobuxus before planting were consistent with the observations reported by these authors.

Field conditions may eliminate or at least reduce populations of opportunistic nursery fungi to select typical forest ECM species [START_REF] Hilszczanska | Persistence of ectomycorrhizas by Telephora terrestris on outplanted Scots pine seedlings[END_REF]. At thirteen months after transplantation, the number of OTUs associated with T. calobuxus doubled and the number of OTUs shared with A. spirorbis increased from four to 10, also consistent with marked changes in the lineages' composition. Notably, we observed the disappearance of one ruderal lineage: /sebacina. At the same time, in T. calobuxus roots, we observed the appearance of new lineages shared with A. spirorbis such as /boletus, /scleroderma-pisolithus and /russula. These changes in the ECM fungal community associated with T. calobuxus suggest that 13 months after transplantation, T. calobuxus was at least partially integrated with the common ECM network previously established by A. spirorbis. The decrease in the number of nursery OTUs was probably filtered by the in situ soil properties.

Saplings planted at least 10 m away from A. spirorbis seemed to have better growth. Although the decrease in height was not significant in other treatments, two hypotheses could explain this observation: (i) higher competition to access resources under A. spirorbis coverage, and (ii) a shading effect of A. spirorbis on other T. calobuxus.

Considering our results from this field experiment, and our experience in ecological restoration in extreme New Caledonian soils whose aim is to recreate the previous plant diversity (or a similar one) and maintain it through time, we recommend as the first step, planting A. spirorbis with 1 m by 2 m spacing. At this density (5000 trees per hectare), the canopy will rapidly close, litter will begin to accumulate, and a new mycorrhizal network will establish itself.

Between three and five years after planting, thinning should be carried out to leave A. spirorbis with 10 m by 10 m spacing. A range of endemic species should be planted within one to five metres of the remaining A. spirorbis. Beyond the direct influence of the remaining A. spirorbis, it would be better to plant endemic ECM species such as Tristaniopsis spp.

Although there are no OTUs shared between T. guillainii and T. calobuxus at first, this situation may evolve according to time. One example is the dynamic of mycorrhization of T. calobuxus, where more and novel OTUs were found after one year of plantation (Figure 6). Reforestation is a long process, where each species brings their own benefits to the ecosystem. After some time, plants can be interconnected through a mycorrhizal network, increasing their resilience. Even if some species (e.g., T. guillainii and T. calobuxus) do not share OTUs at first, they could still be interconnected later through a mycorrhizal network including A. spirorbis. Moreover, other parameters as important as ECM should also be considered, such as arbuscular mycorrhizal fungi, plant growth promoting bacteria, etc. To conclude, we would recommend the use of both Tristaniopsis species in the reforestation process, even if they do not share common ECM OTUs at first. Sannantha spp. or Melaleuca spp. could even benefit from the established ECM network.

Figure 1 .

 1 Figure 1. View of plantation site and surrounding area on the Koniambo Massif, New Caledonia. Plantation plots are located inside areas surrounded by black lines. The circles inside planting areas represent plots planted in a radius around A. spirorbis (see Figure2a). The squares represent plots at least 10 m away from A. spirorbis (see Figure2b). The circles outside planting areas are the exact positioning of harvested T. guillainii trees in the natural maquis surrounding the planting areas.

Figure 1 .

 1 Figure 1. View of plantation site and surrounding area on the Koniambo Massif, New Caledonia. Plantation plots are located inside areas surrounded by black lines. The circles inside planting areas represent plots planted in a radius around A. spirorbis (see Figure2a). The squares represent plots at least 10 m away from A. spirorbis (see Figure2b). The circles outside planting areas are the exact positioning of harvested T. guillainii trees in the natural maquis surrounding the planting areas.Diversity 2022, 14, x FOR PEER REVIEW 4 of 14

Figure 2 .

 2 Figure 2. Schemes of the planting plot designs. (a) Using present A. spirorbis with the green circle representing the canopy, the target species, represented with grey circles, are planted in a radius around A. spirorbis trees at 3 different distances: the black line represents an area with 6 target species, planted at 1 m from the A. spirorbis canopy limit, the red line is at 2 m, and the blue line is at 3 m. (b) On bare soil at least 10 m away from the closest A. spirorbis, the target species, represented with grey circles, are planted in quadra of 4 plants by 4, spaced 1 m from each other.

Figure 2 .

 2 Figure 2. Schemes of the planting plot designs. (a) Using present A. spirorbis with the green circle representing the canopy, the target species, represented with grey circles, are planted in a radius around A. spirorbis trees at 3 different distances: the black line represents an area with 6 target species, planted at 1 m from the A. spirorbis canopy limit, the red line is at 2 m, and the blue line is at 3 m. (b) On bare soil at least 10 m away from the closest A. spirorbis, the target species, represented with grey circles, are planted in quadra of 4 plants by 4, spaced 1 m from each other.

Figure 3 .

 3 Figure 3. Non-asymptotic approach based on interpolation and extrapolation of species diversity and sample coverage based on iNEXT online software (http://chao.stat.nthu.edu.tw, accessed on 6 July 2021) as reported by Chao et al. [44]. Solid lines represent rarefaction of the sampling curves, the observed sampling effort; dotted lines represent extrapolation of the sampling curves, based on Hill numbers for q = 0.1 and 2. (A) Sample completeness curve according to host tree species. (B) Coverage-based rarefaction and extrapolation sampling curve according to the host tree species. Legend: AS = A. spirorbis; TC = T. calobuxus; TG = T. guillainii.

Figure 4 .

 4 Figure 4. Ectomycorrhizal lineages found in nurse trees (A. spirorbis = AS) and target plants (T. calobuxus = TC) at successive distances (1, 2, 3 and 10 m away from A. spirorbis) after one year of plantation. Bar name corresponds to the plant (AS or TC), the year of sampling (2016) and the minimum distance of sampling from AS (0 m for AS; 1, 2, 3 and 10 m for TC; more than 10 m for TG). Data are presented according to the true abundance of lineages (A), and relative abundance expressed in % (B).

Table 1 .Figure 4 .

 14 Figure 4. Ectomycorrhizal lineages found in nurse trees (A. spirorbis = AS) and target plants (T. calobuxus = TC) at successive distances (1, 2, 3 and 10 m away from A. spirorbis) after one year of plantation. Bar name corresponds to the plant (AS or TC), the year of sampling (2016) and the minimum distance of sampling from AS (0 m for AS; 1, 2, 3 and 10 m for TC; more than 10 m for TG). Data are presented according to the true abundance of lineages (A), and relative abundance expressed in % (B).

Figure 5 .

 5 Figure 5. Venn diagrams of the ectomycorrhizal communities (based on OTUs) of A. spirorbis and Tristaniopsis species, in relation to host and year. Data are presented in multiple ways, as follows: (i) evolution of OTUs from 2015 to 2016 in A. spirorbis (A) and T. calobuxus (B), (ii) shared and unique OTUs between A. spirorbis and T. calobuxus in 2015 (C) and 2016 (D), (iii) shared and unique OTUs found between T. calobuxus planted inside a 3 m radius around A. spirorbis, versus T. calobuxus

Figure 5 .

 5 Figure 5. Venn diagrams of the ectomycorrhizal communities (based on OTUs) of A. spirorbis and Tristaniopsis species, in relation to host and year. Data are presented in multiple ways, as follows: (i) evolution of OTUs from 2015 to 2016 in A. spirorbis (A) and T. calobuxus (B), (ii) shared and unique OTUs between A. spirorbis and T. calobuxus in 2015 (C) and 2016 (D), (iii) shared and unique OTUs found between T. calobuxus planted inside a 3 m radius around A. spirorbis, versus T. calobuxus planted in square plots at 10 m away from A. spirorbis (E), and (iv) the shared and unique OTUs at the beginning of the experiment in 2015, between A. spirorbis, T. calobuxus, and T. guillainii (F).

  Diversity 2022, 14, x FOR PEER REVIEW 10 of 14 planted in square plots at 10 m away from A. spirorbis (E), and (iv) the shared and unique OTUs at the beginning of the experiment in 2015, between A. spirorbis, T. calobuxus, and T. guillainii (F).

Figure 6 .

 6 Figure 6. Non-metric multidimensional scaling representation of the ectomycorrhizal lineages (red cross and text) found in all samples. Hosts are symbolized as follows: A. spirorbis with circles, T. calobuxus with triangles, and T. guillainii with a diamond. Two different years of sampling (2015 and 2016) are indicated in blue and orange, respectively.

Figure 6 .

 6 Figure 6. Non-metric multidimensional scaling representation of the ectomycorrhizal lineages (red cross and text) found in all samples. Hosts are symbolized as follows: A. spirorbis with circles, T. calobuxus with triangles, and T. guillainii with a diamond. Two different years of sampling (2015 and 2016) are indicated in blue and orange, respectively.

Acknowledgments:

The authors wish to thank the Plate Forme du Vivant IRD Nouméa for access to their laboratory and IAC for the supply of T. calobuxus saplings. This work was conducted as part of a PhD supported by KNS in the framework of a CIFRE agreement (CIFRE 2013/1434).

Funding: This research was funded by the projects ANR-12-ADAP-0017 ADASPIR and GIP Nickel and its Environment GIPCNRT98 BIOINDIC. This work was conducted as part of a PhD supported by KNS in the framework of a CIFRE agreement (CIFRE 2013/1434).

Institutional Review Board Statement: Not applicable.

Data Availability Statement: ITS sequences of ectomycorrhizal root tips were deposited on DDBJ (http://www.ddbj.nig.ac.jp/, accessed on 5 September 2017).

Supplementary Materials:

The following are available online at https://www.mdpi.com/article/ 10.3390/d14020107/s1, Table S1: Chemical characteristics of soil from the plantation site located on the top of the Koniambo Massif, New Caledonia; Table S2: Coordinates of the 9 T. guillainii sampled for the characterization of their associated ECM fungal partners; Table S3: Average plant growth and survival of T. calobuxus according to plantation plots and distance from the A. spirorbis canopy; Table S4: Taxonomic and lineage identification of all OTUs according to UNITE database. Non-ectomycorrhizal lineages are not included in ECM community analyses; Table S5: Summary data of ITS sequences of ectomycorrhizal root tips deposited on DDBJ (http://www.ddbj.nig.ac.jp/, accessed on 5 September 2017); Table S6: Results of Chi square tests multiple comparisons of ectomycorrhizal lineages. Several data pools were created, to determine if the lineage community differs in relation to multiple factors: (A) host specificity (AS = Acacia spirorbis; TG = Tristaniopsis guillainii; TC = T. calobuxus), (B) the year of sampling (2015 versus 2016), (C) the distance of sampling from AS, expressed in meters (the "none" level corresponds to TG grown in nursery), and (D) the combination of all factors mentioned above (host + year + distance). For each

Conflicts of Interest:

The authors declare no conflict of interest.


Hybrid control of self-oscillating resonant converters Nicola Zaupa, Luis Martínez-Salamero, Senior Member, IEEE, Carlos Olalla, Member, IEEE, and Luca Zaccarian, Fellow, IEEE Abstract-We describe parallel and series resonant converters via a unified set of input-dependent coordinates whose dynamics is intrinsically hybrid. We then propose a hybrid feedback showing a self-oscillating behavior whose amplitude and frequency can be adjusted by a reference input ranging from zero to π. For any reference value in that range we give a Lyapunov function certifying the existence of a unique nontrivial hybrid limit cycle whose basin of attraction is global except for the origin. Our results are confirmed by experimental results on a series resonant converter prototype.

Index Terms-Lyapunov methods, Stability of hybrid systems, Resonant converters, Power electronics, Power conversion.

I. INTRODUCTION

R ESONANT converters are well-known in the power elec- tronics industry due to numerous advantages, including soft-switching and high power densities, as confirmed by the including inductive heating and battery charging [START_REF] Lucía | Efficiency-oriented design of ZVS half-bridge series resonant inverter with variable frequency duty cycle control[END_REF], [START_REF] Park | Design and control of a bidirectional resonant DC-DC converter for automotive engine/battery hybrid power generators[END_REF] context, to cite a few. Their conventional control includes frequency and amplitude modulation, typically relying on small signal models, which are dependent on the operating point. Early works focus their attention on unifying the characterization of several topologies using high-order models, which are then analyzed using first-harmonic approaches or linearization [START_REF] Bhat | A unified approach for the steady-state analysis of resonant converters[END_REF], [START_REF] Cutrona | A unified approach to series, parallel and series-parallel resonant converters[END_REF]. As an alternative, several state planebased approaches have been proposed, where the input switching is derived from current and voltage measurements in the converter. In state-plane control approaches, the dynamics can be simpler, streamlining the design of controllers, which can show improved regulation performance and a more immediate hardware realization. Some of these approaches drive the resonant converter in a self-oscillating way, without the need of external oscillators. One of the inherent advantages of selfoscillation is that the converter can be operated at the resonant frequency with no compensation for parameters variations in the components of the resonant tank. State-plane trajectories were used in [START_REF] Sooksatra | A unified approach to the classification and analyses of resonant converters[END_REF] to describe the steady-state operating point with conventional controllers, but the potential of state planebased switching was only explored in follow up works. A This work has been partially sponsored by the Spanish Ministry of Science and Innovation under project PID2019-111443RB-100, by the international exchange program of the University of Trento (Italy), and by ANR via grant HANDY, number ANR-18-CE40-0010. (Corresponding author: Nicola Zaupa.)

Nicola Zaupa was with Department of Industrial Engineering, Università di Trento, 38122 Trento, Italy. He is now with CNRS-LAAS, Université Toulouse III, 31400 Toulouse, France. (e-mail: nicola.zaupa@laas.fr).

Luis Martínez-Salamero and Carlos Olalla are with the Departament d'Enginyeria Electrònica, Elèctrica i Automàtica, Escola Tècnica Superior d'Enginyeria, Universitat Rovira i Virgili, 43007 Tarragona, Spain (e-mail: luis.martinez@urv.cat; carlos.olalla@urv.cat).

Luca Zaccarian is with CNRS-LAAS, Université de Toulouse, 31400 Toulouse, France, and also with Department of Industrial Engineering, Università di Trento, 38122 Trento, Italy (e-mail: zaccarian@laas.fr). trajectory tracking approach is used in [START_REF] Silva-Ortigoza | Control of a parallel resonant inverter[END_REF], where no selfoscillating mechanism is exploited. Self-oscillation is instead adopted in [START_REF] Sira-Ramírez | On the control of the resonant converter: a hybrid-flatness approach[END_REF], where a state plane-based solution is proposed, based on two controllers: one for the startup and one for the steady-state. The stability of the induced limit cycle with the switching mechanism for the start-up is analyzed in [START_REF] Hernández | On the stability of limit cycles in resonant dc-to-dc power converters[END_REF] with a Poincaré map. Further developments in that direction include [START_REF] Molla-Ahmadian | Hybrid modeling of a DC-DC series resonant converter: Direct piecewise affine approach[END_REF]- [START_REF] Afshang | Hybrid control of the DC-DC series resonant converter operating below resonance[END_REF] that analyze series resonant converters with switched affine models, followed by [START_REF] Bonache-Samaniego | Design of self-oscillating resonant converters based on a variable structure systems approach[END_REF]- [START_REF] Aroudi | Suppression of undesired attractors in a selfoscillating H-Bridge parallel resonant converters under zero current switching control[END_REF] that demonstrate that state-plane based methods can outperform the dynamics of conventional approaches. Nonetheless, the above state-plane control solutions also have limitations, one of them (resolved here) being the assumption that the resonant tank has a sufficiently high quality factor Q [START_REF] Bonache-Samaniego | Design of self-oscillating resonant converters based on a variable structure systems approach[END_REF], [START_REF] Aroudi | Suppression of undesired attractors in a selfoscillating H-Bridge parallel resonant converters under zero current switching control[END_REF].

For providing rigorous guarantees, a full nonlinear dynamics perspective on these ON-OFF type of feedbacks naturally calls for the use of a hybrid dynamical systems formalism: an area where powerful stability analysis tools have been recently developed in [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF] with a Lyapunov approach. Several relevant power electronics challenges have been addressed with these tools recently, such as the hybrid control of inverters in [START_REF] Chai | A robust hybrid control algorithm for a singlephase DC/AC inverter with variable input voltage[END_REF], [START_REF] Torquati | A hybrid predictive control algorithm for tracking in a single-phase DC/AC inverter[END_REF] and DC-DC converters in [START_REF] Theunisse | Robust global stabilization of the DC-DC boost converter via hybrid control[END_REF]- [START_REF] Sferlazza | A hybrid control strategy for quadratic boost converters with inductor currents estimation[END_REF]. Nonetheless, these approaches are not applicable to resonant conversion, where the switching frequency does not correspond to a small ripple, but to the main AC component of the power transfer.

In our preliminary conference results [START_REF] Zaupa | Results on hybrid control of self-oscillating resonant converters[END_REF], we proposed a novel unifying second-order input-dependent (or hybrid) coordinate transformation to analyze the dynamics of secondorder parallel and series resonant converters (PRC, SRC). Differently from the phasor circuit-based steady-state analysis leading to the high-order unifying models in [START_REF] Bhat | A unified approach for the steady-state analysis of resonant converters[END_REF], [START_REF] Cutrona | A unified approach to series, parallel and series-parallel resonant converters[END_REF], our stateplane low-dimensional coordinates allow specifying a stateplane control law based on a reference input corresponding to an angle θ ∈ (0, π], identifying suitable sectors in this new hybrid coordinate system, independently of the load. It is worth pointing out that thanks to the proposed approach the dynamics are unified, but not in the sense of a unique model that can be particularized for different cases. The proposed change of coordinates describes both converters with the same dynamics equations and without increasing their order. Different from the two-controllers scheme of [START_REF] Sira-Ramírez | On the control of the resonant converter: a hybrid-flatness approach[END_REF], our self-oscillating solution only requires one controller. Moreover, our self-oscillation mechanism overcomes the stringent bound on the quality factor assumed in [START_REF] Bonache-Samaniego | Design of self-oscillating resonant converters based on a variable structure systems approach[END_REF], [START_REF] Aroudi | Suppression of undesired attractors in a selfoscillating H-Bridge parallel resonant converters under zero current switching control[END_REF]: our only mild requirement on the quality factor is that the dynamics of the resonant tank must be underdamped. In [START_REF] Zaupa | Results on hybrid control of self-oscillating resonant converters[END_REF], we also discussed in depth the PRC configuration, showing a desirable dependence of the output amplitude and frequency on the reference input θ. We also proved rigorously the existence of a unique nontrivial hybrid limit cycle, but only for the specific case θ = π 2 , leaving the same result for the whole range θ ∈ (0, π] as a conjecture.

v s C L H-BRIDGE R V g Resonant Tank i C v C i L v s C L H-BRIDGE R V g v C Resonant Tank i C
In this work we extend [START_REF] Zaupa | Results on hybrid control of self-oscillating resonant converters[END_REF] in several important directions: first, we generalize the main theorem of [START_REF] Zaupa | Results on hybrid control of self-oscillating resonant converters[END_REF] about the existence of the unique limit cycle and our new proof, based on a different Lyapunov argument, applies to any reference θ ∈ (0, π]. Then, while [START_REF] Zaupa | Results on hybrid control of self-oscillating resonant converters[END_REF] only reported simulation results, we report here on the development of an experimental device implementing our feedback on a SRC architecture. Finally, since only the PRC solution was simulated in [START_REF] Zaupa | Results on hybrid control of self-oscillating resonant converters[END_REF], we perform SRC simulations here and show desirable matching of the simulation and experimental results.

The paper is structured as follows. The unifying coordinate representation is presented in Section II and the related hybrid system in Section III. Section IV states and proves our main theoretical result and Section V contains the controller implementation. The experimental prototype is described in Section VI with comparative simulations and experiments.

Notation. R (R >0 ) [R ≥0 ] and Z (Z >0 ) [Z ≥0 ] are the sets of (positive) [non-negative] real and integer numbers. R n denotes the n-dimensional Euclidean space. Given two vectors u ∈ R n and w ∈ R m , u ⊤ denotes the transpose of u, and (u, w) := u ⊤ w ⊤ ⊤ denotes their stacking. Given a (continuous, discrete, or hybrid) signal x, ẋ denotes its derivative with respect to continuous time t, while x + denotes its next value with respect to discrete time j.

II. UNIFYING INPUT-DEPENDENT COORDINATE CHANGES

We address parallel and series resonant converters (PRC and SRC, respectively), whose circuits are shown in Fig. 1, where v C and i C denote the voltage and the current in the capacitor. Both configurations exhibit a (parallel or series) resonant tank driven by an H-bridge applying a supply voltage v s equal to either V g or -V g to the left terminal, where V g is an external DC supply. The H-bridge is modeled here by a binary variable σ ∈ {-1, 1} describing the switch position, so that v s = V g when σ = 1 and v s = -V g when σ = -1 (in summary v s = σV g ).

The linear equations governing the current and voltage evolution of v C and i C for the parallel configuration at the top of Fig. 1 are the following ones:

L di L dt = σV g -v C , C dv C dt = i L - v C R . (1)
We emphasize that the last term, i L -v C R , is the current i C flowing in the capacitor. Similarly, the linear equations governing the series configuration at the bottom of Fig. 1 correspond to:

L di L dt = σV g -v C -Ri L , C dv C dt = i C . (2)
Note that in this case i L = i C . The novel approach proposed there stems from introducing the next input-dependent quantities for the converters

z 1 := v C V g -σ, z 2 := 1 V g L C i C , (3)
the first one clearly corresponding to a transformed voltage and the second one being a transformed current. Keeping in mind that any variation of σ ∈ {-1, 1} must be instantaneous, so that σ = 0, we may compute the differential equations governing the evolution of variables z := (z 1 , z 2 ) in (3). As shown in [START_REF] Zaupa | Results on hybrid control of self-oscillating resonant converters[END_REF], the dynamics is the same for the two circuits and corresponds to the following damped oscillator

ż1 = ωz 2 , ż2 = -ωz 1 -βz 2 , (4)
where

ω := √ LC -1
is the natural frequency and β > 0 is the inverse of the time constant of the exponential decay associated with each one of the linear circuits:

β PRC := 1 RC , β SRC := R L . (5)
Since the coordinates (z 1 , z 2 ) depend on the input σ, they experience an instantaneous change when σ is toggled. In particular, since σ toggles between -1 and +1, namely σ + = -σ, then (3) provides, for both circuits:

(z + 1 , z + 2 , σ + ) = (z 1 + 2σ, z 2 , -σ), (6)
where we emphasize that σ represents the switch position before the update and σ + represents its position after the update (and similarly for the other variables).

III. HYBRID MODEL AND PROPOSED SWITCHING LAW

Summarizing the previous section we may build a hybrid model of both PRC and SRC, with the notation of [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF] where z = (z 1 , z 2 ) ∈ R 2 is a physical state related to the current and voltage in the circuit, and σ ∈ {-1, 1} is a logical state representing the position of the switch:

ż σ = f (z, σ) := Az 0 (z, σ) ∈ C(θ) (7a)
z + σ + = g(z, σ) :=   z + 2σ 0 -σ   (z, σ) ∈ D(θ), (7b)
where

A := 0 ω -ω -β , ω := ( √ LC) -1 > 0 is the natural frequency of the converter and β > 0 suitably defined based on σ z 1 z 2 D1(3π/4) D-1(3π/4) 2 2 z 1 z 2 σ z 1 z 2 z 1 z 2 θ 2 2 θ D1(π/4) C-1(π/4) C1(π/4) D-1(π/4) C1(3π/4) C-1(3π/4)
Fig. 2. Two possible solutions of system ( 7), [START_REF] Hernández | On the stability of limit cycles in resonant dc-to-dc power converters[END_REF]

with θ < π 2 (left) and θ > π 2 (right).
the PRC or SRC configuration (see ( 5)) represents the internal dissipation.

The sets C and D in ( 7) are called, respectively, "flow set" and "jump set" and they are suitable subsets of R 2 × {-1, 1} to be designed in this section. Their intuitive meaning is that whenever the (augmented) state ξ := (z, σ) belongs to D, it is time to change the switch position in the H-bridge driving the converter, whereas as long as ξ ∈ C, one may let the converter evolve continuously without changing the switch position.

As a consequence, the selection of C and D implicitly defines a feedback controller because it characterizes under what conditions the converter input should be toggled (or left unchanged). In this section we design C and D as in [START_REF] Zaupa | Results on hybrid control of self-oscillating resonant converters[END_REF], based on a reference input θ ∈ (0, π] (called "switching angle" in [START_REF] Zaupa | Results on hybrid control of self-oscillating resonant converters[END_REF]) capable of inducing different output amplitudes and frequencies in the self-oscillating behavior. We select the jump and flow sets as:

C(θ) := C 1 (θ) ∪ C -1 (θ), D(θ) := D 1 (θ) ∪ D -1 (θ). (8a)
where each set C 1 and C -1 denotes a half plane and D 1 and D -1 are two half lines delimiting the flow sets, namely for each q ∈ {1, -1},

C q (θ) := {(z, σ) : σ = q, σ(z 1 sin θ + z 2 cos θ) ≤ 0}, (8b) D q (θ) := {(z, σ) : σ = q, σz 2 ≥ 0, z 1 sin θ + z 2 cos θ = 0}. (8c)
Two possible closed-loop solutions induced by ( 8) are illustrated in Fig. 2, where they evolve in the three-dimensional state-space (z 1 , z 2 , σ). The left figure represents a scenario with the reference θ < π 2 while the right figure corresponds to θ > π 2 . During flowing (in C(θ)), the continuous evolution revolves in the clockwise direction. Switching always occurs when the continuous motion hits the tilted solid line, because from that line flowing in C(θ) is not possible, unless the state σ is toggled. When a switch occurs, the z 1 voltage is shifted horizontally by two units (see ( 6)), and the specific choice of C(θ) and D(θ) ensures that those shifts always map to the interior of C(θ), from where flowing should occur. The choice to split D in two half lines regularizes the domain avoiding Zeno solutions with θ = π. The angle θ controls the tilting of the solid line, namely the subspace where the switch takes place. It is apparent that with small values of θ (left case in Fig. 2) solutions flow only in a small portion of the phase plane while for larger values of θ flowing is allowed in a larger range of phases, up to the limit case θ = π, where the solution spans all the possible phases while flowing.

IV. ASYMTOTICALLY STABLE HYBRID LIMIT CYCLE

A. Main Stability Theorem

According to [START_REF] Bisoffi | Relay-based hybrid control of minimal-order mechanical systems with applications[END_REF], the notion of periodicity for a hybrid trajectory is reported below. Alternative equivalent definitions are given in [START_REF] Lou | Robust stability of hybrid limit cycles with multiple jumps in hybrid dynamical systems[END_REF].

Definition 1 Given an hybrid system H = (C, f, D, g), a nontrivial hybrid periodic trajectory φ is a complete solution (namely, a solution that evolves forever) that is not identically zero and for which there exists a pair (T, J) ∈ R ≥0 × Z ≥0 satisfying T + J > 0, such that (t, j) ∈ dom(φ) implies (t + T, j + J) ∈ dom(φ) and, moreover,

φ(t, j) = φ(t + T, j + J) (9)
The image of φ is a nontrivial hybrid periodic orbit.

The next assumption on the parameters of ( 7) is necessary for the existence of a nontrivial hybrid periodic trajectory.

Assumption 1 Parameters ω and β are strictly positive reals. Moreover, the relation β < 2ω is satisfied, namely the resonant tank is underdamped. Equivalently, the roots of s 2 +βs+ω 2 = 0 are complex conjugate.

The following theorem, whose proof is reported in Section IV-B, provides a justification for the proposed selfoscillating control law. In our preliminary work [START_REF] Zaupa | Results on hybrid control of self-oscillating resonant converters[END_REF] we reported its proof only for the case θ = π 2 . We prove it here for any value of θ ∈ (0, π].

Theorem 1 Under Assumption 1, for each selection of θ ∈ (0, π], the closed loop (7), ( 8) has a unique nontrivial hybrid periodic orbit O θ that is stable and almost globally attractive (its basin of attraction includes all points such that z ̸ = 0).

Moreover, the nontrivial hybrid periodic trajectories of (7), [START_REF] Hernández | On the stability of limit cycles in resonant dc-to-dc power converters[END_REF] are characterized by J = 2 and T (θ) > 0, and exhibit periodic jumps interlaced by flowing intervals of length T (θ)/2.

Remark 1 Assumption 1 imposes constraints on the physical components to ensure that a natural oscillatory motion occurs, as certified by Theorem 1. For the two circuit configurations in Fig. 1, the constraint β < 2ω corresponds to:

P RC : 2R > L C SRC : R 2 < L C . (10)
Requirements [START_REF] Molla-Ahmadian | Hybrid control of DC-DC series resonant converters: The direct piecewise affine approach[END_REF] are reasonable since the effect of the load R must be sufficiently small to not destroy the natural oscillatory behavior of the LC resonant network. Ideally one would want, R → ∞ (open circuit) for the PRC and R → 0 (short circuit) for the SRC. Both conditions in [START_REF] Molla-Ahmadian | Hybrid control of DC-DC series resonant converters: The direct piecewise affine approach[END_REF] can be stated in terms of quality factor Q. Considering that

Q PRC = R C L and Q SRC = 1 R L C
we have that:

Q = ω β > 0.5, (11)
which interestingly corresponds to the same requirement for both architectures. Constraint [START_REF] Afshang | Hybrid control of the DC-DC series resonant converter operating below resonance[END_REF] immediately shows the advantage of our novel switching law, as compared to the alternative solutions of [START_REF] Bonache-Samaniego | Design of self-oscillating resonant converters based on a variable structure systems approach[END_REF], [START_REF] Bonache-Samaniego | Dynamic modeling and control of self-oscillating parallel resonant converters based on a variable structure systems approach[END_REF]. Indeed, a nontrivial discussion is present in [START_REF] Bonache-Samaniego | Design of self-oscillating resonant converters based on a variable structure systems approach[END_REF] showing for the PRC case that with Q < 3.15 there is no guarantee of a self-oscillating behavior (the controller in [START_REF] Bonache-Samaniego | Design of self-oscillating resonant converters based on a variable structure systems approach[END_REF] may reach an equilibrium). The result of Theorem 1, only requiring the mild assumption [START_REF] Afshang | Hybrid control of the DC-DC series resonant converter operating below resonance[END_REF] provides an important improvement.

•

Remark 2

The flow and jump sets of hybrid dynamics [START_REF] Sira-Ramírez | On the control of the resonant converter: a hybrid-flatness approach[END_REF], [START_REF] Hernández | On the stability of limit cycles in resonant dc-to-dc power converters[END_REF] are closed, and the flow and jump maps are continuous functions, therefore [START_REF] Sira-Ramírez | On the control of the resonant converter: a hybrid-flatness approach[END_REF], ( 8) enjoys the so-called hybrid basic conditions of [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF]As. 6.5]. This, among other things, implies robustness of asymptotic stability of compact attractors, as characterized in [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF]Ch. 7]. A consequence of robustness is that one expects a graceful degradation of the closedloop stability properties (the so-called semiglobal practical robustness): an important feature for the experimental results discussed in Sections V and VI. •

B. Hybrid Lyapunov function and proof of Theorem 1

In this section we prove Theorem 1, based on hybrid Lyapunov theory [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF]. Our proof shares interesting similarities with the approach reported in [START_REF] Bisoffi | Global results[END_REF] (see also [START_REF] Bisoffi | Relay-based hybrid control of minimal-order mechanical systems with applications[END_REF]), which address relay-based control of mechanical systems. In particular for the case θ = π 2 it is shown in [START_REF] Zaupa | Results on hybrid control of self-oscillating resonant converters[END_REF] that Theorem 1 immediately follows from [START_REF] Bisoffi | Global results[END_REF]. Here we provide nontrivial derivations to allow extending the result to any θ ∈ (0, π].

Before proving Theorem 1, let us recall from [26, Def. 5.1] that a function ψ : R → R is α-strongly convex if and only if there exists α > 0 such that for each a, b ∈ R it holds that

ψ(a) -ψ(b) ≥ σ(a -b) + α 2 |a -b| 2 , ∀σ ∈ ∂ψ(b), ( 12
)
where ∂ψ(b) ⊂ R is the subdifferential of ψ at b. Based on [START_REF] Bonache-Samaniego | Design of self-oscillating resonant converters based on a variable structure systems approach[END_REF] we can prove the following lemma, which is instrumental for the proof of Theorem 1.

Lemma 1 Consider two continuous functions of the scalar variable ξ ∈ R ≥0 . A linear function ξ → ψ 1 (ξ) = ψ 0 + γξ, with ψ 0 , γ ∈ R, and a strongly convex function ψ 2 with ψ 2 (0) < ψ 0 . Function ξ → ψ 1 (ξ) -ψ 2 (ξ) grows unbounded as ξ → +∞ and has exactly one zero for ξ ≥ 0.

Proof. Denote ψ(ξ) := ψ 2 (ξ) -ψ 1 (ξ) and note that it is strongly convex because, for each a, b ∈ R and each σ ∈ ∂ψ 2 (b), it satisfies, from [START_REF] Bonache-Samaniego | Design of self-oscillating resonant converters based on a variable structure systems approach[END_REF]

applied with ψ = ψ 2 , ψ(a) -ψ(b) = ψ 2 (a) -ψ 2 (b) -γ(a -b) (13) ≥ σ(a -b) + α 2 |a -b| 2 ,
where σ = σ-γ characterizes any vector in the subdifferential of ψ at b. Let us first prove that function ψ has at least one zero for ξ ∈ [0, +∞). From ψ 1 (0) = ψ 0 > ψ 2 (0), we have ψ(0) < 0. Applying (13) at the unique global minimum ξ * (wherein σ = 0 belongs to ∂ ψ(ξ * )), we have ψ(ξ) ≥ ψ(ξ * )+ α 2 |ξ-ξ * | 2 , which proves that lim ξ→+∞ ψ(ξ) = +∞, showing unboundedness of ψ(ξ) as ξ → +∞. Moreover, with ψ(0) < 0 and continuity, there exists at least one ξ 0 where ψ(ξ 0 ) = 0.

Let us now prove that ξ 0 is unique. Assume that there are two points 0 < ξ 0 < ξ 1 where ψ is zero and fix a = ξ 1 , b = ξ 0 . Then from [START_REF] Bonache-Samaniego | Dynamic modeling and control of self-oscillating parallel resonant converters based on a variable structure systems approach[END_REF] with these selections, we get:

0 > σ(ξ 1 -ξ 0 ) + α 2 |ξ 1 -ξ 0 | 2 , (14)
which clearly implies σ < 0. Since σ ∈ ∂ ψ(ξ 0 ) and ψ(ξ 0 ) = 0, by definition of subdifferential, using 0 < ξ 0 , we have ψ(0) ≥ ψ(ξ 0 ) -σξ 0 = -σξ 0 > 0, which is a contradiction, because we proved above that ψ(0) < 0. ■ For proving Theorem 1, we formulate the following corollary of [25, Lemma 1], relating the dissipated energy along a flowing solution of ( 7), [START_REF] Hernández | On the stability of limit cycles in resonant dc-to-dc power converters[END_REF] to the hatched area in Fig. 3

(right).

Lemma 2 Consider any solution z to (7), ( 8) flowing from G 1 at ordinary time t 1 to D 1 at ordinary time t 2 > t 1 and define energy-like function

E(z) = 1 2 z 2 1 + 1 2 z 2 2 = 1 2 |z| 2 . The dissipated energy E(z(t 2 )) -E(z(t 1 )) is equal to β ω Π
, where Π is the (unsigned) area hatched between the graph of the trajectory z(t), t ∈ [t 1 , t 2 ] and the coordinate axis z 2 = 0.

Proof. Consider the following selection of the parameters and states in [START_REF] Bisoffi | Global results[END_REF] x

1 = z 1 , x 2 = ωz 2 , m = 1, c = β, k = ω 2 . ( 15
)
From [25, Lemma 1] the function [START_REF] Sferlazza | A hybrid control strategy for quadratic boost converters with inductor currents estimation[END_REF] = βΠ [START_REF] Sferlazza | A hybrid control strategy for quadratic boost converters with inductor currents estimation[END_REF] between t 1 and t 2 . From [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF], in the z coordinates we have Π [START_REF] Sferlazza | A hybrid control strategy for quadratic boost converters with inductor currents estimation[END_REF] = ωΠ and since

E [21] (x) = 1 2 ω 2 x 2 1 + 1 2 x 2 2 = ω 2 E(z) dissipates ∆E
∆E [21] = ω 2 ∆E, we have ∆E = 1 ω 2 ∆E [21] = β ω 2 Π [21] = β ω Π as to be proven. ■ z 2 z 1 G -1 D 1 D -1 G 1 C -1 C 1 z 2 z 1 G -1 D 1 D -1 G 1 z e AτD z e AτG z ψ up 2 2 2
ψ lw 2 Fig. 3. Evolution of solutions and sets G 1 (in green) and D 1 (in red) discussed in the proof of Theorem 1.

To construct a Lyapunov function proving Theorem 1, let us only consider the half-space C 1 , which also contains D 1 (parallel definitions apply to C -1 ) and introduce the set

G 1 :={z ∈ C 1 : z 2 ≤ 0, (z 1 + 2) sin θ + z 2 cos θ = 0}∪ {z ∈ C 1 : z 1 ∈ [-2, 0], z 2 = 0}, (16)
which corresponds to the green-colored half line parallel to D -1 in Fig. 3. Then, for each point z ∈ C 1 \ {0}, denote the compact time interval associated with the unique backward and forward flowing solutions of [START_REF] Sira-Ramírez | On the control of the resonant converter: a hybrid-flatness approach[END_REF], flowing in C 1 , as

T 1 (z) := {τ ∈ R : e As z ∈ C 1 , ∀s ∈ [τ, 0] ∪ [0, τ ]}, (17)
where [t 2 , t 1 ] should be understood as the empty set when t 2 > t 1 , so that [τ, 0]∪[0, τ ] always describes an interval containing zero, whether τ is positive or negative. Clearly, 0 ∈ T 1 (z) for all z ∈ C 1 \ {0}. Based on T 1 (z), define the following two times, which exist and are unique for each z ∈ C 1 \ {0} due to the revolving nature of solutions, as per Assumption 1:

τ G (z) := {τ ∈ T 1 (z) : e Aτ z ∈ G 1 }, (18)
τ D (z) := {τ ∈ T 1 (z) : e Aτ z ∈ D 1 }. ( 19
)
The Lyapunov function proposed in this proof, which is represented in Fig. 4 for the special case θ = 3π/4, is based on ( 18), ( 19) and corresponds to

V (z) := max{τ G (z), 0} + (δ V (z) -β ω Π(z)) 2 ([ 1 0 ] e Aτ G (z) z) 2 , ( 20
)
δ V (z) := 2 1 + σ [ 1 0 ] e Aτ D (z) z , (21)
where Π(z) has been defined in Lemma 2.

The first term of V (z) in ( 20) is positive when z lies in the stripe between the set D -1 and G 1 of Fig. 3, wherein τ G (z) > 0, but it is zero in the remaining points of C 1 , wherein τ G (z) ≤ 0. The second term of V (z) is inspired by [START_REF] Bisoffi | Global results[END_REF]. Referring to the energy E defined in Lemma 2, its numerator corresponds to the difference between the dissipated energy

δ V (z) = 1 2 ((z + 1,D ) 2 -z 2 1,D ) = 2 (1 + σz 1,D ) , (22)
as defined in [START_REF] Sferlazza | A hybrid control strategy for quadratic boost converters with inductor currents estimation[END_REF]. Note that by construction Π and δ V are constant along flowing solutions, therefore, whenever τ G (z) ≤ 0, V remains constant along flowing solutions. Comparing δ V (z) with β ω Π(z), an energy balance between flows and jumps emerges when δ V (z) -β ω Π(z) = 0, that is when V (z) = 0. The denominator of the right term in [START_REF] Sferlazza | Min-type control strategy of a DC-DC synchronous boost converter[END_REF] simply ensures that, close to the origin, V blows up to infinity, as it should because the origin is a weak equilibrium (it admits a constant flowing solution not converging to the hybrid periodic orbit) and cannot belong to the basin of attraction.

Defining V (z) in a parallel way for z ∈ C -1 , we prove below that the following set, the zero level set of V , corresponds to the set O θ characterized in Theorem 1:

A := {(z, σ) : δ V (z) = β ω Π(z), and τ G (z) ≤ 0}, ( 23
)
with δ V defined in [START_REF] Sferlazza | A hybrid control strategy for quadratic boost converters with inductor currents estimation[END_REF]. We are now ready to prove Theorem 1. such that e Aτ G (z0) z 0 = -2 0 , namely the point where G 1 has a kink, in Fig. 3. Due to homogeneity of the linear solutions (larger solutions are scaled versions of the smallest ones), it is immediate to see that ψ lw 2 (|z|) is (non-strictly) convex and ψ up 2 (|z|) is strongly convex. Therefore their sum is strongly convex. Let us continue by observing that for each z ∈ D 1 , we can express the injected energy as By uniqueness of z * , the nontrivial hybrid periodic orbit O θ is unique and coincides with set A in [START_REF] Bisoffi | Relay-based hybrid control of minimal-order mechanical systems with applications[END_REF].

Proof of

ψ 1 (|z|) := |z + | 2 -|z| 2 = 2(1 -|z| cos θ) = δ V (z).
Let us now prove the asymptotic stability of A with basin of attraction B A = (R 2 \ {0}) × {-1, 1}. To this end, let us first note that V is zero in A, positive in B A and, from Lemma 1, radially unbounded, relative to the open set B A , with respect to A. Moreover, since the points e Aτ D (z) z and e Aτ G (z) z remain constant along flowing solutions and τ G (z) is a decreasing function of time (due to the revolving nature of the solutions stemming from Assumption 1), then V in ( 20) is nonincreasing when flowing in C. Finally, from Lemma 1, and using similar arguments to those of [START_REF] Bisoffi | Global results[END_REF]Lemma 2], the Lyapunov function decreases across jumps. More specifically, the following weak Lyapunov properties hold

V := ⟨∇V (z), f (z, σ)⟩ ≤ 0, ∀(z, σ) ∈ C \ ({0} × {-1, 1}), ∆V := V (g(z, σ)) -V (z) < 0, ∀(z, σ) ∈ D \ A.
As in [START_REF] Bisoffi | Global results[END_REF], the asymptotic stability of A with basin of attraction B A then follows from the nonsmooth hybrid invariance principle of [27, Thm 1].

The proof is completed by noting that, except for the trivial flowing solution at zero, the hybrid limit cycle whose orbit is A, is globally attractive and therefore it is the only possible one. Moreover, due to the symmetry of the flow/jump sets and maps, this cycle is associated with periodic jumps, where the period T (θ) of the jumps is given by the time it takes for a solution to reach D 1 from G 1 , along the periodic orbit. The period of the limit cycle is (2T (θ), 2) because, by construction, it takes two half revolutions for the periodic trajectory starting in from G 1 to revisit the same point in G 1 . ■

V. CONTROLLER IMPLEMENTATION

This section discusses the practical implementation of the hybrid switching law, which has been verified with high fidelity simulations in PSIM software.

Implementing the hybrid feedback requires making a decision about whether a switch should be performed or not, based on the jump/flow sets. An analog-based solution could check the flow/jump conditions in the continuous time domain, as in [START_REF] Sferlazza | Min-type control strategy of a DC-DC synchronous boost converter[END_REF], such that there is no information loss, at the expense of a relatively complex tuning and a rigid design reuse. On the other hand, a digital alternative would facilitate complex calculations and reprogramming capability, at the expense of some information loss due to the analog-to digital converters (ADCs) and the presence of a time-delay in the control loop. Information loss in digitally controlled power converters is caused by the digital pulse width modulator (PWM) and the quantization error of the analog to digital converter (ADC), which may generate undesired oscillations [28]. In turn, timedelay negative effects can be mitigated by performing the sampling/conversion process at a fast sampling rate, up to more than 30 times the switching frequency [29]. In this work, the hybrid control is digitally implemented because the two above mentioned drawbacks are minimized. Namely, no numerically-induced oscillations emerge because the control strategy does not require a PWM element, and no undesirable delay effects emerge because of the low latency of the FPGA and the high sampling frequency of the ADCs. This last statement is confirmed by the simulation results reported in Fig. 5, where a delay in the order of 2% (200 ns) of the half period (10 µs) is acceptable and does not affect significantly the closed-loop response. Fig. 5 reports simulation results for trajectories in the phase-plane (z 1 , z 2 ) when a delay is introduced in the acquisition chain, namely this delay can account for: acquisition time, latency, internal ADC timing. Our discussed theoretical framework actually justifies these robustness features in a rigorous way, thanks to the properties discussed in Remark 2. The results of Fig. 5 confirm that the asymptotic stability is preserved despite the introduction of these perturbations in the feedback loop. For the measurements of the state z = (z 1 , z 2 ), needed by the feedback controller, it is enough to measure a scaled version of the current and the voltage on the capacitor, as discussed in [START_REF] Zaupa | Results on hybrid control of self-oscillating resonant converters[END_REF].

The core task of the controller is evaluating the jump condition, to decide whether to switch or not. The introduced jump set formulation (8c) requires checking a zero crossing conditions that can be conveniently extended to a half plane. Moreover, in order to cope with the disturbances arising at the switching instant (which could generate multiple consecutive jumps), time regularization is introduced, which inhibits jumps for t R seconds, so that the trajectory moves away from the jump set border. We can model this behavior by introducing a timer variable λ > 0 that is reset to 0 at every jump and linearly increases during flow. Then a jump is allowed only if λ ≥ t R . The dynamics for λ, which can be added to [START_REF] Sira-Ramírez | On the control of the resonant converter: a hybrid-flatness approach[END_REF], is

λ = 1 (z, σ, λ) ∈ C R (θ), (25a) λ + = 0 (z, σ, λ) ∈ D R (θ), (25b)
where the time-regularized flow/jump sets are C R (θ) = C(θ) ∪ R ≥0 and D R (θ) = D(θ) ∪ [t R , +∞). A graphical representation of the time-regularized sets is reported in Fig. 6, where the gray area represents the area spanned by the solution while jumps are inhibited. The angular aperture can be approximated with φ = ωt R . As long as t R is sufficiently small, the results of Theorem 1 also apply with the regularized sets.

VI. EXPERIMENTAL VALIDATION A. Prototype design

An experimental prototype of an SRC has been developed with target parameters: f 0 = ω 2π = 50 kHz, V g = 24 V, and operating power of 100 W. The prototype can be seen in Fig. 7, its schematic is shown in Fig. 8 and the list of components is given in Table I. Frequency is limited to reduce the effect of the sampling-induced delay, even though also industrial prototypes work in this range up to a few hundreds of kHz.

The power stage consists of an H-bridge and a resonant tank. The H-bridge comprises two MOSFET power modules (APTC60AM242G) driven by the IC IR2110. The resonant tank employs a multi-layer ceramic (X7R) 100 nF capacitor and a 94.3 µH inductor. The sensing is performed with a Halleffect sensor (CQ-3200) for the current measurements and a differential operational amplifier (NCS2007) for the voltage measurements. Both sensing stages have a low-pass filter to The controller is implemented in a Stratix IV GX (EP4SGX230KF40C2) FPGA in the DE4 board from Terasic, and its daughter board (AD/DA Data Conversion Card) equipped with AD9254 ADCs operating at 100 MSPS. The high sampling frequency reduces the delay to a negligible value estimated in 200 ns, which is in accordance with the simulation results of Fig. 5.

The converter has been tested with two different loads of 10.1 Ω and 22 Ω, i.e., for quality factors of Q = 3 and Q = 1.4 respectively. Note that, according to Remark 1, both these situations cannot be addressed with the approach in [START_REF] Bonache-Samaniego | Design of self-oscillating resonant converters based on a variable structure systems approach[END_REF].

APTC60AM242G APTC60AM242G M 1 M 2 Cin Vg + - R 1 R 1 R 2 R 2

B. Experimental results

The steady-state input-output waveforms are reported in Fig. 9, showing a desirable output oscillation for various values of θ, thus confirming the result of Theorem 1. Fig. 10 illustrates several simulation (top row) and experimental (bottom row) responses, represented in the (z 1 , z 2 ) phase-plane. In the phase-plane representations, the dotted horizontal segments (yellow and purple) represent the jumps in the trajectories.

Additionally, the converter behavior has been experimentally characterized with respect to the input signal θ. Fig. 11 and Fig. 12 report experimental results compared with the simulation for the two loads. The results show a good match for large values of θ, near the resonance. This is expected because this type of converter typically operates around that region. Fig. 11 and Fig. 12 report the peak of the normalized voltage and the current measured on the capacitor, which have a more meaningful physical interpretation. Indeed, z 2 corresponds to the normalized current ((z 2 ) peak ) while z 1 , without the offset introduced by σ, corresponds to the normalized voltage ((z 1 ) peak -1). The reported resonant frequencies (dotted black line) are evaluated based on a characterization of the inductance value of the prototype's resonant tank for the two different load conditions. The different values are due to the nonlinear effects of frequency, current and circuit layout on the value of the inductance.

The experiments show good matching with the simulation results for large values of θ, while the discrepancy increases as θ decreases. Experiments with small θ have not been performed due to the high frequency shown by the converter, which makes the self-oscillation harder to sustain with this controller. Moreover, for small values of θ, the acquired amplitudes were too low with respect to the ADCs input range. These issues reduced the set of angles tested in the prototype, but practically this is not problematic since the converter is supposed to work not far from the resonant frequency. If a larger amplitude variation is needed, other types of controllers should be used, possibly including also operating with zero voltage input (v S = 0).

VII. CONCLUSIONS AND FUTURE DIRECTIONS

We have proposed a unifying coordinate representation for PRC and SRC resonant converters, which leads to a natural feedback control law inducing a unique nontrivial hybrid periodic orbit in a self-oscillating resonant behavior. This resulting hybrid limit cycle has been proved to be almost globally asymptotically stable via a hybrid dynamical systems representation. With the proposed controller, the generation of the limit cycle is ensured in underdamped conditions for any reference input θ ∈ (0, π], providing a large range of output amplitudes and frequencies, whose trends, as a function of θ, have been characterized. Unlike the self-oscillating mechanism proposed in [START_REF] Bonache-Samaniego | Design of self-oscillating resonant converters based on a variable structure systems approach[END_REF] no high values of Q are required, apart from the underdamped nature of the flow dynamics. Indeed, while in [START_REF] Bonache-Samaniego | Design of self-oscillating resonant converters based on a variable structure systems approach[END_REF] a minimum value of Q was needed to ensure trajectory switching between two equilibrium points, our hybrid switching law only requires complex conjugate poles in the characteristic equation of the linear continuous dynamics. Moreover, the reference input θ is bounded, while in [START_REF] Bonache-Samaniego | Dynamic modeling and control of self-oscillating parallel resonant converters based on a variable structure systems approach[END_REF] the reference input k goes to infinity for amplitudes near zero. These properties have been experimentally validated on a SRC prototype with a digital implementation of the controller in an FPGA.

Although a feedback loop has been closed to ensure selfoscillation, no regulation of the output current or voltage has been considered. This could be a future direction to investigate, that would require also a dynamic characterization of the hybrid relation between θ and the output amplitudes, in addition to the static characterization done here. Additional future work includes industrial applications such as, for example, battery charging or induction heating. Finally, the hybrid formulation could be extended to higher order resonant converters following similar formulations to those in [START_REF] Bonache-Samaniego | Design of self-oscillating resonant converters based on a variable structure systems approach[END_REF].

Fig. 1 .

 1 Fig. 1. Circuits associated with the parallel resonant converter (PRC, top) and series resonant converter (SRC, bottom).

  β ω Π(z) (sampled in a Poincaré fashion) along the flow from G 1 to D 1 with the increase of energy across the jump from z 1,D = [ 1 0 ] e Aτ D (z) z, which belongs to D 1 , namely

Theorem 1 Fig. 4 .

 14 Fig. 4. Logarithmic representation of the Lyapunov function V in (20), for the value θ = 3π/4.

( 24 )

 24 Then, from Lemma 1, there exists only one positive valueξ * = |z * | of |z| leading to the energy balance ψ 2 (|z * |) = ψ 1 (|z * |).In particular, the hybrid periodic orbit O θ corresponds to the image of the hybrid periodic trajectory starting at the unique point z * ∈ D 1 .

Fig. 5 .

 5 Fig. 5. Comparison between the simulated response with an analog implementation (top-left) with the sampled-data implementation, with different delays t D and using a constant switching angle θ = 3π/4.

Fig. 6 .

 6 Fig. 6. Representation of the jump sets. The theoretical formulation (left) and the experimental one (right) corresponding to half-plane with time regularization. The shaded areas show the effects of time regularization.

Fig. 7 .

 7 Fig. 7. Picture of the experimental prototype.

Fig. 8 .Fig. 9 .

 89 Fig.8. Schematic of the electronics involved in the prototype including: H-bridge, resonant tank, sensing, controller and drivers.

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Comparison between the experiments and the simulations with R = 10.1 Ω. The black dotted line corresponds to the estimated resonant frequency f0 = 50 kHz.

TABLE I SRC

 I PROTOTYPE PARAMETERS.

	Parameter	Component	Value
	Vg	-	24 V
	C in	X7R	1 µF
	M 1-2	APTC60AM242G	-
	L	AGP4233	100 µH
	C	X7R	100 nF
	C LPF	poly	10 nF
	C HPF	poly	100 nF
	R LPF	trough-hole resistor	50 Ω
	R HPF	trough-hole resistor	1 kΩ
	R 1	trough-hole resistor	270 kΩ
	R 2	trough-hole resistor	1.2 kΩ
	Driver	IR2110	-
	OPAMP	NCS2007	-
	Current Sensor	CQ-3200	-
	ADC	AD9254	-
	FPGA	EP4SGX230KF40C2	-
	reduce the noise, additionally the current sensing has a high-
	pass filter to remove the DC component introduced by the
	current sensor, which would hinder the operation of the ADC
	stage.

  on reset-induced periodic trajectories of planar systems," in European Control Conference, Jun. 2016, pp. 2644-2649. [26] A. Y. Aravkin, J. V. Burke, and D. Drusvyatskiy, "Convex analysis and nonsmooth optimization," 2017. [Online]. Available: https: //sites.math.washington.edu/ ∼ burke/crs/516/notes/graduate-nco.pdf [27] A. Seuret, C. Prieur, S. Tarbouriech, A. Teel, and L. Zaccarian, "A nonsmooth hybrid invariance principle applied to robust event-triggered design," IEEE TAC, vol. 64, no. 5, pp. 2061-2068, 2019. [28] A. Peterchev and S. Sanders, "Quantization resolution and limit cycling in digitally controlled pwm converters," IEEE Transactions on Power Electronics, vol. 18, no. 1, pp. 301-308, 2003. [29] A. Costabeber, L. Corradini, P. Mattavelli, and S. Saggini, "Time optimal, parameters-insensitive digital controller for DC-DC buck converters," in 2008 IEEE Power Electronics Specialists Conference, 2008, pp. 1243-1249.