Chia-Hao Chang

Marie Mayer

Georgina Rivera-Ingraham

Eva Blondeau-Bidet

Wen-Yi Wu

Catherine Lorin-Nebel
email: catherine.lorin@umontpellier.fr

Tsung-Han Lee
email: thlee@email.nchu.edu.tw

Effects of temperature and salinity on antioxidant responses in livers of temperate (Dicentrarchus labrax) and tropical (Chanos Chanos) marine euryhaline fish

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

Introduction

Estuarine and lagoon environments are highly dynamic and characterized by fluctuations in temperature and oxygen levels. In these habitats, salinity also drastically changes within hours [START_REF] Freire | Oxidative stress in estuarine and intertidal environments 'temperate and tropical[END_REF], imposing physiological, metabolic, and molecular adjustments to organisms inhabiting such unstable habitats. In temperate areas such as the Mediterranean Basin, coastal lagoons are often warmer in spring and summer, sometimes reaching up to 28 • C (Ifremer, 2013), compared to the much more stable and generally lower temperatures recorded in the open sea during the summer months. The European sea bass (D. labrax) is a temperate marine species that migrates to such lagoon habitats in spring to use them as feeding grounds or nurseries, and sometimes migrate to river mouths [START_REF] Barnabé | L'élevage du loup et de la daurade[END_REF][START_REF] Dufour | Identification of sea bass (Dicentrarchus labrax) nursery areas in the north-western Mediterranean Sea[END_REF]. The salinity of lagoon environments can strongly vary and sometimes reach levels close to those of fresh water (FW), notably following extreme precipitation events in spring [START_REF] Mariotti | Mediterranean water cycle changes: transition to drier 21st century conditions in observations and CMIP3 simulations[END_REF]. In autumn, when temperatures reach levels below 10 • C, D. labrax migrates back to deeper warmer seawater (SW) [START_REF] Pickett | The patterns of recruitment of sea bass, Dicentrarchus labrax L. from nursery areas in England and Wales and implications for fisheries management[END_REF]. Previous studies have shown that the standard metabolic rate increased in D. labrax between 10 and 25 • C [START_REF] Claireaux | Influence of temperature, oxygen and salinity on the metabolism of the European sea bass[END_REF], and maximal growth rate was reported at 26 • C (Person-Le [START_REF] Ruyet | Effects of temperature on growth and metabolism in a Mediterranean population of European sea bass, Dicentrarchus labrax[END_REF]. In addition, compared to SW at 18 • C, exposure to 24 • C causes a higher nitrogen excretion and increased branchial mRNA expression in genes involved in the maintenance of hydromineral balance and nitrogen excretion (Person-Le [START_REF] Ruyet | Effects of temperature on growth and metabolism in a Mediterranean population of European sea bass, Dicentrarchus labrax[END_REF][START_REF] Masroor | Effect of salinity and temperature on the expression of genes involved in branchial ion transport processes in European sea bass[END_REF]. D. labrax has been reported to tolerate full FW [START_REF] Nebel | Differential freshwater adaptation in juvenile sea-bass Dicentrarchus labrax: involvement of gills and urinary system[END_REF]; however, due to osmoregulatory requirements, sea basses must adjust their physiological machinery to hyperosmoregulate, which might be energetically costly. The above-mentioned studies refer to increased metabolic and physiological activity at temperatures up to 24 • C, particularly in FW.

Milkfish (C. chanos), in contrast, is a tropical marine teleost of Southeast Asia and Taiwan, where it is an important economic species. It reproduces in shallow marine areas of coastal waters (lagoons and atolls) and uses FW habitats as nursery grounds [START_REF] Bagarinao | Systematics, distribution, genetics and life history of milkfish, Chanos chanos[END_REF]. According to the Aquaculture Condition Reports (Fisheries Agency, Council of Agriculture, Taiwan), the percentage of FW milkfish ponds has increased in this decade, probably linked to FW-acclimated milkfish having a better feed conversion rate than SW-acclimated milkfish [START_REF] Martinez | Milkfish (Chanos chanos) culture: situations and trends[END_REF]. The optimal temperature range for milkfish aquaculture is 24-33 • C with limits at 15 • C during cold snaps [START_REF] Hanke | Thermal stress response of juvenile milkfish (Chanos chanos) quantified by ontogenetic and regenerated sacle cortisol[END_REF]. In winter, high mortalities have been recorded during cold snaps, which have caused enormous economic losses in Taiwan [START_REF] Liao | Cold snap causes mass death of milkfish in Tainan[END_REF]. Previous studies have shown that SW-acclimated milkfish exhibit a better cold tolerance than FW-acclimated fish [START_REF] Kang | Seawater-acclimation abates cold effects on Na + , K + -ATPase activity in gills of the juvenile milkfish, Chanos chanos[END_REF], notably through differential energy utilization pathways [START_REF] Chang | Changes in hypothermal stress-induced hepatic mitochondrial metabolic patterns between fresh water-and seawater-acclimated milkfish[END_REF][START_REF] Chang | Differential effects of hypothermal stress on lactate metabolism in fresh water-and seawater-acclimated milkfish, Chanos chanos[END_REF].

Exposure to different environmental factors such as temperature, salinity, and dissolved oxygen can influence the oxidative status and cellular production of reactive oxygen species (ROS). These can trigger the activation of antioxidant defense mechanisms in different organs to prevent and intercept such oxidative species [START_REF] Martínez-Álvarez | Antioxidant defenses in fish: biotic and abiotic factors[END_REF]. Among antioxidant enzymes that intercept ROS, superoxide dismutase (SOD) and catalase (CAT) form an important part of the antioxidant responses in fish [START_REF] Lesser | Oxidative stress in marine environments: biochemistry and physiological ecology[END_REF]. Temperature increase or decrease has been shown to induce oxidative stress and trigger antioxidant defenses in fish, including the two analyzed species [START_REF] Malek | The effects of temperature reduction on gene expression and oxidative stress in skeletal muscle from adult zebrafish[END_REF][START_REF] Lushchak | Environmentally induced oxidative stress in aquatic animals[END_REF][START_REF] Vinagre | Effect of temperature on oxidative stress in fish: lipid peroxidation and catalase activity in the muscle of juvenile seabass, Dicentrarchus labrax[END_REF][START_REF] Chang | Differential effects of hypothermal stress on lactate metabolism in fresh water-and seawater-acclimated milkfish, Chanos chanos[END_REF]. The liver is an essential organ involved in antioxidative mechanisms [START_REF] Martínez-Álvarez | Antioxidant defenses in fish: biotic and abiotic factors[END_REF]. It is not yet clear how temperature alone and synergically combined with salinity affects antioxidant mechanisms in these two species, which are both extremely euryhaline but originate from habitats characterized by different temperature regimes and different temperature tolerances. Increased gene expression does not always lead to enhanced protein activity because of the translational and posttranslational mechanisms of enzyme activity regulation [START_REF] O'brien | Mitochondrial biogenesis in cold-bodies fishes[END_REF][START_REF] Song | Overwinter mortality in yellow drum (Nibea albiflora): insights from growth and immune responses to cold and starvation stress[END_REF]. It is thus essential to simultaneously analyze the responses at the gene expression and protein activity levels.

In this study, the hepatic redox status was analyzed by investigating two key antioxidants at the molecular and protein levels in order to compare the responses in the livers of a temperate and a tropical euryhaline aquaculture fish to warm and cold temperatures combined with salinity changes.

Material and methods

Experimental design

Juvenile sea bass (D. labrax) from a West Mediterranean population was brought to the Montpellier University and maintained for one week in 3500-L tanks containing natural seawater (SW) from the Mediterranean Sea at 38‰ and 18 • C according to the natural habitat in a temperate climate. Fish were transferred to 200-L tanks (14 fish/tank, density of 6-7 kg/m 3 , two replicates per condition) to be acclimated either to 24 • C or kept at 18 • C as a control condition, as in [START_REF] Masroor | Effect of salinity and temperature on the expression of genes involved in branchial ion transport processes in European sea bass[END_REF]. Milkfish (C. chanos) were maintained in 100-L tanks holding SW at 35‰ prepared from dechlorinated tap water (fresh water; FW) with proper amounts of Blue Treasure Sea Salts (New South Wales, Australia) at 28 • C according to natural habitat in subtropical climate. Fish (N = 6 fish/tank) were then acclimated progressively to either 18 • C or kept at 28 • C as a control. In both cases, temperature acclimation was progressive (2 • C/h) either by cooling (C. chanos) (PF-225M, PRINCE, Tainan, Taiwan) or warming (D. labrax) (Eheim Thermocontrol) the water. After two weeks of temperature acclimation, fish were transferred directly either to FW or kept in SW (sea bass: 7 fish/tank, two replicates for each condition; milkfish: 6 fish/tank, two replicates for each condition) at the respective temperature and maintained in these temperature-salinity conditions for two weeks before sampling (Fig. 1). For both species, a 12 h light/12 h dark photoperiod was used, and water was aerated and mechanically/biologically filtered (Eheim System, Lens, Pas-de-Calais, France). Temperature (28 ± 0.5 • C and 18 ± 0.3 • C), salinity (0‰ and 35 ± 0.5‰), oxygen, and nitrogen levels were checked daily. A quarter of the water volume was changed every two days. Fish were fed with fish granules (for sea bass, Aphymar feed, Meze, Herault, France; for milkfish, FwuSow, Taichung, Taiwan) twice a week (for sea bass) or every day (for milkfish) until 2 days before sampling. At the end of the experiments, fish were anesthetized in a solution of phenoxy-2-ethanol (240 ppm) prior to tissue collection. European sea bass and milkfish used in the experiments had a length of 19.2 ± 1.3 cm (mean±SD) and 10.2 ± 0.2 cm and an average weight of 86.87 ± 20.23 g and 13.7 ± 1.3 g, respectively. Four groups were compared: SW at 18 • C, SW at 24 • C or 28 • C for sea bass and milkfish, respectively, FW at 18 and FW at 24 • C or 28 • C for sea bass and milkfish, respectively. Following anesthesia, liver tissues were immediately collected from SWand FW-exposed sea bass and milkfish acclimated at 18 • C and 24 • C or 28 • C, respectively. Liver samples were immersed in Trizol® reagent (for gene expression analyses) or flash frozen in liquid nitrogen (for quantification of antioxidant activities). All tissues were stored at -80 • C until analysis.

These experiments respected the guidelines of the European Union (directive 86/609), the French law (decree 87/848) regulating animal experimentation, and Institutional Animal Care and Use Committee (IACUC) of the National Chung Hsing University (IACUC Approval No. 105-024 to T.H.L.).

RNA extraction and reverse transcription

Total RNA was extracted using Trizol® reagent according to the manufacturer's instructions. RNA quantity and purity were assessed by measuring the A260/A280 ratio using a NanoDrop® ND-1000 V3300 spectrometer (Nanodrop Technology Inc., Wilmington, Delaware, USA). RNA quality was checked in some samples using an Agilent bioanalyzer (Agilent) using the electrophoretic trace method. One microgram of total RNA was treated with DNase I amplification grade (Invitrogen™, Life Technologies). Reverse transcription was performed using 200 U M-MLV reverse transcriptase (Invitrogen™), and the first strand of complementary DNA (cDNA) was generated using 250 ng of random primers (Invitrogen™), dNTPs (10 mM), and 40 U of RNase OUT (Invitrogen™), following the manufacturer's instructions.

Gene expression analyses

All primer pairs used for qPCR are listed in Table 1. The amplification efficiencies of the primer pairs used in the qPCR that utilized serial cDNA dilution were predicted to be 92-104%. The PCR products were doublechecked by melting curve, 1.5% agarose gel electrophoresis, and sequencing. The qPCR mixture contained 8 μL of cDNA, 2 μL of 1 μM primer pairs, and 10 μL of 2 × KAPA SYBR® FAST qPCR Master Mix (Kapa Biosystems, Wilminton, MA, USA). The qPCR was performed using the MiniOpicon Real-Time PCR system (Bio-Rad, Hercules, CA, USA). The cDNA samples in all experimental groups were mixed as an internal control to reduce artificial effects in every reaction. The formula 2 -[(Ct target, n -Ct ef1α, n) -(Ct target, ic -Ct ef1α, ic)] was used for calculating relative mRNA expression, where "Ct" is the threshold cycle number, "n" indicates each sample, and "ic" indicates the internal control.

Antioxidant enzyme activities

Frozen liver samples were maintained at -80 • C until analysis of SOD and CAT activities using a TECAN spectrophotometer (Infin-ite®200 PRO, Männedorf, Switzerland). Tissues were homogenized in KPi buffer (50 mM KH 2 PO 4 , 50 mM K 2 HPO 4 , pH = 7.3) containing Triton X-100, 120 mM KCL, and protease inhibitors (1 μg/mL) using a Mixer Mill 400 (Retsch GmbH, Haan, Germany, 30 s at 30 beats s -1 ) with steel balls (Retsch). Following centrifugation at 14,000 rpm for 3 min (4 • C), the supernatant was divided into two parts to assess SOD and CAT activities.

SOD activity was measured using the protocol of [START_REF] Mccord | Superoxide dismutase: an enzymic function for enythrocuprein (Hemocuprein)[END_REF], modified by [START_REF] Livingstone | Antioxidant enzymes in the digestive gland of the common mussel Mytilus edulis[END_REF]. CAT activity was determined by measuring the decrease in absorbance at 240 nm resulting from H 2 O 2 consumption, as described by [START_REF] Aebi | Catalase in vitro[END_REF]. All measurements were carried out in triplicate. Throughout the procedure, samples were kept on ice and the results were expressed per unit (U) of activity per mg of protein, following protein quantification according to the Bradford method [START_REF] Bradford | A rapid and sensitive method for the quantitation of microgram quantities of protein utilizing the priniciple of protein-dye binding[END_REF] using bovine serum albumin (BSA) as standard. One U SOD corresponds to the amount of enzyme causing a 50% inhibition of nitroblue tetrazolium reduction under assay conditions. One U CAT is defined as the amount of enzyme required to reduce 1 μmol of H 2 O 2 per min.

Statistical analyses

Statistical analyses were performed using GraphPad Prism (version 6, GraphPad Software Incorporated, La Jolla, CA, USA). Normality and homogeneity of variance were checked using the D'Agostino-Pearson test and Bartlett test. A two-way factorial analysis of variance with temperature and salinity as the main factors was performed. Critical differences between groups were evaluated using the Fisher's leastsquare difference test. A non-parametric Mann-Whitney test was performed for SOD and CAT protein activity data. Data are presented as box and whisker plots showing median, minimum, and maximum values. The level of statistical significance was set at p < 0.05.

Results

The mRNA levels and protein activity of SOD in livers

In livers of D. labrax, a salinity effect was observed for Dlsod1 (superoxide dismutase 1 of D. labrax) and Dlsod2 (superoxide dismutase 2 of D. labrax) expression (Table 2, two-way ANOVA, p < 0.05). D. labrax challenged to FW at 18 • C displayed 37% and 46% significantly lower Dlsod1 and Dlsod2 mRNA expression levels compared to the SW group at 18 • C (Fig. 2A andB). At 24 • C, a significant 32% lower Dlsod2 mRNA expression was measured following FW transfer; no change in Dlsod1 mRNA expression was observed at this temperature. In livers of C. chanos, a highly significant salinity and temperature effect was detected for Ccsod1 (superoxide dismutase 1 of C. Chanos; Table 2, twoway ANOVA, p < 0.0001). The interaction between temperature and salinity had a significant effect on Ccsod1 expression (p < 0.01). C. chanos challenged from SW 28 • C to SW 18 • C showed a 208% higher Ccsod1 mRNA expression compared to the control but no change in Ccsod2 (superoxide dismutase 2 of C. chanos) mRNA levels. Following FW transfer, no difference in mRNA expression was observed at 28 • C (Fig. 2C). At 18 • C, however, Ccsod1 mRNA levels were 46% lower in FW fish than in SW individuals (Fig. 2C). We did not observe any significant differences in Ccsod2 mRNA levels between the different salinity and temperature conditions in C. chanos (Fig. 2D).

In D. labrax, hepatic SOD activity was increased following acclimation to 24 • C under SW conditions (Fig. 3A). FW transfer did not change SOD activity levels at either temperature, but a higher variability of the data was detected in FW conditions at both temperatures. In C. chanos, SOD activity in the livers was significantly increased following temperature transfer from 28 • C to 18 • C in SW (Fig. 3B). FW acclimation did not affect SOD activity levels, regardless of the temperature.

Catalase mRNA levels and protein activity

In D. labrax, the effect of temperature on Dlcat (catalase of D. labrax) mRNA expression was measured (Table 2, p < 0.05), but there was no effect of salinity or the interaction between both parameters. The mRNA levels of Dlcat in livers were significantly different only between FW conditions, with a 52% higher expression at 24 • C than at 18 • C (Fig. 4A). In C. chanos, Cccat (catalase of C. chanos) mRNA expression in the livers did not show an effect of temperature, salinity, or the interaction between both parameters (Table 2). No significant differences were observed between temperature and salinity conditions (Fig. 4B).

CAT activity did not change in livers of D. labrax following temperature transfer in SW. Following FW transfer, however, a 30% decrease in CAT activity was observed at 18 • C, whereas no change in mRNA levels was observed at 24 • C (Fig. 5A). In C. chanos, a temperature effect was observed regarding hepatic CAT activity (Table 2, p < 0.05). CAT activity decreased by 47% following a temperature challenge from 28 • C to 18 • C in SW. Following FW transfer, however, no change in hepatic CAT activity was observed regardless of the temperature (Fig. 5B).

Discussion

This study compared antioxidant genes and enzyme activities in two important aquaculture species from different climate zones under different salinity environments and nonlethal temperature stress.

Antioxidant responses to hypoosmotic transfer

Oxidative stress can be categorized according to endogenous or exogenous sources. Both the endogenous source mainly from mitochondria and the exogenous source via cytosolic pathways lead to cytosolic oxidative stress [START_REF] Chowdhury | Oxidative stress in fish: a reivew[END_REF]. Therefore, mRNA expression of cytosolic superoxide dismutase (sod1) and mitochondrial superoxide dismutase (sod2) are indicators of oxidative stress from the cytosol (exogenous source) and mitochondria (endogenous source), respectively [START_REF] Wang | Superoxide dismutase: dual roles in controlling ROS damage and regulating ROS singaling[END_REF]. Acute responses of the marine euryhaline European sea bass and milkfish when transferred to FW have been reported in previous studies focusing on osmoregulatory ability in gills as well as responses to oxidative stress in the liver [START_REF] Kang | Seawater-acclimation abates cold effects on Na + , K + -ATPase activity in gills of the juvenile milkfish, Chanos chanos[END_REF][START_REF] Sinha | Nutritional status as the key modulator of antioxidant responses induced by high environmental ammonia and salinity stress in European sea bass (Dicentrarchus labrax)[END_REF][START_REF] Masroor | Effect of salinity and temperature on the expression of genes involved in branchial ion transport processes in European sea bass[END_REF][START_REF] Chang | Differential effects of hypothermal stress on lactate metabolism in fresh water-and seawater-acclimated milkfish, Chanos chanos[END_REF][START_REF] Islam | Effects of extreme ambient temperature in European seabass, Dicentrarchus labrax acclimated at different salinities: growth performance, metabolic and molecular stress responses[END_REF]. [START_REF] Sinha | Nutritional status as the key modulator of antioxidant responses induced by high environmental ammonia and salinity stress in European sea bass (Dicentrarchus labrax)[END_REF] reported that SOD and catalase (CAT) activities in the livers of European sea bass were not changed and downregulated, respectively, under short-term hypoosmotic challenge from 32 to 2.5‰. Synergic effects were observed when the hypoosmotic challenge was combined with ammonia treatment. In this study, hepatic SOD and CAT activity of the European sea bass acclimated to hypoosmotic environments revealed the same patterns. The gene expression of sod isoforms, however, was downregulated, which might have resulted from the negative feedback mechanisms that regulate the expression of sod isoforms [START_REF] Wang | Superoxide dismutase: dual roles in controlling ROS damage and regulating ROS singaling[END_REF]. In milkfish, the acute salinity challenge did not affect oxidative stress or antioxidant mechanisms [START_REF] Chang | Hypothermal stress induced differential expression profiles of the immune response gene, warm-temperature-acclimation associated 65-kDa protein (Wap65), in the liver of fresh water and seawater milkfish, Chanos chanos[END_REF]2020). Increased antioxidant responses, however, were usually found in the other euryhaline species upon hypoosmotic challenge at normal temperature. In the black porgy (Acanthopagrus schlegeli), genes and enzyme activities of SOD and CAT were upregulated after 1 day of hypoosmotic challenge [START_REF] An | Profiles of antioxidant gene expression an physiological changes by thermal and hypoosmotic stresses in black porgy (Acanthopagrus schlegeli)[END_REF].

In the cinnamon clownfish (Amphiprion melanopus), enzyme activities of SOD, CAT, and glutathione peroxidase (GPx) were upregulated during two days of hypoosmotic challenge [START_REF] Park | Effect of hypoosmotic and thermal stress on gene expression and the activitiy of antioxidant enzymes in the cinnamon clownfish, Amphiprion melanopus[END_REF]. In the rainbow trout (Oncorhynchus mykiss), hepatic CAT activity elevated but SOD activity did not change under two-days hypoosmotic challenge [START_REF] Blewett | The effect of hydraulic flowback and produced water on gill morphology, oxidative stress and antioxidant responses in rainbow trout (Oncorhynchus mykiss)[END_REF]. Moreover, all antioxidant responses in livers of the marine olive flounder (Paralichthys olivaceus) was increased under one-and two-week hypoosmotic stress [START_REF] Kim | Salinty-mediated changes in hematological parameters, stress, antioxidant responses, and acetylcholinesterase of juvenile olive flounders (Paralichthys olivaceus)[END_REF]. The long-term salinity changes in European seabass and milkfish did not affect their oxidative status, according to the enzyme activities of SOD and CAT. Compared to the other euryhaline species, the antioxidant capacity in the liver of European sea bass and milkfish was thus suggested to have the ability to maintain homeostasis when transferred to hypoosmotic environments.

Long-term temperature effects on antioxidant responses

Temperature is an important abiotic factor that leads to oxidative stress in aquatic organisms [START_REF] Chowdhury | Oxidative stress in fish: a reivew[END_REF]. [START_REF] Pörtner | Thermal limits and adaptation in marine Antarctic ectotherms: an integrative view[END_REF] reported that aquatic organisms under hyperthermal or hypothermal challenges increase their oxidative stress levels and induce antioxidant mechanisms due to the conceptual model of thermal limitation [START_REF] Pörtner | Thermal limits and adaptation in marine Antarctic ectotherms: an integrative view[END_REF]. In studies on teleosteans, the gene expression or enzyme activity of antioxidant mechanisms or oxidative damage markers are common indicators used to determine the redox status under short-term or long-term temperature treatments. The cold-water fish Schizothorax prenanti acclimated at 11 • C was determined to have an optimal raising temperature of 21 • C based on SOD and CAT activity as well as on the contents of MDA (Malondialdehyde; a marker for lipid peroxidation) [START_REF] Yang | Effects of temperature on activities of antioxidant enzymes and Na + /K + -ATPase, and hormone levels in Schizothorax prenanti[END_REF]. The mRNA expression profiles of the antioxidant mechanisms, including sod and cat in the liver of the pufferfish (Takifugu obscurus) under 1 h hot-or cold-temperature stress were upregulated, leading to apoptotic responses (Cheng et al., 2018a(Cheng et al., , 2018b)). In the livers of the yellow drum (Nibea albiflora), the mRNA expression profiles of total sod isoforms (cytosolic superoxide dismutase, sod1; mitochondrial superoxide dismutase, sod2, extracellular superoxide dismutase, sod3) revealed that the elevated mRNA expression of sod1 was highly related to upregulated SOD activity under hypothermal stress [START_REF] Song | Overwinter mortality in yellow drum (Nibea albiflora): insights from growth and immune responses to cold and starvation stress[END_REF]. The thermal sensitivity of mitochondria in teleosts has been suggested to increase oxidative stress under hypothermal and hyperthermal stress [START_REF] O'brien | Mitochondrial biogenesis in cold-bodies fishes[END_REF][START_REF] Banh | Mitochondrial reactive oxygen species production by fish muscle mitochondria: potential role in acute heat-induced oxidative stress[END_REF]. Most studies on temperature-induced oxidative stress in teleosts only determined one sod isoform as an indicator, either cytosolic or mitochondrial [START_REF] Yang | Effects of temperature on activities of antioxidant enzymes and Na + /K + -ATPase, and hormone levels in Schizothorax prenanti[END_REF][START_REF] Chang | Hypothermal stress induced differential expression profiles of the immune response gene, warm-temperature-acclimation associated 65-kDa protein (Wap65), in the liver of fresh water and seawater milkfish, Chanos chanos[END_REF][START_REF] Yu | Antioxidant defenses of Onychostoma macrolepis in response to thermal stress: insight from mRNA expression and activity of superoxide dismutase and catalase[END_REF]Cheng et al., 2018b). However, in the present study, we highlight the interest in simultaneously carrying the analyses of both sod isoforms to provide a better understanding of the source of oxidative stress [START_REF] Song | Overwinter mortality in yellow drum (Nibea albiflora): insights from growth and immune responses to cold and starvation stress[END_REF][START_REF] Maslanka | Linkage between carbon metabolism, redox status and cellular physiology in the yeast Saccharomyces cerevisiae devoid of SOD1 or SOD2 gene[END_REF].

During the four-week temperature treatments, the cytosolic and mitochondrial isoforms of sod had different responses in European sea bass and milkfish. In European sea bass, the expression of sod isoforms did not change under four-week hyperthermal stress, but SOD activity was upregulated under long-term heat stress in SW. Although the differences in the FW group were not significant, this might be due to the different ability of individuals to tolerate long-term hyperthermal challenge. Upregulation of CAT and glutathione peroxidase (GPx) activity as well as enhanced lipid peroxidation were reported in the liver of European sea bass under heat stress (25 • C) for 30 days [START_REF] Almeida | Effects of temperature in juvenile seabass (Dicentrarchus labrax L.) biomarker responses and behavior: implications for environment monitoring[END_REF]. In the present study, however, CAT activity did not change during four weeks (28-days) of hyperthermal stress (24 • C). The results suggested that 24 • C might be a critical temperature for European sea bass or dependent on fish size, leading to different abilities of temperature tolerance [START_REF] Dahlke | Thermal bottlenecks in the life cycle define climate vulnerability of fish[END_REF]. In addition, short-term experiments might provide some information about the source of ROS for future studies. In milkfish, the mRNA expression of cytosolic sod showed synergistic interaction with temperature and salinity factors, while the mitochondrial sod did not show any change even upon acute hypothermal challenge [START_REF] Chang | Hypothermal stress-induce salinity-dependent oxdiative stress and apoptosis in the livers of euryhaline milkfish, Chanos chanos[END_REF]. This finding suggested that the source of ROS was from the cytosol, because ROS induced extrinsic pathway of apoptosis in milkfish livers [START_REF] Chang | Hypothermal stress-induce salinity-dependent oxdiative stress and apoptosis in the livers of euryhaline milkfish, Chanos chanos[END_REF]. The cytosol contains NADPH oxidase, the major ROS-producing enzyme correlated with immune responses under hypothermal stress [START_REF] Singh | NAPDH oxidase 4 modulates hepatic responses to lipopolysaccharide mediated by toll-like receptor-4[END_REF][START_REF] Kaushal | Molecular interactions between reactive oxygen species and autophagy in kidney disease[END_REF]. Therefore, immune responses of both FW and SW milkfish were enhanced by the upregulation of cytosolic SOD and the extrinsic apoptosis pathway from one week to one month under hypothermal stress [START_REF] Chang | Hypothermal stress induced differential expression profiles of the immune response gene, warm-temperature-acclimation associated 65-kDa protein (Wap65), in the liver of fresh water and seawater milkfish, Chanos chanos[END_REF][START_REF] Chang | Differential effects of hypothermal stress on lactate metabolism in fresh water-and seawater-acclimated milkfish, Chanos chanos[END_REF]. The cat expression profiles and CAT activity did not change after weeks of hypothermal stress. However, similar to the pattern of the European sea bass, some individual differences in enzyme activity under low-temperature treatment were found in the FW group.

Taken together, the results of this study revealed that changes in ambient temperatures enhanced responses of antioxidant enzymes in the livers of two marine euryhaline aquaculture species in different climate zones. After hypoosmotic challenge, variability in the responses was found in the FW group of both milkfish and European sea bass. Although synergistic responses of temperature and salinity were not found, ambient temperatures appeared to be the major factor regulating the activities of SOD in these two species. Meanwhile, milkfish may be more sensitive to the regulation of gene expression of antioxidative enzymes than European sea bass under long-term temperature stress, and the ROS may be derived from the cytosol. These two species have similar patterns of antioxidative mechanisms under four-week hyper-or hypo-thermal stress. The European sea bass, however, has more oxidative damage upon hyperthermal stress when acclimated to FW. This could be due to differential acclimation capacities in European sea bass from the same genetic lineage [START_REF] Nebel | Differential freshwater adaptation in juvenile sea-bass Dicentrarchus labrax: involvement of gills and urinary system[END_REF]. This study compared temperature-induced oxidative stress in the livers of two euryhaline species by elevated antioxidant mechanisms, providing a better understanding of future applications in their preferred aquaculture environments.

Declaration of competing interest

No conflicts of interest are declared by the authors.

Fig. 1 .

 1 Fig. 1. Experimental design for temperature and salinity treatments of European sea bass (D. labrax) and milkfish (C. chanos).

Fig. 2 .

 2 Fig. 2. Box and whiskers plot showing the median, minimum, and maximum mRNA expression levels of sod 1 (A, C) and sod 2 (B, D) in livers of D. labrax (A, B) and C. chanos (C, D) exposed to SW and FW at 18 • C and 24 • C (A, B) or 28 • C (C, D). mRNA levels were normalized to ef1α. Different letters indicate significant differences between conditions (two-way ANOVA followed by a Fisher Least Significant Difference (LSD) post hoc test, p < 0.05). FW, fresh water; SW, seawater.

Fig. 3 .

 3 Fig. 3. Box and whiskers plot showing the median, minimum, and maximum SOD activity levels (U/mg proteins) in D. labrax (A) and C. chanos (B) exposed to SW and FW under hyperthermal acclimation (A) or hypothermal acclimation (B). Different letters indicate significant differences between conditions (Mann-Whitney test, p < 0.05). FW, fresh water; SW, seawater.

Fig. 4 .

 4 Fig. 4. Box and whiskers plot showing the median, minimum, and maximum catalase mRNA expression in D. labrax (A) and C. chanos (B) exposed to SW and FW at 18 • C and 24 • C (A) or 28 • C (B). mRNA levels were normalized to ef1α. Different letters indicate significant differences between conditions. FW, fresh water; SW, seawater. Two-way ANOVA followed by a Fisher Least Significant Difference (LSD) post hoc test, p < 0.05).

Fig. 5 .

 5 Fig. 5. Box and whiskers plot showing the median, minimum, and maximum CAT activity levels (U/mg proteins) in D. labrax (A) and C. chanos (B) exposed to SW and FW at 18 • C and 24 • C (A) or 28 • C (B). Different letters indicate significant differences between conditions (Mann-Whitney test, p < 0.05). FW, fresh water; SW, seawater.

Table 1

 1 Primer sequences used in the gene expression analysis.

	Target gene	Primer name	Sequence ID	Sequence (from 5' to 3')	Efficiency
	sod 1 sod 2 cat sod 1 sod 2 cat	D.l. sod 1-F D.l. sod 1-R D.l. sod 2-F D.l. sod 2-R D.l. cat-F D.l. cat-R C.c. sod 1-F C.c. sod 1-R C.c. sod 2-F C.c. sod 2-R C.c. cat-F C.c. cat-R	DLA_LG14_005480 DLAgn_00071530 DLAgn_00171080 XP_030648381 XP_030627172 XP_030641298	AACCATGGTGATCCACGAGA ATGCCGATGACTCCACAGG TGCCCTCCAGCCTGCTCT CTTCTGGAAGGAGCCAAAGTC GGCTGGGAGCCAACTATCTG GGAGCTCCACCTTGGTTGTC GCATGTTGGGGACCTTGG TCAGCCTTCTCATGGATCACC GGCTTTGAAAAGGAGAGTGG CTGCAGATAATAAGCGTGCTC ATGCCCCCTGGCATTGAG GGCCCTGAAGGGACAGTT	98.8% 92.0% 94.8% 98.3% 102.3% 104.1%

Table 2

 2 Two-way ANOVA results of different gene expressions analyzed with salinity and temperature as the main factors. ns: not significant, *p < 0.05, **p < 0.01, ***p < 0.001, ****p < 0.0001. N = 8-13 per condition.

	Genes	Interaction	Salinity	Temperature
	Dlsod 1 Dlsod 2 Dl SOD activity Ccsod 1 Ccsod 2 Cc SOD activity Dlcat Dl CAT activity Cccat Cc CAT activity	ns ns ns ** ns ns ns ns ns ns	* * ns **** ns ns ns ns ns ns	ns ns ns **** ns * * ns ns *

Acknowledgement

This study was supported financially by grants from the France CNRS -Taiwan MOST PRC projects to C.L.N. (n • 27609) and T.H.L. (MOST-105-2911-I-005-501). This work was also financially supported in part by the iEGG and Animal Biotechnology Center from The Feature Area Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE), Taiwan (MOE 109-S-0023-A) to T.H.L. C.H.C was supported by postdoctoral fellowships from the MOST (108-2811-B-005-520).


T Simon

J-F Tr Égou Ët

H Morel

X Lin-Shi

M Giaccagli
email: mattia.giaccagli@univ-lyon1.fr

D Astolfi

V Andrieu

Robust Regulation of a Power Flow Controller via Nonlinear Integral Action

Tanguy Simon, Student Member, IEEE, Mattia Giaccagli, Student Member, IEEE, Jean-Franc ¸ois Tr égou ët, Daniele Astolfi, Member, IEEE, Vincent Andrieu, Herv é Morel, Senior Member, IEEE and Xuefang Lin-Shi

Abstract-This paper considers a robust output setpoint tracking problem for a m-terminal power flow controller (PFC) for meshed DC micro-grids. The PFC is a power electronics device used to control the power flow at a node in the meshed grid, and may act as a DC circuit breaker. The system is modelled by state-space bilinear dynamics coupled with a polynomial output. In the proposed design, the plant is first extended with an integral action processing the regulation error. The cascaded model composed of the plant and the integrator is then stabilised using a saturated state-feedback law, designed with a forwarding approach. An anti-windup function is added to cope with transient saturations. A tuning method is proposed to set the controller gains along the available degrees of freedom with respect to a cost function. The stability of the closed-loop is guaranteed for any compact set of initial conditions and for small parametric variations around the nominal set-point. Experiments have been carried out and these properties are successfully assessed on a tenth-scale experimental setup.

Index Terms-Bilinear systems, integral action, meshed micro-grids, power flow controllers, robust control, forwarding

I. INTRODUCTION

I n the light of the current climate breakdown, it is of paramount importance to cut global greenhouse gas emissions. The access to electrical energy holds an important place in these discussions, and direct current (DC) micro-grids respond favourably to those issues. This type of electrical power network enhances the penetration of small renewable energy generators and lowers the energy losses, while helping the clean access to electrical energy or the transition to a more energy-frugal lifestyle (see, e.g., [START_REF] Sen | Microgrid control: A comprehensive survey[END_REF], [START_REF] Simon | Modelling and Control of a Power Flow Controller for DC Microgrids[END_REF] and references therein for a broader discussion on the subject). The meshed structure of micro-grids improves this result by reducing the amount of copper needed (because there can be multiple paths between two points and the average power in each line in a building is low), as well as improving the reliability, modularity and efficiency of the system [START_REF] Mackay | Decentralized current limiting in meshed dc distribution grids[END_REF].

To control a meshed DC grid, a DC Power Flow Controller (PFC) is required. It is a multi-terminal DC-DC converter located at a node in the mesh, sometimes called a smart-node [START_REF] Zafeiratou | Dynamical modelling of a dc microgrid using a port-hamiltonian formalism[END_REF]. Its objective is the regulation of the power in each line of the node (see Fig. 1), despite the high intermittency of renewable generators.

Although PFCs for high voltage DC applications (HVDC) have received a strong academic interest (see, e.g., the recent survey [START_REF] Balasubramaniam | Series current flow controllers for dc grids[END_REF]) very little has been done for low-voltage applications (LVDC). The lower voltage rating leads to a completely different converter topology, and therefore to different control schemes. Among them, recall a multi-terminal PFC with a compensation node [START_REF] Takahashi | A multi-terminal power flow control method for next-generation dc power network[END_REF] improved by removing the compensation node [START_REF] Natori | A novel control approach to multiterminal power flow controller for next-generation dc power network[END_REF], a PFC made of two separate Split-PI converters [START_REF] Barara | Control strategy scheme for consistent power flow control in meshed dc micro-grids[END_REF], and a three-terminal PFC [START_REF] Simon | Modelling and Control of a Power Flow Controller for DC Microgrids[END_REF], [START_REF] Simon | Robust Output Set-Point Tracking for a Power Flow Controller via Forwarding Design[END_REF]. The main shortcomings of [START_REF] Takahashi | A multi-terminal power flow control method for next-generation dc power network[END_REF]- [START_REF] Barara | Control strategy scheme for consistent power flow control in meshed dc micro-grids[END_REF] are the absence of dynamic model and their control strategies, which fail to give any proof of stability or robustness. No direct control of the power is achieved, and the control laws are applied to two-terminal devices, which do not constitute a node. In [START_REF] Takahashi | A multi-terminal power flow control method for next-generation dc power network[END_REF], the control law is a constant ratio determined by a look-up table, based on the knowledge of the voltage at the end of the line, an uncertain parameter in practice. In [START_REF] Natori | A novel control approach to multiterminal power flow controller for next-generation dc power network[END_REF], a PI controller is used to regulate the current instead of the power, whose reference is again computed using the voltage at the end of the line. Moreover, the reservoir voltage, which is the voltage on a capacitor inside the converter, is not controlled and can drift outside the physical boundaries. Finally, in [START_REF] Barara | Control strategy scheme for consistent power flow control in meshed dc micro-grids[END_REF], the authors propose a current-limited voltage controller using the RST technique with hysteresis switching. The reservoir voltage is properly controlled but power flow control is not achieved.

In this study, a m-terminal power flow controller is considered. Assume a synchronous PWM switching scheme on each branch, and suppose that the grid's dynamics are partially unknown to take into account its high variability. In this context, the PFC in the grid has been recently modeled using a state-space approach [START_REF] Simon | Modelling and Control of a Power Flow Controller for DC Microgrids[END_REF]. The resulting continuous-time and finite-dimensional model is uncertain and bilinear. Moreover, the output to be regulated is a second-order polynomial. Indeed, this signal corresponds to the electrical power in all the lines, and this quantity is the product of couples of state variables, i.e. the voltage and current in each line.

The corresponding control problem is challenging as it is associated with a system which in non-linear, uncertain and with a non-linear output. While many papers dealing with stabilisation problems for bilinear systems can be found in the literature (see, for instance, [START_REF] Koditschek | Stabilizability of second-order bilinear systems[END_REF]- [START_REF] Andrieu | Global asymptotic stabilization for a class of bilinear systems by hybrid output feedback[END_REF]), very few addressed the more general problem of output regulation, e.g., [START_REF] Grasselli | Output regulation of a class of bilinear systems under constant disturbances[END_REF], and more recently [START_REF] Cisneros | Global tracking passivity-based pi control of bilinear systems: Application to the interleaved boost and modular multilevel converters[END_REF], [START_REF] Tang | Optimal output tracking control for bilinear systems[END_REF], and all of them are focused on control designs for systems having a pure linear output and are therefore not directly applicable to the problem explained above. Despite this lack of literature on output regulation, bilinear systems are a class of systems commonly employed to model physical systems, such as a heat exchanger [START_REF] Zitte | Robust control of a class of bilinear systems by forwarding: Application to counter current heat exchanger[END_REF], hydraulic systems [START_REF] Guo | A control scheme for bilinear systems and application to a secondary controlled hydraulic rotary drive[END_REF], power factor compensators and HVDC converters [START_REF] Cisneros | Global tracking passivity-based pi control of bilinear systems: Application to the interleaved boost and modular multilevel converters[END_REF], microbial cell growth [START_REF] Williamson | Observation of bilinear systems with application to biological control[END_REF] and many others [START_REF] Pardalos | Optimization and Control of Bilinear Systems: Theory, Algorithms, and Applications[END_REF].

Nevertheless, direct power flow control is achieved in [START_REF] Simon | Modelling and Control of a Power Flow Controller for DC Microgrids[END_REF] on a three-terminal PFC, using a state feedback on the linearised dynamics after adding integrators. Yet, this first attempt only gives local stability results in the state-space, and although its robustness has been tested, no proofs are given for local stability in the parametric space. Furthermore, the saturation of the duty cycles is not taken into account when designing the controller.

The authors' contributions with respect to the literature are the following. (i) The proposed controller achieves semi-global asymptotic stability in the state-space and provides robustness with respect to small parameter variations. To this end, model uncertainties and non-linearities are directly taken into account at the design step. In a nutshell, the PFC model is first extended with an integral action processing the regulation error. Then, a stabiliser for the extended system is derived following the so-called "forwarding approach" (see, e.g., [START_REF] Mazenc | Adding integrations, saturated controls, and stabilization for feedforward systems[END_REF]- [START_REF] Poulain | Robust asymptotic stabilization of nonlinear systems by state feedback[END_REF] or [START_REF] Giaccagli | Sufficient conditions for output reference tracking for nonlinear systems: a contractive approach[END_REF]- [START_REF]Incremental stabilization of cascade nonlinear systems and harmonic regulation: a forwarding-based design[END_REF] for an incremental version). This control law inherits stability properties with respect to small parameter variations as shown in [START_REF] Astolfi | Integral Action in Output Feedback for Multi-Input Multi-Output Nonlinear Systems[END_REF], [START_REF] Praly | Stabilization of nonlinear systems via forwarding mod {LgV }[END_REF]. (ii) An arbitrary number of terminals is considered, i.e. m ≥ 2 can be any integer. (iii) As the control inputs are physically represented by the duty cycles of each terminal, a saturation is applied to the control action to meet such constraints. Comparing to the unconstrained case, the closed-loop basin of attraction is preserved through the implementation of an anti-windup correction term to deal with (possible) unstable behaviours of the integrator dynamics with such input saturation. (iv) A tuning procedure for the controller gains is offered. (v) All these achievements are successfully assessed on an real tenth-scale test-bench for m = 3, and via simulations for a larger number of terminals.

A preliminary conference version of this paper was presented in [START_REF] Simon | Robust Output Set-Point Tracking for a Power Flow Controller via Forwarding Design[END_REF]. Here, as compared with [START_REF] Simon | Robust Output Set-Point Tracking for a Power Flow Controller via Forwarding Design[END_REF], 1) a controller for the m-terminal PFC is proposed to improve the applicability of the proposed method, whereas [START_REF] Simon | Robust Output Set-Point Tracking for a Power Flow Controller via Forwarding Design[END_REF] is concerned with the particular case of m = 3; 2) The saturation of the duty cycles is taken into account, not only in the simulation but also in the theoretical proofs, and an anti-windup design is proposed; 3) Additional degrees of freedom are provided in the control design, and a tuning procedure is proposed; 4) All these extensions have been successfully implemented on a real tenth-scale test-bench for m = 3 and via simulations for a larger number of terminals, while in [START_REF] Simon | Robust Output Set-Point Tracking for a Power Flow Controller via Forwarding Design[END_REF] only simulations on a 3-terminal PFC have been implemented to validate the proposed control design. The paper is organised as follows. In Section II, the model of the power flow controller is presented and the studied control problem is formalised. In Section III, the forwarding control approach is reviewed and specialised to the case of bilinear systems having a second-order polynomial output. Section IV contains the main theoretical results of the paper. The control law design is presented and it is shown that such a design guarantees global asymptotic stability of the nominal equilibrium of the closed-loop system. Such design is proven to be (locally) robust, i.e. it ensures that the regulation is achieved even in the case of sufficiently small parameter uncertainties. Finally, a tuning procedure is proposed for the control law. In Section V, the experimental measurements are presented on a 3-terminal PFC, along with simulations on a 5-terminal one. Finally, conclusions are given in Section VI. Proofs of technical Lemmas are postponed to the Appendix.

I Symbols Electrical C f Filter capacitance C R Reservoir capacitance i k , i k Current in L f i Gk , i Gk Grid current L f Filter inductance L Gk Grid inductance P k Power in line k R Gk Grid resistance v k , v k Line voltage (on C f ) V Gk Grid voltage v R , v R Reserv. volt. (on C R ) State-space
Notation. The operator "diag {}" builds a diagonal matrix from entries of the input vector argument. Given a vector a ∈ R n , the notation a k refers to the k-th element of a, with 1 being the index of the first element. The symbol Id m stands for the identity matrix of size m × m and 1 m for the column vector of size m in which each element is 1. The null matrix of size m × n is denoted by 0 m×n while 0 m denotes column vectors. Dimensions are omitted when obvious from the context. Note 1, m as every integer between 1 and m included, i.e. 1, m = {1, . . . , m}. Given a square matrix A, det(A) indicates its determinant. Given a set S, card(S) denotes its cardinality and int(S) its interior. Given

a set [a 1 , a 2 ] ⊂ R, the notation [a 1 , a 2 ] n denotes the set [a 1 , a 2 ] n := [a 1 , a 2 ]ו • •×[a 1 , a 2 ] ⊂ R n .
dz d (s m )    , (1)
for some vector s = s 1 . . . s m ∈ R m , i.e. as the functions that implement element-wise, respectively, the sat and dz function defined before. Note that the saturation constants in Sat s s are different element-wise. A summary of the notation used in this article is given in Table I to aid the reader.

II. PROBLEM STATEMENT

The Power Flow Controller (PFC) is an electrical device whose objective is to control the electric power in the lines it is connected to. As shown in Fig. 1, in this article, a general m-terminal version is considered. The chosen electrical circuit to achieve this function is made of m identical buck-boost converters whose high-side are connected in parallel to a unique reservoir capacitor noted C R . Each branch (buck-boost) is depicted on the left of Fig. 2. The grid connected to terminal k, as seen by the PFC, is modelled by a Thevenin equivalent circuit, as drawn on the right-hand side of the same figure. This equivalent circuit aims at capturing the dynamics of the network, originating from the interplay between the impedance of the conductors and the dynamics of the loads/generators connected to the grid. This circuit cannot operate properly if the reservoir voltage is not controlled 1 . This adds another control objective which can be solved knowing that such a voltage is constant if and only if the sum of average powers is equal to zero. Consequently, if the reservoir voltage is regulated to a constant value v R r and all the lines but one are regulated to a constant power reference P r k for k ∈ 1, m -1 , the power in the last line naturally converges to the overall power balance, i.e. P r m tends to -m-1 k=1 P r k . The problem tackled in this paper is the following: design a state-feedback controller delivering the duty ratios for the pulse-width modulation (PWM) switching of the transistors to achieve power control in each line of the node, while maintaining the reservoir voltage to a fixed given constant value, despite the uncertainty of the parameters characterising the grid. Moreover, to ensure flexibility and modularity, assume limited knowledge on the grid model at the end of each line.

C R v R L f i k C f v k k L Gk i Gk R Gk V Gk PFC Grid k

A. Model of The Power Flow Controller

In Fig. 2, the currents are denoted by i and measured in Amps and the dynamic voltages by v and measured in Volts. Note that the line voltage should be positive in a grid (more specifically within a precise tolerance of the nominal voltage), therefore every steady-state voltage v k will be considered nonnegative. The component parameters are denoted by L f /L Gk , C f /C R , R Gk and V Gk for, respectively, inductors (Henry), capacitors (Farad), resistors (Ohm) and constant voltages (Volts). The dynamic model of the system can be derived by using Kirchhoff's and Ohm's electrical laws, along with the dynamic electrical laws for inductors and capacitors, while assuming ideal components, see, [START_REF] Simon | Modelling and Control of a Power Flow Controller for DC Microgrids[END_REF] for a more detailed construction of the PFC's model. A synchronous PWM switching is implemented, and the dynamics are averaged over a switching period. The input vector u = [u 1 , . . . , u m ] ∈ R m is made of the duty ratio of each terminal and hence each of the u i (t) must be included in the set [0, 1] ⊂ R for all t ≥ 0. Therefore, the set of admissible inputs is defined as

U := 0, 1 m (2)
The dynamic variables are all gathered in the state vector x ∈ R n with n = 3m + 1, and the uncertain model parameters are collected in the vector θ ∈ R 3m , as shown in Table II.

The output vector corresponds to the control objectives y =

[P 1 , . . . , P m-1 , v R ] ∈ R m , with P k = i Gk v k .
A statespace model can then be obtained for the PFC, following the results in [START_REF] Simon | Modelling and Control of a Power Flow Controller for DC Microgrids[END_REF]. The system can be represented as a bilinear system. The states are the currents i k , i Gk and the voltages v k for k ∈ 1, m and v R . Additionally, the PFC is coupled with a polynomial output on which the regulation task has to be achieved. Such an output is composed by the powers P k for k ∈ 1, m and the reservoir voltage v R . In details, it can be described by a set of dynamical equations of the form

ẋ = A(θ)x + N(x)u + q(θ), y = Cx + H(x)x, u ∈ U, (3)
with

A(θ) = J(θ) -1     0 0 m 0 m 0 m 0 m 0 Id m 0 0 m -Id m 0 Id m 0 m 0 -Id m -diag {R G }     (4a) N(x)u = m j=1 (N j u j )x = J(θ) -1   0 u 0 2m -u 0 0 0 2m 0 0   x, (4b) q
(θ) = J(θ) -1 [0 n-m , V G ] , (4c) J(θ) = diag {C R , 1 m L f , 1 m C f , L G } , (4d)
C = 0 m-1 0 1 0 n-1 , (4e)
H(x) = 0 1 2 diag {i G1 , . . . , i Gm-1 } 0 m-1 0 1+m 0 m-1 0 1 2 diag {v 1 , . . . , v m-1 } 0 m-1 0 m-1 0 (4f)
where, for each j = 1, . . . , m, the N j are constant n × n matrices defined as N j = J(θ) -1 N j where N j are m different matrices full of zeros with a 1 in the first row and j + 1 column and a -1 in the first column and j + 1 row, and with

L G = [L G1 , . . . , L Gm ], R G = [R G1 , . . . , R Gm ] and V G = [V G1 , . . . , V Gm ].
Let Θ ⊂ R 3m be the non-empty set of possible system parameters that are compatible with the physics of the system. It is defined as

Θ := θ ∈ R 3m : L Gk > 0, R Gk > 0, V Gk ≥ 0, k ∈ 1, m . (5)
There is no loss of generality in these constraints since L G , R G represent physical properties (inductance and resistance) which are always strictly positive. As shown later in (42a),

P r k = 0 implies lim t→+∞ v k (t) = v k = V Gk , i.e.
when the power reference is null, the line voltage tends to V Gk . As stated before, this line voltage should always be non-negative, then so should V Gk .

The vector of references corresponds to the control objectives, i.e. r = [P r 1 , . . . , P r m-1 , v R r ] , and the non-empty set

L G1 . . . L Gm R G1 . . . R Gm V G1 . . . V Gm of possible references R ⊆ R m is defined as R := r = (P r 1 , . . . , P r m-1 , v R r ) ∈ R m : (P r 1 , . . . , P r m-1 , ) = (0, . . . , 0), v R r > 0 . (6)
where, as stated before, v R r should be sufficiently high for the device to operate properly and hence only strictly positive values have been taken into consideration. The case in which all the reference powers P r j are null has not been taken in consideration, as it makes the control objective structurally impossible, as shown later on.

B. Control problem

This article focuses on a robust regulation problem for the PFC represented by the model [START_REF] Mackay | Decentralized current limiting in meshed dc distribution grids[END_REF]. Define beforehand the compact notation δ := (θ, r) and D := Θ × R. The tackled problem is stated as follows.

Problem. Given a nominal set of parameters D and δ nom ∈ D, find functions ξ : R n × R nz → R nz and α : R n × R nz → U such that for any arbitrarily large compact set of initial conditions X × Z ⊆ R n × R nz there exists δ > 0 such that, for any δ satisfying δδ nom ≤ δ, the resulting trajectories of system (3) in closed-loop with the regulator

ż = ξ(x, z), u = α(x, z)
are bounded forward in time and lim t→∞ y(t) = r. Following an internal model based design (see, e.g., [START_REF] Astolfi | Integral Action in Output Feedback for Multi-Input Multi-Output Nonlinear Systems[END_REF], [START_REF] Francis | The internal model principle of control theory[END_REF]), the system (3) is first extended with an integral action processing the regulation error as follows

ż = y -r . (7)
Indeed, as shown in [START_REF] Astolfi | Integral Action in Output Feedback for Multi-Input Multi-Output Nonlinear Systems[END_REF], the use of the integral action ( 7) is necessary to achieve asymptotic tracking in the presence of (sufficiently small) perturbations of the plant model in every direction. The key feature of the integral action is that if there exists an equilibrium for the closed-loop system, then the regulation error is necessarily equal to 0 at this equilibrium. Furthermore, the control law is saturated since u ∈ U, and this physical constraint may lead to unstable behaviours in the dynamics of the integrator. To deal with this issue, an anti-windup design is implemented (see [START_REF] Tarbouriech | Anti-windup design: an overview of some recent advances and open problems[END_REF]- [START_REF] Galeani | A tutorial on modern anti-windup design[END_REF]) to mitigate the effects of uncontrollable integral action. This leads to the following integral dynamics

ż = ξ(x, z) := y -r + ζ(x, z) (8)
where the function ζ represents such an anti-windup. The overall proposed control scheme is depicted in Fig. 3. The design of the function α(•) that has been implemented, follows the so-called forwarding approach. The next section specialises and enriches such a technique to the class of bilinear systems having a second-order polynomial output.

III. FORWARDING DESIGN FOR A CLASS OF BILINEAR

SYSTEMS WITH A POLYNOMIAL OUTPUT Consider a system composed by a cascade of a bilinear system having a second-order polynomial output feeding an integrator of the form

ẋ = Ax + (N (x) + B)u, ż = Cx + H(x)x, (9)
where Standing Assumption. The number of inputs is not smaller than the number of outputs, i.e m ≥ p.

(x, z) ∈ R n+p is the state, u ∈ R m is the control input, A, B
The following result specialises the forwarding control design (see, e.g., [START_REF] Astolfi | Integral Action in Output Feedback for Multi-Input Multi-Output Nonlinear Systems[END_REF]- [START_REF]Sufficient conditions for global integral action via incremental forwarding for input-affine nonlinear systems[END_REF], [START_REF] Praly | Stabilization of nonlinear systems via forwarding mod {LgV }[END_REF] and the references therein) for systems of the form of (9). Proposition 1. Consider system [START_REF] Simon | Robust Output Set-Point Tracking for a Power Flow Controller via Forwarding Design[END_REF]. Suppose that A is Hurwitz and that the matrix CA -1 B is full rank. Select P = P 0 and M i = M i such that

P A + A P ≺ 0, ( 10
) M i A + A M i = 1 2 (H i + H i ) , ∀i ∈ 1, p , (11)
and let M 0 = CA -1 . Then, for any matrix Ω = Ω 0, the origin of system (9) in closed-loop with u = ψ(x, z), with the function ψ : R n+p → R m defined as

ψ(x, z) = -x P (N (x) + B) -(z -M (x)) Ω (M 0 + 2R(x)) (N (x) + B) , (12) with R(x) := M 1 x . . . M p x (13) M (x) := M 0 x + R(x)x , (14)
is globally asymptotically stable and locally exponentially stable.

Proof. The proof is based on a Lyapunov function construction which follows the results presented in [START_REF] Poulain | Robust asymptotic stabilization of nonlinear systems by state feedback[END_REF]. In particular, let W : R n+p → R be defined as

W (x, z) = 1 2 x P x + 1 2 (z -M (x)) Ω(z -M (x)) , (15)
with P and M defined respectively by ( 10) and ( 14) as in the statement of the proposition. Note that W is proper and positive definite. By construction, the function M satisfies

∂M ∂x (x)Ax = M 0 Ax +    x (M 1 A + A M 1 )x . . . x (M p A + A M p )x    , = Cx + 1 2    x (H 1 + H 1 )x . . . x (H p + H p )x    = Cx + H(x)x . (16)
Hence, the time derivative of W along the solutions of system (9) satisfies

Ẇ (x, z) = x (P A + A P )x + [x P (N (x) + B) -(z -M (x)) Ω (M 0 + 2R(x))(N (x) + B)] u ,
where the relation ( 16) has been used. Using again the definition of M in ( 14) and the definition of ψ in (12) yields

Ẇ (x, z) = x (P A + A P )x -ψ (x, z)ψ(x, z) ≤ 0 . Furthermore, note that {(x, z) : Ẇ (x, z) = 0} = {(x, z) : x = 0, ψ(x, z) = 0}.
(17) Moreover, using the definition of M 0 , ψ (0, z) = z ΩCA -1 B is obtained. Since by assumption, CA -1 B and Ω are full rank, the set {(x, z) : Ẇ (x, z) = 0} coincides with the origin and therefore (x, z) → Ẇ (x, z) is negative definite. Consequently, W is a Lyapunov function of the closed-loop system and the origin is globally asymptotically stable. Finally, employing the same method, note that the quadratic function

W 0 (x, z) = 1 2 x P x + 1 2 (z -M 0 x) Ω(z -M 0 x),
is a Lyapunov function for the first-order approximation

ẋ = Ax + Bψ 0 (x, z), ż = Cx,
with ψ 0 (x, z) being the first-order approximation of ψ. Hence, local exponential stability of the equilibrium is obtained.

IV. MAIN RESULTS

In this section, the main theoretical results of the paper are stated. Given the PFC model (3), a first detailed analysis of the set of solutions that solve the problem is given. Then, the controller design shown in Fig. 3 is presented. It is defined as an integral action processing the tracking error implemented with an anti-windup design, and a state-feedback stabiliser for the closed-loop based on forwarding design. It will also be shown that the presented design is able to achieve output regulation even in the case of (sufficiently small) parameter uncertainties. Moreover, a tuning procedure for the control gains is provided.

A. Set of solutions

The considered problem may not be solvable for all δ = (θ, r) ∈ D since for some values of δ there may not exist an equilibrium pair (x , u ) satisfying y = r within the input constraints. Therefore, for a given δ ∈ D, let E(δ) be the set of admissible equilibrium points, namely the set of steady-state solutions on which output regulation is achieved, i.e.

E(δ)

:= (x , u ) ∈ R n × U : A(θ)x + N(x )u + q(θ) = 0, Cx + H(x )x = r . (18)
The set S is then defined as the set of admissible parameters and references δ for which there exists such equilibrium points:

S := δ ∈ D : card(E(δ)) > 0 . (19)
A characterisation of S can then be provided through model inversion as shown below, whose proof has been postponed to the appendix.

Proposition 2. Consider system (3). Then

S = δ ∈ D : ∆ k (δ) ≥ 0, 0 ≤ 1 2v R r V Gk ± ∆ k (δ) ≤ 1, k ∈ 1, m , (20)
where

∆ k (δ) := V 2 Gk -4R Gk P r k ( 21
)
with P r m := -m-1 k=1 P r k . Moreover, for a given δ ∈ S, there exists from one to 2 m pairs (x , u ) ∈ E(δ).

B. Controller design

This section aims to solve the regulation problem presented in Section II-B for the PFC (3) following the control structure presented in Fig. 3. The controller is designed for some known nominal parameters and references δ nom := (θ nom , r nom ). Furthermore, it is assumed that this pair belongs to int(S). This allows to prove that output set-point tracking is still achieved for δ distinct from δ nom but sufficiently close to it, as specified later on. The main result of the paper is stated as follows.

Theorem 1. Consider the set S defined in [START_REF] Tang | Optimal output tracking control for bilinear systems[END_REF]. Select any δ nom = (θ nom , r nom ) ∈ int(S) and any corresponding (x nom , u nom ) ∈ E(δ nom ). Then, for d > 0 sufficiently high, the Problem for system (3) is solved by the control law

ż = ξ(x, z) := y -r -Dz d (z -M (x -x nom )) u = α(x, z) := u nom + Sat 1-u nom -u nom (ψ(x -x nom , z)) ( 22
)
where the functions Sat and Dz are defined in (1), the function ψ is chosen as [START_REF] Quinn | Stabilization of bilinear systems by quadratic feedback controls[END_REF] with the matrices A, B, C and functions N, H defined as

A := A(θ nom ) + m j=1 N j u nom,j , B := N(x nom ) C := C + 2H(x nom ), N (x)
:= N(x), H(x) := H(x), P, M defined as in [START_REF] Koditschek | Stabilizability of second-order bilinear systems[END_REF], [START_REF] Banks | Stabilizability of finite-and infinite-dimensional bilinear systems[END_REF] and any Ω = Ω 0.

Remark 1. Note that the integral action dynamics is designed in the form of (8) with the anti-windup term that takes the form of a dead-zone function. The function α is selected as a first (nominal) feed-forward action u nom plus a second term made by saturating the function ψ derived from the forwarding approach in Proposition 1. Thanks to the saturation, the stabiliser satisfies the input constraints as α takes only values in U, ensuring the validity of the control law with respect to the model (3). Indeed the following holds

0 ≤ u = u nom + Sat 1-u nom -u nom (ψ(x, z)) ≤ 1
In this sense, note that the control law ψ in Proposition 1 is of infinite gain margin, i.e. κψ(•) is still a stabiliser for (9) for all gains κ > 0, and hence the saturation does not restrict the set of solutions. To this end, recall the well-known link between forwarding and small-input control ( [START_REF] Kaliora | On the stabilization of feedforward systems with bounded control[END_REF]) .

Proof. The proof is divided in three parts. It is first shown that the origin of the closed-loop system (3), ( 22) is globally asymptotically stable for δ = δ nom . Then, it is proven that for d sufficiently large, the anti-windup effect disappears in the target equilibrium (and so regulation is achieved). Finally it is shown that the proposed design is robust to sufficiently small model parameter variations and semi-global asymptotic stability on an equilibrium is guaranteed (on which the regulation objective is satisfied).

Part 1: Global output regulation. Consider δ = δ nom ∈ int(S) and (x , u ) ∈ E(δ). Define the following change of coordinates:

  u x z   →   u x z   :=   u -u nom x -x nom z -z nom  
in which z nom = 0. In these coordinates, system (3) reads

ẋ = A(θ)(x + x nom ) + N(x + x nom )(u + u nom ) + q(θ), = A(θ) + m j=1 N j u nom,j x + N(x + x nom )u (23)
while the z-dynamics reads

ż = (C + 2H(x nom ))x + H(x)x,
for which the relations Cx nom + H(x nom )x nom = r and H(x)x nom = H(x nom )x have been used. By selecting the matrices A, B, C and the functions N, H as in the statement of the theorem, a system of the form of ( 9) is obtained.

The following two technical lemmas show that A is Hurwitz and that the non-resonance condition CA -1 B full rank holds. Their proofs are postponed to the Appendix.

Lemma 1. Pick any δ ∈ int(S). Then for all (x , u ) ∈ E(δ), the matrix A = A(θ) + m j=1 N j u j is Hurwitz. Lemma 2. Pick any δ ∈ int(S). Then for all (x , u ) ∈ E(δ) the matrix CA -1 B is full rank.

Remark 2. Lemma 1 can be understood as the natural stability of the system: in practice, for any constant duty ratio, the system stabilises to a steady-state equilibrium point. Concerning Lemma 2, it has been shown in [START_REF] Astolfi | Integral Action in Output Feedback for Multi-Input Multi-Output Nonlinear Systems[END_REF] that such condition is necessary to achieve output set-point tracking in case of (sufficiently small) parametric uncertainties in every direction. Since the linearized model around the equilibrium point is stabilizable, this condition implies controllability of the extended (plant and integral action) linearised system and it is commonly referred to as "non-resonance condition" It follows from the proof of Lemma 2 that the points δ ∈ ∂S := S \ int(S) do not satisfy such a condition.

Following the proof of Proposition 1, consider the Lyapunov function

W (x, z) = 1 2 x P x + 1 2 (z -M (x)
) Ω(z -M (x)) where P = P 0 verifies P A + A P ≺ 0, the function M is defined as in [START_REF] Banks | Stabilizability of finite-and infinite-dimensional bilinear systems[END_REF] and Ω is any symmetric positive definite matrix. Note that P always exists in view of Lemma 1. Computing the time derivative and following the same steps of the proof of Proposition 1, it follows that

Ẇ (x, z) = -1 2 x (P A + A P )x -Ψ(x, z) -(z -M (x)) Ω Dz d (z -M (x)) (24)
with Ψ(x, z) := ψ(x, z) Sat

1-u nom -u nom (ψ(x, z))
with ψ defined as in the statement of the theorem, i.e. in [START_REF] Williamson | Observation of bilinear systems with application to biological control[END_REF]. The function Ψ verifies

Ψ(x, z) > 0 ∀ (x, z) = (0, 0). ( 25
)
To show the previous inequality, let ψ i (x, z) denote the i-th element of ψ(x, z). By definition of the saturation function in (1),

Ψ(x, z) = m i=1 Ψ i (x, z) = m i=1 ψ i (x, z) sat 1-u nom,i -u nom,i (ψ i (x, z))
and, for any i ∈ 1, m , the following conditions hold: 1) if ψ i (x, z) < -u nom,i , then [START_REF] Astolfi | Integral Action in Output Feedback for Multi-Input Multi-Output Nonlinear Systems[END_REF] since each element of the sum is positive. Moreover, since s Dz d (s) ≥ 0 for any s, d, it follows from (24) that Ẇ (x, z) ≤ 0. Since this derivative is a sum of nonnegative terms, Ẇ (x, z) = 0 if and only if each term is null. Moreover, x (P A + A P )x = 0 if and only if x = 0. It follows that

Ψ i (x, z) = -ψ i (x, z)u nom,i ≥ 0; 2) if -u nom,i ≤ ψ i (x, z) ≤ 1 -u nom,i , then Ψ i (x, z) = ψ i (x, z) 2 ≥ 0; 3) if ψ i (x, z) > 1 -u nom,i , then Ψ i (x, z) = ψ i (x, z)(1 -u nom,i ) ≥ 0, proving
Ψ(0, z) = 0 ⇐⇒ ψ i (0, z) sat 1-u nom,i -u nom,i (ψ i (0, z)) ∀i ∈ 1, m ⇐⇒ ψ(0, z) = 0 ⇐⇒ z = M (0) = 0,
hence that Ẇ (x, z) = 0 ⇐⇒ (x, z) = (0, 0) and that the origin of the closed-loop system is globally asymptotically stable.Following the proof of Proposition 1, it is possible to show that it is also locally exponentially stable. Part 2: Anti-windup function design. The dead-zone function Dz d has been introduced to implement an anti-windup in the integral action. It follows from the Lyapunov analysis that such function does not compromise the stability of the nominal closed-loop. However, the constant d must be chosen sufficiently large so that when trajectories are close to the equilibrium point for which y = r, the anti-windup has no effect (i.e. the dead-zone is equal to zero) and set-point tracking is achieved. The dead-zone constant d must therefore satisfy

d ≥ d, d := sup (x,z)∈Dz×D M {||z -M (x)||}
where D z , D M are defined as the sets containing all possible equilibrium of (x, z). On one hand, when δ = δ nom , i.e. the system converges to the nominal equilibrium, d can be taken equal to 0 as the feed-forward action u = u nom is sufficient to bring the system to the equilibrium where regulation is achieved. On the other hand, when δ = δ nom , d must be sufficiently high: the dead-zone constant must be chosen with respect to the plant's uncertainties, so that the anti-windup effect vanishes on the (new) equilibrium point. Giving a detailed formulation of d is cumbersome and out of the scope of the paper, as its value affects the largest admissible bounds on the set-points δ (see Part 3 of the proof of this Theorem) and reciprocally. Its computation has therefore been omitted. For practical implementations, once a bound on the possible uncertainties for the parameters and references is known (and hence a neighbourhood of the nominal equilibrium), taking d sufficiently high will guarantee that the anti-windup disappears on the new target equilibrium.

Such aspects have been remarked also in the experimental part in Section V. Taking d too high will not compromise the stability or the regulation, but will simply result in a delay of the effect of the anti-windup. Part 3: Robustness analysis. To conclude the proof of the theorem, we aim to show that for every compact set of initial conditions, there exists a bound of the parameters' uncertainty δ such that, if |δδ nom | ≤ δ, then the following can still be guaranteed: i) that an equilibrium point for the closed-loop system exists; ii) that such an equilibrium is asymptotically and locally exponentially stable; iii) that the regulation task is still achieved. In this aim, let w := (x, z) and let the nominal closed-loop (3), ( 22) be defined by

ẇ = ϕ(w, δ nom ) . (26)
From the Lyapunov analysis in Part 1, there exist a radially unbounded Lyapunov function W and a positive definite function V such that ∂W ∂w (w)ϕ(w, δ nom ) ≤ -V (w) < 0 where w = (xx nom , zz nom ), for which the origin of the closed-loop is globally asymptotically stable and locally exponentially stable . Therefore, for each compact set D, there exists two compact sets of initial conditions containing the origin and denoted C and C, both in X × Z, such that C is forward invariant 2 for the closed-loop system [START_REF] Poulain | Robust asymptotic stabilization of nonlinear systems by state feedback[END_REF]. Therefore, by [START_REF] Astolfi | Integral action in output feedback for multiinput multi-output nonlinear systems[END_REF]Lemma 5] there exists ρ > 0 such that, for each C 1 vector field ϕ p satisfying

|ϕ p (w, δ) -ϕ(w, δ nom )| ≤ ρ, ∀ w ∈ C, (27)
∂ϕp ∂x (w, δ) -∂ϕ ∂x (w, δ nom ) ≤ ρ, ∀ w ∈ C, (28)
there exists an exponentially stable equilibrium of

ẇ = ϕ p (w, δ) ,
whose basin of attraction contains the compact set C. Let us define now the function ρ :

D → R >0 as ρ(δ) := max w∈ C, |ϕ(w, δ) -ϕ(w, δ nom )| , ∂ϕ ∂x (w, δ) -∂ϕ ∂x (w, δ nom ) .
Such a function is continuous and satisfies ρ(δ nom ) = 0. Any positive real number δ > 0 can now be selected such that, |δδ nom | ≤ δ implies ρ(δ) ≤ ρ. This parameter δ > 0 is a solution to the third part of the proof. Indeed, for each δ such that |δδ nom | ≤ δ, then the closed loop system ẇ = ϕ(w, δ) , satisfies ( 27) and ( 28) and consequently admits an exponentially stable equilibrium with a basin of attraction containing D × {0}. Thanks to the integral action, the output set-point tracking is achieved on this equilibrium, which concludes the proof.

Remark 3. A more precise characterisation of the robustness bound δ can be given by explicitly computing its value. Indeed

2 Simply pick C = {(x, z) : W (x, z) ≤ c 0 } for sufficiently large c 0 .
the closed-loop is a polynomial system, hence computation solvers or polynomial optimisation tools (see [START_REF] Henrion | Gloptipoly: Global optimization over polynomials with matlab and sedumi[END_REF], [START_REF] Henrion | Gloptipoly 3: moments, optimization and semidefinite programming[END_REF]) could be used.

C. Tuning of the control gains

The proposed control law α(x, z) in ( 22) admits some freeto-choose parameters. In particular, any matrices P = P satisfying [START_REF] Koditschek | Stabilizability of second-order bilinear systems[END_REF] and Ω = Ω 0 could be selected. The aim of this section is to give guidelines for choosing the matrices P and Ω with regard to some cost function around the nominal equilibrium point. Consider the nominal system (3), ( 22) with δ = δ nom . Since only local performance is sought around the nominal equilibrium, the tuning procedure is designed without taking into account the saturation function and the dead-zone anti-windup, as these functions have no effect sufficiently close to the equilibrium point. Taking the nominal error coordinates

(x, z) → (x, z) = (x -x nom , z -z nom ), the closed-loop system writes ẋ = Ax + (N (x) + B)u ż = Cx + H(x)x u = ψ(x, z) (29)
where u = uu nom , A, B, N, C and H are taken as in Theorem 1 and ψ(•) is taken accordingly to Proposition 1. Consider the cost function J : R n × R m → R >0 defined as

J := ∞ 0 w (t) Qw(t) + u (t) Ru(t) dt (30)
for some given matrices Q = Q 0 and R = R 0, where w(t) = (x(t), z(t)) is the trajectory of the closed-loop at time t for some initial conditions. Consider the linearisation of the closed-loop system (29) around w = (x, z) = (0, 0):

ẇ = Ã w + B ũ ũ := K w ( 31
)
where w ∈ R n+m , ũ ∈ R m and

à := A 0 C 0 , B := B 0 K := -B P + B M 0 ΩM 0 -B M 0 Ω (32)
For a linear system of the form (31), a linear state-feedback optimal control law ([39, Section 3.12]) is given by

ũ = Kopt w = -R-1 B S w (33)
where S = S 0 is the solution of the algebraic Riccati equation

S Ã + Ã S -S B R-1 B S = -Q (34)
The idea is to choose the control law degrees of freedom such that its linearisation K is as close as possible to the optimal control law for linear systems Kopt . In this sense, it can be rewritten as an optimisation problem of the form min

P,Ω,ε ε s.t. ( K -Kopt )( K -Kopt ) -εId m 0 P 0, Ω 0, ε ≥ 0 A P + P A ≺ 0 (35)
where K depends on P and Ω, as seen in [START_REF] Tarbouriech | Anti-windup design: an overview of some recent advances and open problems[END_REF]. This problem can be expressed in LMI form using the Schur complement:

min P,Ω,ε ε s.t. -εId m ( K -Kopt ) ( K -Kopt ) -Id 0 P 0, Ω 0, ε ≥ 0 A P + P A ≺ 0
which is a semi-definite program for which efficient solvers exist, see for instance [START_REF] Henrion | Gloptipoly: Global optimization over polynomials with matlab and sedumi[END_REF], [START_REF] Henrion | Gloptipoly 3: moments, optimization and semidefinite programming[END_REF], [START_REF] Vanantwerp | A tutorial on linear and bilinear matrix inequalities[END_REF].

Remark 4. As the system's dynamics involve the plant and the integrator, in most applications there is no interest in choosing the cost function to be dependent on the full state space, i.e.

Q is generally positive semi-definite. Hence one can select Q := C C for some matrix C of appropriate dimensions, where ( Ã, C) is assumed to be detectable to still provide convergence of the closed-loop system towards an equilibrium point. Moreover, note that the plant's uncertainties play a role in this tuning, suggesting that more advanced techniques could be used, such as robust optimal control or stochastic optimal control. As this focus is out of the main scope of this work, the details will not be explained, and the interested readers may refer for instance to [START_REF] Van Handel | Stochastic calculus, filtering, and stochastic control[END_REF]- [START_REF] Chen | The robust optimal control of uncertain systems-state space method[END_REF] and the references therein.

V. EXPERIMENTATIONS

This section presents tenth scale experimental measurements of the PFC in closed-loop with the proposed controller. The experimental setup is presented in Fig. 4 and5. A dSPACE MicroLabBox rapid prototyping system (µLB) is used to control the PFC. A voltage-controlled electronic load in series with a resistor (EL+R) is connected to the first terminal of the PFC through thirty metres of standard U1000 RV2 cable (L G ). The two other terminals are connected to a resistor (R) and a power supply (PS), also through thirty metres of the same cable.

The proposed controller is tested through different scenarios. First, it is verified that the control objectives can be properly reached, despite perturbations of the uncertain parameters. Second, a test is performed to show the efficiency of the anti-windup function. Third, an experiment illustrates the effect of the proposed tuning of the parameters. Finally, the generalisation to m-terminal is illustrated by a simulation on MATLAB Simulink, for m = 5.

Unless otherwise stated, the parameters of control law (22) are selected as follows: the matrices P and M i , i ∈ 1, p are computed using CVX, a package for specifying and solving convex programs [START_REF] Grant | CVX: Matlab software for disciplined convex programming, version 2.1[END_REF], [START_REF]Graph implementations for nonsmooth convex programs[END_REF], such that 22) is multiplied by a scalar positive gain κ = 0.01 inside the saturation (see Remark 1) and the tuning matrix Ω is chosen equal to diag {[1, 1, 5]}. These values have been found by experimentally tuning the parameters to reach satisfactory dynamics.

A(θ nom ) P + P A(θ nom ) -Id n , M i A(θ nom ) + A (θ nom )M i = 1 2 (H i + H i ), the function ψ in (

A. Robust Regulation

The system is initialised on the nominal set-point δ nom = (θ nom , r nom ), whose numerical values are displayed in Table III. At t = 21 ms, the reference is changed to r a , and at t = 61.5 ms, the system is perturbed with a voltage step on the electronic load, from 2 V to 10 V, so that θ = θ a (see Table III). From Fig. 6, it can be concluded that the objectives are properly reached in 20 ms, so that modelling and measurement errors are properly compensated for. Moreover, observe that although no reference is tracked on the third line, a constant power value is asymptotically reached. This asymptotic value is determined by the total power balance, so that it equals -P r 1 -P r 2 plus the power losses in the converter.

B. Saturation and anti-windup

Since the control law does not exhibit large oscillations with the proposed tuning, it not possible to illustrate the effect of transient saturations and of the anti-windup term. Moreover, it is assumed in the theoretical developments that all given references are reachable (i.e. belong to S). However,bearing in mind that in practice the power references are given by a higher-layer control algorithm, observe that if a constant perturbation occurring on a line makes the given reference

-50 W v R r 50 V ra P r 1 -60 W P r 2 -60 W v R r 50 V
unreachable, it will induce a saturation of the duty cycle and lead to an integral wind-up, until the higher-layer controller is able to deliver a new attainable reference. This scenario has been tested and the most relevant signals are displayed in Fig. 7. P r 1 is set to -85 W, which is attainable when V G1 = 2 V, but not when V G1 = 10 V. Such a step on V G1 is implemented via the electronic load at t = 0.1 s. This prevents the power reference to be reached due to duty cycle constraints. Assume that the higher-layer controller delivers a new reachable reference P r 1 = -75 W at t = 0.5 s. Without anti-windup action, i.e. ζ = 0 on (8), the value of z 1 increases continuously, and the integral wind-up problem is clearly illustrated as the tracking capability is only recovered after 600 ms (in blue). This is in contrast with z 1,AW which saturates around 0.2 J, leading to a much faster recovery of the tracking capability (in red).

C. Evaluation of the tuning method

In Section IV-C, a procedure is given to assist the tuning of the control parameters: define a cost function in [START_REF] Praly | Stabilization of nonlinear systems via forwarding mod {LgV }[END_REF], solve the Riccati equation in [START_REF] Galeani | A tutorial on modern anti-windup design[END_REF] and deduce the control gain using [START_REF] Zheng | Anti-windup design for internal model control[END_REF]. Then, chose P and Ω along the optimisation problem [START_REF] Kaliora | On the stabilization of feedforward systems with bounded control[END_REF]. This procedure has been followed: for simplicity, the cost function is chosen using the matrices Q = Id 13 and R = 0.01 × Id 3 . Indeed, since the inputs are saturated duty cycles, there is no need to minimise their magnitude, so the associated cost is small. The Riccati equation has been solved using the icare(•) function in MATLAB.

In order to evaluate the impact of Ω, matrix P is kept unchanged while different values for Ω are tested with the optimisation problem. One then notices that for Ω = Id 3 , smaller values of lead to a smaller ε, i.e. K is closer to Kopt . This is verified during a startup test on the nominal system. Considering (θ nom , r nom ) given by Table III at t = 2 ms, the reservoir voltage transients are displayed in Fig. 8 for different . It can clearly be noticed that decreasing the value of Ω improves the dynamics on the nominal set-point. However, note that this matrix multiplies integral part z the current law (see [START_REF] Quinn | Stabilization of bilinear systems by quadratic feedback controls[END_REF]). Therefore, the selection of Ω appears to be a trade-off between nominal performance, suggesting Ω = 0, and accuracy of the steady-state, requiring Ω 0.

D. Simulation of a 5-terminal PFC

This paper generalises the control law from [START_REF] Simon | Robust Output Set-Point Tracking for a Power Flow Controller via Forwarding Design[END_REF] to m branches. A MATLAB-Simulink simulation validates this extension for m = 5. The two new branches have the same parameters and references as branches one and two and have been inserted between the second and the third ones, i.e. they are numbered 3 and 4. Thus, the numerical values are L G1,2,3,4,5 = 18 µH, R G1,3 = 21.7 Ω, R G2,4 = 24.5 Ω, R G5 = 1.2 Ω, V G2,4 = 0 V and V G5 = 40 V. V G1,3 is initially set to 2 V and changed to 10 V at t = 61.5 ms. The power references are all initially at P The resulting reservoir voltage and power signals are drawn in Fig. 9. Their comparison to those of Fig. 6, shows that the proposed model is able to accurately capture the dynamics of the system. Furthermore, this simulation suggests that the generalisation of the control law to more than three terminals maintains both the stability and the performance of the system.

VI. CONCLUSION

A robust controller has been designed for an m-terminal PFC ensuring the regulation of the electrical power in each line of the node, while explicitly taking into account the saturation constraints of the duty cycles. The control law is made of an integral action processing the regulation error, and a stabiliser for the resulting cascade system is designed via forwarding techniques. An input saturation function is included to meet the physics of the system and an anti-windup design is added to prevent a possible instability caused by this saturation. A procedure to tune the controller with respect to a cost function is then presented. An experimental setup for a 3-branch PFC has been built to show the validity of the proposed control law. The experiments show the robustness of the proposed design, as well as the performance of the anti-windup action and the tuning method. Moreover, simulations on a 5-branch PFC have been implemented to shown the validity of the extension of the control law to any m-terminal PFC.

The application of the control problem requires a stability of the closed-loop system for a very large domain in the parametric and references spaces. Future studies will involve finding other methods to enhance the robustness of the proposed design, involving the choice of an optimal nominal setpoint and numerical simulations.

Using (36d): -R Gk i Gk = -V Gk + v k then gives (-V Gk + 2v k )i Gk = 0, k ∈ 1, m -1 and, from (42a): v k = 1 2 V Gk ± ∆ k (δ) , one finds ±i Gk ∆ k (θ, r) = 0, k ∈ 1, m -1 .

In int(S), ∆ j (θ, r) > 0, hence i Gk = 0. Remembering that by definition of Θ and R, R Gk > 0 for k ∈ 1, m and v R r > 0 and using [START_REF] Grant | CVX: Matlab software for disciplined convex programming, version 2.1[END_REF], it can be concluded that v k = i k = u k = 0, k ∈ 1, m -1 . Using (43a), yields

1 C R (u m i m + i m u m ) = 0, (45) (44)
⇐⇒

u m i m -i m R Gm v R r i m = 0, (46)
(42d),(42b)

⇐⇒ v m v R r i m -1 v R r (V Gm -v m )i m = 0, ( 47
) (42a) ⇐⇒ 1 v R r ∆ k (δ) = 0. (48)
Again, since v R r > 0 and ∆ k (δ) > 0 ∀δ ∈ int(S), it follows that i m = 0, hence that i Gm = v m = u m = 0 by [START_REF] Grant | CVX: Matlab software for disciplined convex programming, version 2.1[END_REF], and [x , u ] = 0. It can therefore be concluded that T is full rank for all (θ, r) ∈ int(S).

Fig. 1 .

 1 Fig. 1. m-terminal PFC at a node in the grid.

Fig. 2 .

 2 Fig. 2. Detail of the k-th branch of the PFC (left) and the Thevenin grid model as seen by this terminal (right)

Fig. 3 .

 3 Fig. 3. Proposed control structure for the PFC

Fig. 5 .

 5 Fig. 4. Picture of the experimental setup

Fig. 6 .

 6 Fig. 6. Experimental measurements of the PFC in closed-loop with control Reservoir voltage and power in each line with their references

Fig. 7 .Fig. 8 .

 78 Fig.7. Experimental measurements illustrating the efficiency of the dead-zone function method to prevent an integral wind-up when the setpoint is unreachable. Power on the first line with its reference (top) and corresponding integrator (bottom)

r 1 , 2 , 3 , 4 =

 1234 -50 W, and stepped to -60 W at t = 21 ms. The reservoir voltage reference v R r remains at 50 V.

Fig. 9 .

 9 Fig. 9. Simulation of the PFC model (3) for m = 5, in closed-loop with control (22). Reservoir voltage and power in the m -1 first lines with their references

TABLE

  Then, given any two vectors s = s1 . . . sm ∈ R m , s = s 1 . . . s m ∈ R m and any scalar d ∈ R, define the following (vector) saturation function Sat s s : R m → R m and the following (vector) deadzone function Dz d : R m → R m as

		Define the following
	asymmetrical scalar saturation function sat s s : R → R as  s if s < s   sat s s (s) =   s s if s ≤ s ≤ s if s > s
	for some constants s ≤ s ∈ R and the following scalar dead-zone function dz d : R → R as
	dz d (s) = s -sat d -d (s)
	for some constant d > 0. Sat s s (s) =    sat s1 s 1 (s 1 ) . . .    ,	Dz d (s) =	  	dz d (s 1 ) . . .
	sat sm s m (s m )

  , C are matrices of suitable dimensions, and functions

H : R n → R p×n and N : R n → R n×m defined as H(x) = H 1 x . . . H p x and N (x) = N 1 x . . . N m x for some n × n matrices H 1 , . . . , H p , N 1 , . . . , N m .

If v R < v k for some k ∈ 1, m , the current flows freely through the diode in the upper transistor of the k th branch, which cannot be controlled by the PWM switching (see Fig.

[START_REF] Simon | Modelling and Control of a Power Flow Controller for DC Microgrids[END_REF]

APPENDIX

A. Proof of Proposition 2

By considering [START_REF] Mackay | Decentralized current limiting in meshed dc distribution grids[END_REF] and posing ẋ = 0, the following set of equations is found

where k ∈ 1, m , and

Solving (36b) and (36c) results in

where v R = v R r > 0. Hence the first condition for the existence of solutions is 0 ≤

Taking [START_REF] Henrion | Gloptipoly: Global optimization over polynomials with matlab and sedumi[END_REF] in [START_REF] Kirk | Optimal control theory: an introduction[END_REF] results in

from which P r m := -m-1 k=1 P r k = i Gm v m is defined. Then, multiplying (36d) by v k and using (37a), it follows that

hence the second condition for the existence of real solutions is

≥ 0, the characterisation of E(δ) can be given using [START_REF] Astolfi | Integral action in output feedback for multiinput multi-output nonlinear systems[END_REF] and [START_REF] Henrion | Gloptipoly: Global optimization over polynomials with matlab and sedumi[END_REF] as

B. Proof of Lemma 1

To show that A is Hurwitz, it is sufficient to show that the origin of the system ẋ = Ax is globally asymptotically stable. Taking the quadratic Lyapunov function

by definition of Θ. By applying LaSalle's invariance principle, the system converges to the following set {x : VJ (x) = 0} = Im{[Id n-m 0 (n-m)×m ] }. Pick any element x so that i Gk ≡ 0, for all k ∈ 1, m . Plugging it in the i Gk -dynamic, see (4a) resuts in

and so i k ≡ 0. Finally, by looking at the i k -dynamic, the following holds

Before concluding that v R ≡ 0, it is necessary to prove that (u 1 , . . . , u m ) = (0, . . . , 0). From (42d) and the definition of R, a necessary and sufficient condition is (v 1 , . . . , v m ) = (0, . . . , 0). From (42a), and the definition of Θ, v k equals zero on two conditions: The first one is V Gk = 0 and ∆ k = 0, which is excluded by taking (δ) ∈ int(S). The second one is V Gk -V 2

Gk -4R Gk P r k = 0, which occurs if and only if P r k = 0, by definition of Θ. Yet, the definition of R ensures that (P r 1 , . . . , P r m-1 ) = (0, . . . , 0), meaning there is at least one u k = 0 and so v R ≡ 0 for all (θ, r) ∈ int(S), which concludes the proof.

C. Proof of Lemma 2

As the number of inputs equals the number of outputs, CA -1 B is full rank if and only if the following matrix is injective:

Using (43b),(43c),(43d) and (43f), one gets

Placing the latter in (43e) yields