Anitha Muralidhara

Anne Bado-Nilles

Guy Marlair

Victor Engelen

Christophe Len

Pascal Pandard

Humins in the environment: early stage insights on ecotoxicological aspects

Keywords: Humins, furanics, ecotoxicity, immunomarkers, biodegradability, byproduct valorization

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INTRODUCTION

Furanic compounds have recently drawn noticeable techno-economic interest due to the their production from non-edible parts of lignocellulosic biomass to produce fuels, chemicals and materials. [START_REF] Jong | Furanics: A novel diesel fuel with superior characteristics[END_REF] For this very reason, some of these furanic compounds have been listed by the U.S Department of Energy as one of the top 12 and top 30 high potential chemical building blocks. [START_REF] Werpy | Top Value Added Chemicals from Biomass: Volume I --Results of Screening for Potential Candidates from Sugars and Synthesis Gas[END_REF] In a similar context, Avantium's YXY® process produces 2,5-Furandicarboxylic acid (FDCA) for the development of bio-plastics. 3 FDCA production process primarily involves the dehydration of carbohydrates (C6 and C5 sugars) into alkoxymethyl furfural (RMF). [START_REF] Jong | Furanics: A novel diesel fuel with superior characteristics[END_REF][START_REF] Jong | Furandicarboxylic Acid (FDCA), A Versatile Building Block for a Very Interesting Class of Polyesters[END_REF] F o r P e e r R e v i e w 2 RMF is the furanic intermediate mixture essentially composed of methoxymethyl furfural (MMF) and 5-hydroxymethyl furfural (HMF). Besides leading to various furanics and other platform chemicals, the Acid Catalysed Dehydration (ACD) process also leads to the production of an unavoidable side stream residue called humins. [START_REF] Van Zandvoort | Formation, molecular structure, and morphology of humins in biomass conversion: Influence of feedstock and processing conditions[END_REF] Humins are polyfuranic macromolecule mixtures with minor quantities of furanic derivatives retained in their structure. [START_REF] Synvina | Personal communication Humins[END_REF][START_REF] Pin | Valorization of biorefinery side-stream products: Combination of humins with polyfurfuryl alcohol for composite elaboration[END_REF] The chemical structure of humins is highly complex and largely depends on the type of feedstock, operating conditions and the functional groups associated with it. [START_REF] Wang | Structural characterization and pyrolysis behavior of humin by-products from the acid-catalyzed conversion of C6 and C5 carbohydrates[END_REF][START_REF] Van Zandvoort | Structural characterization of 13 C-enriched humins and alkali-treated 13 C humins by 2D solid-state NMR[END_REF] Despite existing for many decades, humins, until today, have been used only as residues in low value applications such as combustion and gasification. [START_REF] Hoang | Valorization of humin-based byproducts from biomass processing -A route to sustainable hydrogen[END_REF] With the primary aim of upgrading biorefinery side streams to improve bio-based economics in the recent past, innovative potential application for humins as renewable raw materials have been identified mainly in catalysis, water purification, matrix of impregnation materials, CO 2 sequestration and energy storage. [START_REF] Gajic | HTC-biochar as a soil conditioner -first results from a field trial with sugar beet[END_REF][START_REF] Hu | Engineering Carbon Materials from the Hydrothermal Carbonization Process of Biomass[END_REF][START_REF] Mija | Humins as promising material for producing sustainable carbohydrate-derived building materials[END_REF][START_REF] Filiciotto | Catalytic insights into the production of biomass-derived side products methyl levulinate, furfural and humins[END_REF] The complex structure of humins certainly indicates highly variable physico-chemical properties, and in turn gives an indication of the potentially varying risk profiles that need to be anticipated for their sustainable new use. Besides understanding these aspects, it is equally important to recognize the nature and properties of the most important compounds present throughout the entire production chain of humins.

Although MMF and HMF have been independently addressed by various researchers so far, their effect as a mixture has not been studied together. Also noteworthy, neither humins nor RMF is yet assessed and classified according to the international Classification and Labelling System of Chemicals such as the CLP regulation in the EU (derived from the Globally Harmonised System of Classification and Labelling of Chemicals (GHS) [START_REF]UNECE, Globally Harmonized System of Classification and Labelling of Chemicals (GHS) Seventh revised edition[END_REF] ) or according to the United Nations Model Regulations for the Transport of Dangerous Goods (UN TDG). [START_REF] Unece | UN Recommendations on the Transport of Dangerous Goods -Model Regulations Nineteenth[END_REF] Thus, very limited or almost no safety related information of humins or RMF mixtures is available for the end user, indicating that the materials are still in the early stage research. [START_REF] Synvina | Personal communication Humins[END_REF] Bio-based nature humins and furanic intermediates and/or platform chemicals does not necessarily provide safe attributes to these materials in all working environments. [START_REF] Ventura | Evaluating the toxicity of biomass derived platform chemicals[END_REF] Considering potentially the vast amounts of production and broad application routes being explored, it is important to strengthen the sustainable application pathways, for which availing information on their safety profiles plays a key role. Currently, there is a lack of information regarding the persistence of these materials in the environment and their possible effects or consequences which seems to be the main concern as perceived from the literature sources. [START_REF] Synvina | Personal communication Humins[END_REF][START_REF] Synvina | Personal Communication Crude MMF 2017[END_REF] The current study as part of the EU funded HUGS project 19 is focusing exclusively on addressing the ecotoxicological and biodegradation properties of humins and other key furanic intermediates namely HMF, MMF and RMF and an important side-stream methyl levulinate (ML). The proposed assessment is based on the output of a dual experimental approach combining conventional OECD ecotoxicity and biodegradability tests as demanded by REACH regulation for a first appraisal of environmental hazards and the examination of several immunomarkers in fish cells to document possible long-term effects of studied substances on the aquatic ecosystem.

MATERIALS AND METHODS

Tests required for the annex VII of REACH regulation (1-10 tonnes/year) (i.e. (a) Algal growth inhibition test (OECD 201), [START_REF]Test No. 201: Freshwater Alga and Cyanobacteria, Growth Inhibition Test[END_REF] (b) Daphnia magna acute immobilisation test (OECD 202), [START_REF]Daphnia sp. Acute Immobilisation Test[END_REF] and (c) ready biodegradability test (OECD 301F) [START_REF]Test No. 301: Ready Biodegradability[END_REF] have been performed for ML, HMF, MMF, RMF and humins. They are presented briefly in Table 1.

Further details on humins samples and other selected compounds have been made available in a parent work of the same team. [START_REF] Muralidhara | Insights on thermal and fire hazards of humins in support of their sustainable use in advanced biorefineries[END_REF] The selected biodegradability test is a stringent test for which a positive result will allow concluding that the chemical will degrade ultimately and quickly in the environment. Water solubility of stock solutions was confirmed by Total Organic Carbon (TOC) analysis, by comparing the TOC measurement with theoretical organic content of the selected compound.

For the Daphnia magna test, results are expressed as EC50 48h (i.e., the concentration that immobilizes 50% of daphnids within 48 hours). In Pseudokirchneriella subcapitata test, results are expressed in terms of EC10 and EC50 (i.e. effective concentration of the substrate at which either 10 % or 50 % of the test population are affected respectively). The algal growth inhibition test is a multi-generation test. However, for risk assessment purposes and derivation of Predicted No-Effect Concentrations (PNECs), the EC50 is treated as a shortterm toxicity value. The EC10 may be used as an additional long term result when other longterm data are available. Effective Concentration (EC) values have been calculated using a logistic Hill model with bootstrap estimation of confidence intervals. Calculations have been performed with REGTOX software. [START_REF]REGTOX Weblet Importer[END_REF] In addition to the tests detailed above, immunotoxicity tests [START_REF] Bado-Nilles | Coupling of OECD standardized test and immunomarkers to select the most environmentally benign ionic liquids option-Towards an innovative "safety by design" approach[END_REF] were also conducted to develop a first approach concerning positive or negative effects on the ecosystem. Forty adult sticklebacks (Gasterosteus aculeatus, 5.3 ± 0.3 cm, 2.0 ± 0.4 g), obtained from one spawn in INERIS husbandry facility, were used. Before experiments, the fish were maintained in a laboratory tank and fed daily with frozen red mosquito larvae and brine shrimp. Each fish was sacrificed by cervical dislocation, measured and weighed. Splenic leucocyte isolation was made following further protocols. [START_REF] Bado-Nilles | Flow cytometry detection of lysosomal presence and lysosomal membrane integrity in the three-spined stickleback (Gasterosteus aculeatus L.) immune cells: applications in environmental aquatic immunotoxicology[END_REF] Each leucocyte suspension was adjusted at 10 6 cells mL -1 before incubation at 4 °C for 16 h with RMF, HMF, Humins b or ML with concentration ranging from 0 to 500 mg L -1 . [START_REF] Bado-Nilles | Detection of immunotoxic effects of estrogenic and androgenic endocrine disrupting compounds using splenic immune cells of the female three-spined stickleback, Gasterosteus aculeatus (L.)[END_REF] Then, leucocyte distribution, cellular mortality (apoptotic and necrotic leucocytes), leucocyte respiratory burst, [START_REF] Bado-Nilles | Applications in environmental risk assessment of leucocyte apoptosis, necrosis and respiratory burst analysis on the European bullhead, Cottus sp[END_REF] lysosomal presence [START_REF] Bado-Nilles | Flow cytometry detection of lysosomal presence and lysosomal membrane integrity in the three-spined stickleback (Gasterosteus aculeatus L.) immune cells: applications in environmental aquatic immunotoxicology[END_REF] and phagocytosis activity [START_REF] Bado-Nilles | In vivo effects of the soluble fraction of light cycle oil on immune functions in the European sea bass, Dicentrarchus labrax (Linné)[END_REF] analyses were carried out on all leucocytes, using a Cyan TM ADP flow cytometer (Beckman Coulter). Statistical analyses were performed using XLStat 2008 (Addinsoft). After verification of normality (Anderson-Darling test) and of homogeneity of covariance matrices (Bartlett test), a one-way ANOVA and a Student Newman-Keuls post-hoc tests were performed to assess the effect of each pollutant at each concentration in relation to control values. All hypotheses were tested for statistical significance at the level of p ≤ 0.05.

RESULTS AND DISCUSSION

The results of the ecotoxicity and biodegradability tests required for the annex VII of REACH regulation are summarized in Table 2. Among all the tested compounds, median effective concentrations highlighted the lack of short-term toxicity of ML, towards Daphnia magna and Pseudokirchneriella subcapitata tests (EC50s > 500 mg.L -1 ). These results are consistent with the daphnia test results conducted by Lomba et al., [START_REF] Lomba | Ecotoxicity studies of the levulinate ester series[END_REF] where the EC50 24h was 2,761±512 mg.L -1 . Irrespective of the duration of exposure, these values confirm that ML is a chemical causing no short-term toxicity effects. Regarding humins, the range of EC50 48h values for the 2 batches (Ha and Hb with compositions C= 53.4 %, H= 5.9 %, O= 39.7% and C= 53.59 %, H= 5.88 % and O= 40.2 % respectively) highlighted no short-term toxicity effects towards Daphnia magna. Despite containing varying quantities of furanics in their structure, humins in general resulted in lack of short-term toxicity compared to other furanics tested independently. Whereas, EC50 value for HMF, MMF and RMF ranged from 13 to 36.5 mg.L - 1 , globally indicating the same range of toxic effects for compounds with a similar chemical structure (i.e. presence of a furan ring and aldehyde as one of the common functional groups). Remarkably, for these three compounds a significant increase in inhibitory effects between 24 hours and 48 hours of exposure is observed. This reveals that comparison of EC50 values, in particular for such compounds, must be based on the same duration of exposure.

In contrast with D. magna mobility test, the EC50 values for P. subcapitata ranged in a significantly narrower interval between the highest and lowest toxic compound (i.e. 51.5 -110.2 mg.L -1 ), making it difficult to draw firm conclusions on the respective toxicity of tested compounds. Such findings were observed also by Lomba et al., [START_REF] Lomba | Ecotoxicity studies of the levulinate ester series[END_REF] and Roberts et al., [START_REF] Roberts | Relative toxicities of chemicals to the earthworm Eisenia foetida[END_REF] where similar compounds tested with different species resulted in varying levels of observed toxicity. Variations in structural and functional properties of chemicals, alkyl chain lengths, responses of similar chemicals between different species were some of the common observations made from their results, which reasonably correlate in our case as well. From this it is evident that, the toxicity profiles of tested compounds can be both species and exposure dependent.

Regarding immune responses of humins and related compounds, major effects concerned inflammatory processes [START_REF] Junek | Bimodal effect of humic acids on the LPS-induced TNF-α release from differentiated U937 cells[END_REF] whereas the innate immune response seems to be weakly impacted. In the present work, for whatever immune parameters tested in stickleback splenocytes, no effect was detected, regardless of products and test concentrations (from 0 to 500 mg.L -1 ). Moreover, the immune values were in line with classic proportions of granulocytesmonocytes (24.4±6.5 %) and lymphocytes (75.6 ± 6.5 %) and to basal phagocytic activity (47.9 ± 4.5 % of phagocytic capacity; 20.2±6.5 % of phagocytic capacity; 1.2 ± 0.8 for respiratory burst index; 4.3 ± 0.7 MFI for lysosomal presence) previously observed. [START_REF] Bado-Nilles | Coupling of OECD standardized test and immunomarkers to select the most environmentally benign ionic liquids option-Towards an innovative "safety by design" approach[END_REF][START_REF] Bado-Nilles | In vivo effects of the soluble fraction of light cycle oil on immune functions in the European sea bass, Dicentrarchus labrax (Linné)[END_REF][START_REF] Bado-Nilles | Detection of immunotoxic effects of estrogenic and androgenic endocrine disrupting compounds using splenic immune cells of the female three-spined stickleback, Gasterosteus aculeatus (L.)[END_REF] As shown in Figure 1, biodegradability data made it possible to distinguish three groups of compounds: i) ML and 5-HMF which can be classified as readily biodegradable as they reached the pass level (i.e. 60% theoretical oxygen demand in a 10-day window within the 28day period of the test); ii) the two batches of humins (Ha and Hb) for which some biodegradation was observed (% biodegradation ranging between 35 and 40%) and iii) MMF and RMF which showed no significant sign of biodegradation. For RMF, the inhibition control, containing both the test compound and aniline (reference compound) confirmed that RMF was not toxic for the activated sludge used as inoculum. This finding allowed concluding that the lack of biodegradation could not be attributed to an inhibitory effect on the activated sludge. The measurements of TOC removal at the end of the test confirmed the results calculated from the oxygen consumption for the whole set of compounds.

Interestingly, similar results have been obtained for each type of ecotoxicity and biodegradation test conducted for the two different batches of humins (Ha and Hb). These findings reveal that the slight variations observed in the composition of humins do not result in a significant variation of their ecotoxicological profile.

Eventually, an attempt was made to summarize the most significant and discriminating results of our study in Figure 2 (i.e. Daphnia magna immobilisation test and biodegradation test). They highlight that MMF and RMF (left part of the figure) could be the compounds of concern as they showed some toxicity for aquatic organisms and a lack of ready biodegradability. Our results are in accordance with the data publicly available in the ECHA's chemicals database for MMF where this compound is classified as toxic to aquatic life with long lasting effects (H411, acute chronic 2). [START_REF]ECHA. 5-(methoxymethyl)-2-furaldehyde -Registration Dossier -ECHA[END_REF] Accordingly, given the fact that these compounds of concern are generally present as impurities in humins [START_REF] Muralidhara | Insights on thermal and fire hazards of humins in support of their sustainable use in advanced biorefineries[END_REF] , better environmental footprint might be targeted by minimizing their presence in crude humins. In this manner, larger portfolio of sustainable uses of these biobased feedstocks could be developed.

CONCLUSIONS

Limited knowledge of the characteristics and physico-chemical properties of humins probably caused them to be used in the past only in low value applications and visibly no efforts were made in providing enough data regarding their environmental attributes. With promises of future commercial products emerging from humins, fulfilling the existing data gaps is therefore a priority from the production, regulatory and end user perspectives. The preliminary assessment shows that industrial humins produced by YXY® process do not display any immediate ecotoxicological concerns so far, thereby displaying some safer margins for further applications. As a word of caution, environmental fate of chemicals is purely species and exposure dependent, as is evidenced in the case of HMF, MMF and RMF. Therefore, due care must be taken before generalizing the hazards of a specific chemical tested with various species. Nevertheless, in the long run, there is a definite scope for in-depth safety-oriented research with further optimization of humins production pathways and newly targeted applications. In this context, research has already started to address thermal and fire hazards of these chemicals to complete the overall assessment of their safety profiles.

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Figure 1 :

 1 Figure 1: Biodegradability profile of Humins a, Humins b, ML, HMF, MMF and RMF.

Figure 2 :

 2 Figure 2: Relationship between biodegradability and toxicity profiles (EC50 48h) of all tested compounds.

Table 1 :

 1 Summary of tests performed for regulatory purposes (REACH regulation Annex VII)

	Organisms	Effect	Endpoints	Expression of results	Test duration	Test methods
	Micro-algae Pseudokirchneriella subcapitata	Chronic	Growth	EC10 EC50	72 hours	OECD 201, 2011
	Micro-crustaceans Daphnia magna	Acute	Mobility	EC50	48 hours	OECD 202, 2004
	Activated sludge receiving predominantly domestic sewage	Ready Biodegradability	Oxygen consumption	% biodegradation	28 days	OECD 301F, 1992

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Table 2 :

 2 Results of ecotoxicity and biodegradability tests

	* highest tested concentration
	** not tested

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Acknowledgements

This work has been funded by the European Commission as part of a H2020 MSCA project named HUGS under the grant agreement # 675325.

The authors are thankful to Synvina CV for supplying humins, RMF, MMF and ML samples.

Bibliography

Conflicts of interest

There are no conflicts to declare.


Adrien Barton
email: adrien.barton@irit.fr

Laure Vieu
email: laure.vieu@irit.fr

Jean-François Ethier
email: jf.ethier@usherbrooke.ca.

Directing Actions

Keywords: Directive information entity, Information content entity, Action

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Introduction

Informational entities, such as words, can be used to represent existing entities, such as material objects, properties, or processes. Some informational entities, however, can direct processes. This happens when they represent how to perform some actions and such actions are then performed on the basis of this description. Consider for example a recipe, a drug prescription or an industrial procedure: those informational entities do not primarily describe actions that have been performed, but explain how to perform actions.

Although some work in applied ontology have been published on aboutness [START_REF] Smith | Aboutness: Towards foundations for the information artifact ontology[END_REF], we are not aware of any that investigate in detail the notion of directedness (but see below 2.2). It is of fundamental importance in ontology, as it connects three realms: informational entities (specification of actions), cognitive entities (intentions) and processes (the actions themselves). This article will concentrate on the connection between the directive informational entities and the actions, skipping over the relevant cognitive entities. Thus, this article aims at providing some bases for representing directedness in applied ontologies.

We will first provide some more specific motivations for such a work, and present some work that have been done in related areas. We will then introduce several relations of directedness that capture different aspects of this notion, as well as the related (but different) notion of compliance. A following section will then present the notions of synonymy and specialization that can hold between directive entities. A summary and presentation of future work will conclude the paper.

Example and Background

A Simple Example

We will use throughout the article the following examples of simplified martini recipes: At some time t1, Mr. Bond makes a martini following the recipe DICE1 from his favorite book. He pours 4 cl of gin in the shaker (action g1); adds 1cl of vermouth in the shaker (action v1); and shakes the shaker (action s1). We will call m1 the mereological sum of g1, v1 and s1, and we will say that DICE1 directed m1, DICE1.1 directed g1, DICE1.2 directed v1 and DICE1.3 directed s1.

Suppose now that in the second edition of the book, the author introduces two modifications, writing in DICE2 'verse' instead of 'pour', and adding 'slowly'. We will say that DICE2.1 is synonymous with DICE1.1: any action compliant with DICE2.1 (that is, of the type specified by DICE2.1) is also compliant with DICE1.1, and vice versa. We will say that DICE2.2 specializes DICE1.2: any action compliant with DICE2.2 is also compliant with DICE1.2, but the reverse does not hold.

To represent such entities and their connections, we need to introduce relations of directedness and compliance between informational entities and actions, and explain their connection with the relations of parthood, synonymy and specialization illustrated above. We also need to represent when such instructions direct successfully such actions, or when they merely attempt to direct them. Finally, we need to differentiate successful and maximal direction, e.g., DICE1.1 successfully directing the part of g1 during which Bond pours the first 2 cl of gin, but maximally directing g1.

Background

This section will describe some relevant former work for the analysis of directedness, showing their limitations and how our present work can complement them.

Information Artifacts in Ontology

There are few proposals addressing the ontological nature of informational artifacts such as recipes. The Informational Artifact Ontology (IAO) [START_REF] Smith | Aboutness: Towards foundations for the information artifact ontology[END_REF] is based on the Basic Formal Ontology (BFO) [START_REF] Arp | Building Ontologies with Basic Formal Ontology[END_REF]. In this framework, informational entities are called "information content entities" (ICE), which have the property of being about something. ICEs can be concretized in some qualities, such as a pattern of ink on a paper, or a mental quality in a cognitive system. An important subclass of ICE is Directive information entity, which encompasses in particular Action specification, which aims at directing actions. However, there is to date no formalized relation of directedness that would relate action specifications with the actions they direct. It is especially important to introduce such a relation for ontologies based on BFO, such as the biomedical ontologies of the OBO Foundry. For example, it is important to be able to relate prescriptions of drugs, prescriptions of laboratory tests [START_REF] Barton | LABO: An ontology for laboratory test prescription and reporting[END_REF] or surgery instructions with the actions they direct.

Such directive informational entities could also be introduced in other upper ontologies. For example, in DOLCE [START_REF] Masolo | The WonderWeb library of foundational ontologies and the DOLCE ontology[END_REF], they might be categorized as a non-agentive social object, but DOLCE doesn't propose either any relation resembling directedness.

Informational objects called "representations" are also addressed in YAMATO [START_REF] Mizoguchi | YAMATO: yet another more advanced top-level ontology[END_REF]. Representations can be realized by a variety of entities. In particular, procedures are realized by actions. But procedures do not appear to have the identity criteria for directive informational entities discussed below in Section 3.1, and, although lacking a formal characterization that would enable to define it unambiguously, it seems the relation "realization_of" in YAMATO is more related to the notion of compliance than to the notion of directedness.

We will in the following use the neutral term "Directive information content entity" (abbreviated "DICE") referring to an informational entity that could direct actions. The formalization we propose can presumably be adapted in various upper ontologies.

Languages for Process Specification

In engineering and conceptual modelling, several representation frameworks have been proposed to specify the dynamic aspects of systems. The theory called Process Specification Language (PSL) represent "activities" in the context of system interoperability in the manufacturing domain [6] [START_REF] Menzel | A formal foundation for process modeling[END_REF] [START_REF] Grüninger | Using the PSL ontology[END_REF]. It evolved into a modular ontology formalized in Common Logic (and OWL) as well as an ISO standard [START_REF] Pouchard | ISO 18629 PSL: A standardised language for specifying and exchanging process information[END_REF].

Activities in PSL may have zero or more "activity occurrences" (events or processes, including, but not limited to, actions [START_REF] Katsumi | Using PSL to extend and evaluate event ontologies[END_REF]), to which they are related with a binary relation named "occurrence_of". Occurrences are situated in time while activities are not, but subactivities of activities can be temporally ordered. PSL is organized around a core ontology of time and events. The occurrence_of relation differs in several ways from the directedness relation we are addressing here. First, activity occurrences are not restricted to actions. Second, this relation appears to be one of type-token. Therefore, activities are reified types or classes, without any characteristics of informational entities (as, e.g., having author and addressee, epistemic and deontic effects, physical concretizations...). Moreover, activity occurrences are all exact occurrences of their type, and so are by definition successful. Attempts are considered in PSL as interrupted sequences of subactivity occurrences; there is no space for failure in the execution of an atomic activity. 2 So, PSL does not provide the kind of relation we are looking for between DICE and the actions they direct (or attempt to direct). It nevertheless encompasses useful features, in particular with respect to the mereological and temporal structure of both activities and activity occurrences.

BPMN is a major framework used to represent industrial processes and their workflows. It is not an ontology, but there are proposals to analyze its ontological foundations [START_REF] Rospocher | An ontology for the Business Process Modelling Notation[END_REF] or to partially integrate within domain ontologies [START_REF] Falbo | A software process ontology as a common vocabulary about software processes[END_REF]. However, crucially for us, within BPMN there is no notion of informational object. In addition, the distinction between the types of processes and their occurrences is taken care of by a semantic interpretation external to the notation language. As a result, there is not much to take inspiration there for the present paper, concerned with axiomatizing directedness between DICEs and actions. Similar observations can be made for another important language for representing dynamic systems, UML activity diagrams in conceptual modelling [START_REF] Börger | An ASM semantics for UML activity diagrams[END_REF]. Of course, extensions of the present work to account for the semantic contents of DICEs are likely to build on such rich and elaborate representation frameworks in the future.

Framework

We will formalize the axioms in FOL, omitting universal quantifiers for free variables. We will admit in the universe of discourse particulars only. Symbols and formulas are written in bold.

The Identity of Informational Entities

We must first clarify that in our framework, the identity of an informational entity (and in particular, of a DICE) derives from the speech act that created it (for a theory of discourse based on this assumption, see the Segmented Discourse Representation Theory or SDRT [START_REF] Asher | Logics of conversation[END_REF]). Consider for example DICE3 (see Figure 1), another very similar recipe of martini, in a different book, created independently of DICE1 by a different author. DICE1.2 and DICE3.1 are not the same entity, as they were created by two different speech acts, in different discursive contexts. However, they have two things in common. First, they are constituted by the same sentence (the same succession of words). Second, their semantic contents is the same action type.

A crucial aspect of the semantic contents of a DICE like DICE1 is the temporal ordering of the action types referred to by its parts DICE1.1, DICE1.2 and DICE1.3. Indeed, when executing DICE1, the shaking should occur after the pourings, something captured among DICE parts through discourse relations in a theory like SDRT. But action types are independent of speech acts, and could be combined temporally in various ways, as seen with DICE1 and DICE3 whose semantic contents are different temporal arrangements of the same action types. That there are different "sums" of these three action types shows that the mereology on action types cannot be a classical extensional mereology. PSL does include a "subactivity" relation to decompose activities into subactivities, and assume it is a partial ordering [START_REF] Grüninger | Using the PSL ontology[END_REF], but does not claim it is a full-fledged mereological relation. In fact, the very notion of mereology on types, or universals, is problematic. There is no consensus on the existence of "structural universals" [START_REF] Fisher | Structural universals[END_REF], as they face critical issues such as that of having a part "several times over" [START_REF] Masolo | Graph-Based Approaches to Structural Universals and Complex States of Affairs[END_REF]. As far as we know, the PSL literature is silent with respect to how to account for multiple subactivity links, such as when the same atomic activity of making one step is repeated in the complex activity of walking.

On the contrary, mereology on DICEs is extensional. There is no other DICE than DICE1 that is only composed by the parts DICE1.1, DICE1.2 and DICE1.3. In particular, there is no other DICE that has as parts DICE1.1 and DICE1.2 with the former following the latter, just as there is no alternative body of mine with my two hands swapped. We will use a classical mereology [START_REF] Varzi | Mereology[END_REF], with the parthood relation part_of and the associated overlaps relation. Parts of a DICE are related by discourse relations, some of which imply temporal constraints on the action types in their semantic contents [START_REF] Asher | Logics of conversation[END_REF].

In this paper, we will concentrate on DICEs and their connections with actions, rather than on the sentences that constitute those DICEs, on the semantic contents of DICEs in terms of action types, or on the discourse relations between DICE parts and their semantic consequences on the temporal order of action types, all of which is left for future extensions of this work.

Directedness

There are two important cases of directedness to distinguish. Suppose for example that Bond is attempting to make a Martini by following DICE1. At t0, he attempts to perform the recipe through his action a0, but gets the proportion wrong: we will say that DICE1 "a-directs" (where "a" stands for "attempt") a0, and write this a-directs(DICE1,a0). Later at t1, Bond attempts again to perform the recipe through his action m1 (mentioned above), and succeeds: we will say that DICE1 "successfully directed" m1, and write this sdirects(DICE1,m1). On this understanding, if a DICE successfully directs an action, then it attempts to direct this action -but the reciprocal implication does not hold, since an attempt can fail: s-directs(d, a) → a-directs(d, a). This paper will clarify the intuition behind those two relations a-directs and s-directs, and which axioms they satisfy. The relation a-directs (and therefore s-directs too) has the class DICE as domain, and a class we can name Action as range:

a-directs(d, a) → DICE(d) ∧ Action(a)

Agents, Intentions and Compliance

We will not enter here into any detailed investigation of the ontology of actions. We will make the following hypotheses: a) several agents can constitute a larger agent (e.g., two humans dancing constitute one agent); b) the mereological sum of several actions made by the same agent is also an action; c) all actions are intentional (see [START_REF] Brand | Intending and acting: toward a naturalized action theory[END_REF] for a discussion). We will not formalize intentions though, since our main goal is to connect informational entities with actions (because of their private, mental character, we typically have less data about the intentions of agents than about the action specifications they are supposed to follow, or about the actions that they indeed performed). We nevertheless suppose that for a DICE to a-direct an action, we need: 1) the cognitive system of the agent representing in some way this DICE (this could be analyzed by the DICE being concretized by a mental quality of the agent, if one follows IAO's ontology); 2) this representation leading to an intention to perform an action as described; 3) and this intention leading to a subsequent action (that can be compliant or not with the instruction, depending on whether it is successful or not). As we said above, we will not represent the notion of intention here; however, the relation of directedness takes into account such intentions.

We will also make use of a relation of "compliance" between an action and a DICE, that we will write compliant(a,d). This is a not a relation of direction, but rather a relation independent of any intentions that the agent of the action may have had and so of the possibly different DICE he may have followed or not. As such, it is akin to the token-type relation "occurrence_of" used in PSL, although DICEs are not action types.

Proposal

Max-directing

In the use case presented above, we have: s-directs(DICE1, m1); s-directs(DICE1.1, g1); s-directs(DICE1.2, v1); s-directs(DICE1.3, s1). Moreover, since g1, v1 and s1 are all essential steps in making martini, we assume that they are also directed by the whole DICE1: s-directs(DICE1, g1); s-directs(DICE1, v1); s-directs(DICE1, s1). (Note that this would not be true in a framework like PSL, which uses a relation occurrence_of that is of a type-token sort; and this is also not true for the notion of compliance: g1 is not compliant with DICE1, but only with its part DICE1.1.)

One proposition, inspired by [START_REF] Schulz | Higgs Bosons, Mars Missions, and Unicorn Delusions: How to Deal with Terms of Dubious Reference in Scientific Ontologies[END_REF], would be to state that if DICE1 successfully directs an action, then this action is an instance of Martini_making. However, this statement would conflict with s-directs(DICE1, g1) and s-directs(DICE1, v1), since g1 and v1 are not instances of Martini_making (rather, they are proper parts of m1, which is an instance of Martini_making).

To avoid this problem, we can introduce, on top of the relation s-directs, the relation max-directs, meaning "maximally successfully directs", connecting a DICE with a compliant action it successfully directs. We can then state: max-directs (DICE1, a) → Martini_making(a) and similar axioms connecting DICE1.1 and Gin_pouring, DICE1.2 and Vermouth_pouring, etc. Thus, we have max-directs (DICE1, m1) but DICE1 does not max-direct g1 nor v1 (it only s-direct them).

A DICE max-directs an action if, by definition, the DICE a-directs this action, and this action is compliant with this DICE: )) Note that one cannot speak of a "maximal direction" for a mere attempt that would not be successful, since an attempted action can be basically anything, as long as the agent has the intention to realize the action described by the relevant DICE.

In particular, this distinction between the relations max-directs and s-directs can account for directive entities that are followed for some time but not fully realized. Suppose that at t2, Bond pours 4 cl of gin in the shaker (g2) but then receives an urgent call at t3 and has to leave to save the world; thus, he never finishes making this martini. Then s-directs(DICE1, g2) although there is no instance of Martini_making maximally directed by DICE1 of which g2 is a part. Therefore, a DICE can successfully direct some action a even if there is no action maximally directed by this DICE of which a is part 3 .

One could also think that if a DICE s-directs some action, then there exists some part of this DICE that max-directs this action. However, this is not true. For example, the instruction 'CBC' ("complete blood count") on a laboratory test prescription could sdirect more than 10 different tests (hematocrit, hemoglobin, etc.); however, there is no part of 'CBC' that max-directs a hematocrit test, a hemoglobin test, etc. 4

Synonymy between DICEs

As mentioned earlier, DICE1.1 and DICE2.1 are synonymous: we write synonymous_with(DICE1.1,DICE2.1) and assume that synonymous_with is reflexive, symmetric and transitive: it is an equivalence relation (note that two DICEs constituted by the same succession of sentences might be different, if created by two different speech-acts; that would be a trivial case of synonymy).

It is important to be able to connect a directed action with the exact DICE that directed it. As a matter of fact, various DICEs may lead to different risk of errors, even if they are synonymous (one might e.g. be more difficult to understand than the other). We define synonymy as related to compliance5 : synonymous_with(d1, d2) ≡def ∀a (compliant(a, d1) ↔ compliant(a, d2))

Suppose that Vesper Lynd reads DICE2.1 and decides to verse 4 cl of gin (g3). Then we have max-directs(DICE2.1, g3), the same way we had max-directs(DICE1.1, g1). However, DICE1.1 does not direct g3 (DICE1.1 does not cause the apparition of an intention in Lynd's mind that leads to g3), g3 is merely compliant with DICE1.1: compliant(g3, DICE1.1) (and compliant(g3, DICE2.1)). For the same reason, DICE2.1 does not direct g1 but we have compliant(g1, DICE2.1). However, all actions compliant with DICE1.1 (and DICE2.1, since they are synonymous) are instances of Gin_pouring: compliant(a, DICEi.1) → Gin_pouring (a) (for i=1 or 2)6

Specialization of a DICE by a DICE

Suppose that Bond reads DICE2.2 and consequently pours slowly 1cl of vermouth in the shaker -that's the action v2. Then we have: max-directs(DICE2.2, v2). Since Bond did not represent in his cognitive system DICE1.2, it did not direct v2. This is not because of the slow character of v2: it is only because Bond did not perform v2 with the intention of following DICE1.2. As a matter of fact, DICE1.2 can direct vermouth pourings which are of any speed. On the other hand, DICE2.2 can only direct slow vermouth pourings. However, there is a connection between DICE1.2 and DICE2.2. We will say that DICE2.2 "specializes" DICE1.2 and write specializes(DICE2.2, DICE1.2). This relation can be expressed in terms of compliance: indeed, the class of actions that are compliant with DICE2.2 is a subclass of the class of actions that are compliant with DICE1.2: specializes(d, d') ≡def ∀a (compliant(a, d) → compliant(a, d')) This definition implies that the relation specializes is reflexive: any DICE specializes itself (under a large understanding of the term "specializes"; one can also introduce the relation strictly specializes to capture a strict, non-reflexive understanding of the term "specializes"). It is also transitive. However, it is not symmetrical: DICE2.2 specializes DICE1.2, but DICE1.2 does not specialize DICE2.2. It is also not antisymmetrical: DICE1.1 and DICE2.1 specialize each other, but they are not identical: they are synonymous, which implies that they share the same compliant actions. Indeed, we obtain as a theorem that two DICEs are synonymous iff they specialize each other:

(specializes(d, d') ∧ specializes(d', d)) ↔ synonymous_with(d, d')

Discussion

Now that the essential formal notions of directedness have been introduced, we can discuss their connection with parthood, synonymy, compliance and specialization.

DICEs part of a DICE

The fact that DICE1 directs not only m1, but also g1, v1 and s1 suggests that if a DICE successfully directs an action, then a larger DICE (by a mereological relation) also successfully directs this action; and the same holds for attempts to direct. That is, with x = 's' or 'a', we would have the following axiom: As a matter of fact, it seems wrong that the mereological sum of DICE1 and DICE3, namely the mereological sum of the first and the last Martini recipes, also directs m1. This intuition is captured by the identity conditions of DICEs essentially relying on speech acts as explained in Section 3.1. Since there is no (complex) speech act leading to a DICE whose parts are exactly DICE1 and DICE3, their mereological sum simply is not a DICE. The speech act identity condition for informational entities on a DICE entails that its parts should be created by (sub)speech acts and with the intention that these parts be related (through discourse relations) into a whole discourse directing actions towards a common goal. Obviously, two different recipes taken from two different books by two different authors are not parts of a whole discourse. The specification of those exact conditions that need to hold for two DICEs to be related by the relation part_of is a complex issue, that should be addressed in future work, probably relying on a rich enough theory of discourse such as SDRT.

A Medical Use Case

We will now illustrate with a medical example of drug prescription why it is important to represent relations of directedness, synonymy and specialization in biomedical applied ontologies [START_REF] Ethier | An ontological analysis of drug prescriptions[END_REF] [START_REF] Barton | A classification of instructions in drug prescriptions and pharmacist documents[END_REF]. Suppose that on 2018/01/01, in Quebec, a doctor deems that her patient should take metoprolol for hypertension control until further notice. She then writes the instructions i1 = 'metoprolol 100 mg PO BID' instructing her patient to take a drug with 100 mg of active ingredient metoprolol twice a day ('BID') per mouth ('PO') until further notice. If he takes metoprolol in such a way (action a1), then i1 s-directs a1.

Suppose instead that the patient loses the prescription, and goes back during the day to his doctor to get a new prescription. This time, the doctor writes i2 = 'metoprolol 100 mg per mouth twice a day'. Then i2 is synonymous with i1 (even if the patient may better understand i2 than i1). However, they do not direct the same actions: if the patient had forgotten what was written on the first prescription and takes the drug following i2, then i2 directs this action, not i1.

Actually, the pharmacist will typically specialize the instruction i2 further for the patient, with an instruction such as i3 = 'Apo-Metoprolol 50 mg, 2 tab orally 8 am and 2 tab orally 8 pm every day'. i3 specializes i2: any action compliant with i3 is also compliant with i2, but the reverse does not hold. Finally, i2 maximally directs the patient's drugtaking until e.g. the patient receives a new instruction to stop taking metoprolol, but merely successfully directs the drug-taking by the patient of one dose of metoprolol in the morning of 2018/03/15.

Conclusion and Future Work

We have proposed bases for the representation of directedness in ontologies. Intuitively, for a DICE to a-direct an action, three conditions (at least) should be satisfied: 1) this DICE is represented in the cognitive system of an agent (which could be formalized, following IAO, as the concretization of the DICE by a mental quality of the agent); 2) it leads to an intention to perform this action; 3) and this intention leads to an action. We have introduced a taxonomy of relations of directedness: attempt to direct (a-directs); successfully directs (s-directs); and maximally directs (max-directs). We have analyzed the connections between those notions of directedness. We have introduced the relations of synonymy and specialization, and defined them in terms of compliance. The relation of synonymy is an equivalence relation, and the relation of specialization is reflexive and transitive.

Future work should address essential points such as a more detailed investigation in the ontology of actions and intentions, as well as the temporal order in which actions specified by instructions should be realized. In particular, this will require to analyze the different kinds of relations that can hold between various parts of a DICE, in light of theories such as SRDT, and relate them with some formal framework specifying processes (such as PSL, BPMN, UML activity diagrams…) in order to characterize the semantics of some specific discourse relations. The connection between directive aspects (that we addressed here) and normative aspects (whether an action is permitted, obligatory, forbidden or recommended) should also be investigated. Two challenges related to each other would be representing the direction of non-actions, such as the kind of entity that would be directed by a DICE 'Do not smoke'; and representing the notion of contradiction between DICEs, such as when a doctor writes to his patient 'Stop metoprolol' after two years of prescription, when realizing that her patient start reacting badly to this drug. Finally, an ontological characterization of the notion of intention might be helpful for use in e.g. robotics. In particular, one would need to characterize the link between the representation of a DICE and the apparition of a corresponding intention; and the link between an intention and an action to which it leads. Like for other cognitive attitudes such as beliefs [START_REF] Barton | First steps towards an ontology of belief[END_REF], dispositional characterization of intention might be developed.

Figure 1 .

 1 Figure 1. Three martini recipes DICE1, DICE2 and DICE3

  max-directs(d, a) ≡def a-directs(d, a) ∧ compliant(a, d) Moreover, if d max-directs a, then d successfully directs a and d does not direct anything larger (in the mereological sense) than a: max-directs(d, a) → (s-directs(d, a) ∧ ∀b ((s-directs(d, a) ∧ overlaps(a, b)) → part_of(b, a)

  (x-directs(d, a) ∧ part_of(d, d') ∧ DICE(d')) → x-directs(d', a) This axiom brings about a clarification on the relation part_of between DICEs.

There is some level of non-determinism captured in occurrence trees, but this helps accounting for underspecified temporal ordering of subactivities and for alternative ways of accomplishing an activity.

The distinction between max-directs on the one hand and s-directs and a-directs on the other hand enables to capture the notion of interruptability of a procedure without relying on complex occurrence trees as in PSL.

There could be, however, a synonymous DICE that has such parts -see below the discussion on synonymy.

Our use of synonymy on DICEs is not a trivial extension of the standard linguistic notion applying to lexical items. We define it here on the basis of compliance only. Other choices could have been made, taking the syntactic, semantic and pragmatic structure of the DICEs into account.

With this kind of axioms and the link between max-directs and compliant, max-directs (DICE1, a) → Martini_making(a) as mentioned above becomes a theorem.


Tonametl Sanchez
email: tonametl.sanchez@ipicyt.edu.mx

Andrey Polyakov
email: andrey.polyakov@inria.fr

Denis Efimov
email: denis.efimov@inria.fr

A Consistent Discretisation method for Stable Homogeneous Systems based on Lyapunov Function

Keywords: Nonlinear systems, discrete-time systems, homogeneous systems

In this paper we propose a discretisation scheme for continuous and asymptotically stable homogeneous systems. This method is based on the dynamics of the system projected on a level surface of a homogeneous Lyapunov function. The discretisation method is explicit and preserves the convergence rate of the continuous-time system.

INTRODUCTION

Homogeneity is a symmetry constraint imposed on the system dynamics that provides useful properties for analysis and design of control systems. For example, this kind of systems can be used to approximate nonlinear models by preserving important nonlinear features of the original dynamics [START_REF] Zubov | Methods of A. M. Lyapunov and their applications[END_REF][START_REF] Hermes | Nilpotent Approximations of Control Systems and Distributions[END_REF][START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF]. Some additional interesting characteristics of homogeneous systems are, the existence of homogeneous Lyapunov functions [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF], and homogeneous controllers [START_REF] Kawski | Stability and nilpotent approximations[END_REF][START_REF] Hermes | Differential Equations, Stability and Control, chapter Homogeneous coordinates and continuous asymptotically stabilizing feedback controls[END_REF][START_REF] Sepulchre | Homogeneous Lyapunov functions and necessary conditions for stabilization[END_REF][START_REF] Grüne | Homogeneous State Feedback Stabilization of Homogenous Systems[END_REF], the direct association of the homogeneity degree with the convergence rates [START_REF] Haimo | Finite time controllers[END_REF][START_REF] Hong | On an Output Feedback Finite-Time Stabilisation Problem[END_REF][START_REF] Nakamura | Smooth Lyapunov functions for Homogeneous Differential Inclusions[END_REF], and the intrinsic robustness properties to exogenous perturbations and delays [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF], [START_REF] Efimov | Weighted Homogeneity for Time-Delay Systems: Finite-Time and Independent of Delay Stability[END_REF]. These features have motivated many authors to develop techniques for analysis and design of homogeneous control systems as homogeneous observers for linear systems [START_REF] Cruz-Zavala | Lyapunov functions for continuous and discontinuous differentiators[END_REF], homogeneous polynomial systems [START_REF] Parrilo | Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization[END_REF], and several high-order sliding mode controllers [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF], to mention a few interesting cases.

On the other hand, with omnipresence of computers and digital devices, discretisation of continuous-time models has become essential to the design of control systems. It is useful, for example, for computer simulation, for implementation in electronic devices, and for the design of digital (or sampled-data) controllers [START_REF] Nesić | Perspectives in Robust Control, chapter Sampled-Data Control of Nonlinear Systems: an Overview of Recent Results[END_REF][START_REF] Goodwin | Sampling and Sampled-Data Models: The Interface Between the Continuous World and Digital Algorithms[END_REF].

This work was supported by the Agence Nationale de la Recherche project ANR-18-CE40-0008 "DIGITSLID", by the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898, and by Government of Russian Federation (Grant 08-08).

Unfortunately, unlike linear systems, nonlinear ones do not have (in general) an exact discretisation. Nevertheless, it is expected that an approximate discretisation (which is frequently obtained by a method derived for linear dynamics) preserves the most relevant features of the continuous-time system.

Although, there exist several methods to discretise nonlinear systems [START_REF] Hairer | Solving Ordinary Differential Equations I[END_REF], it has been shown that for nonlinear homogeneous systems these approaches may guarantee convergence of discretizations only locally and loosing important stability properties [START_REF] Efimov | Realization and Discretization of Asymptotically Stable Homogeneous Systems[END_REF]. The standard discretisation techniques become even less applicable in the case of non-smooth dynamics. This has motivated several authors not only to study the properties of standard discretisation methods but also to design new ones to discretise homogeneous systems [START_REF] Brogliato | The implicit discretization of the super-twisting slidingmode control algorithm[END_REF][START_REF] Koch | Discrete-Time Equivalent Homogeneous Differentiators[END_REF][START_REF] Polyakov | Consistent Discretization of Finite-Time and Fixed-Time Stable Systems[END_REF][START_REF] Efimov | Discretization of homogeneous systems using Euler method with a state-dependent step[END_REF].

In this paper we propose a procedure to discretise asymptotically stable continuous homogeneous systems. Our method relies on the information provided by a homogeneous Lyapunov function, thus, the discrete-time trajectory is guaranteed to converge to the origin. Moreover, the discretisation is consistent in the sense of [START_REF] Polyakov | Consistent Discretization of Finite-Time and Fixed-Time Stable Systems[END_REF], namely, the convergence rate of the continuous-time trajectory is preserved in its discrete-time counterpart. It is also important to emphasise that the proposed method is explicit, therefore, no system of algebraic equations has to be solved at each iteration.

Paper organization:

In Section 2 the definition of homogeneity and some properties of homogeneous systems are recalled. In this section we also give two simple motivational examples and state the problem to be solved in the paper. In Section 3 we study the dynamics of a homogeneous system projected on the unitary level set of the Lyapunov function. In Section 4 we describe the proposed discretisation method for homogeneous systems. Some examples of the discretisation technique are presented in Section 5. In Section 6 we state some final remarks. The proofs of the results are not provided due to space restrictions.

Notation: Real and integer numbers are denoted as R and Z, respectively. R >0 denotes the set {x ∈ R : x > 0}, analogously for the set Z and the sign ≥. For x ∈ R n , |x| denotes the Euclidean norm. For a continuous positive definite function V : R n → R, we denote the set S V = {x ∈ R n : V (x) = 1}. The set of strictly increasing continuous functions η : R ≥0 → R ≥0 with η(0) = 0 is denoted by K. For x ∈ R and p ∈ R ≥0 , x p := |x| p sign(x).

PRELIMINARIES AND PROBLEM STATEMENT

In this section we recall some properties of homogeneous systems and give the statement of the problem to be solved. We consider the following continuous-time system

ẋ(t) = f (x(t)) , x(t) ∈ R n , (1)
where f : R n → R n is continuous on R n . We assume that for each x(0) ∈ R n , the solution of (1) exists and is unique in forward time for all t ∈ R ≥0 .

Homogeneity

Now, let us recall the definition of weighted homogeneity. Definition 1. (Weighted homogeneity, [START_REF] Kawski | Stability and nilpotent approximations[END_REF]). Let Λ r denote the family of dilations given by the square diagonal matrix Λ r = diag( r1 , . . . , rn ), where r = [r 1 , . . . , r n ] , r i ∈ R >0 , and ∈ R >0 . The components of r are called the weights of the coordinates. Thus:

(a) a function V : R n → R is r-homogeneous of degree m ∈ R if V (Λ r x) = m V (x), ∀x ∈ R n , ∀ ∈ R >0 ; (b) a vector field f : R n → R n , is r-homogeneous of degree µ ∈ R if f (Λ r x) = µ Λ r f (x), ∀x ∈ R n , ∀ ∈ R >0 .
System (1) is said to be r-homogeneous of degree µ ∈ R if its vector field f is r-homogeneous of degree µ. Now, we recall some properties of homogeneous systems. Suppose that (1) is r-homogeneous of degree µ, and

V : R n → R is a differentiable r-homogeneous function of degree m. Hence, V = -W (x) , W (x) := -∇V (x)f (x) , (2)
where the function W : R n → R is r-homogeneous of degree m + µ. Note that if the origin of (1) is asymptotically stable, then there exists a continuously differentiable function which is a strict 1 Lyapunov function, and is r-homogeneous of some degree m ∈ R >0 . Indeed, the existence of such a Lyapunov function is guaranteed if the condition m > max i=1,...,n r i hold [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF]. In such a case, W is positive definite, continuous for all x ∈ R n , and there exists α ∈ R >0 such that [START_REF] Hong | On an Output Feedback Finite-Time Stabilisation Problem[END_REF]

, V ≤ -αV m+µ m (x) , (3)
1 In this paper we consider only strict Lyapunov functions i.e. those such that ∇V (x)f (x) < 0 for all x ∈ R n \ {0}.

where the constant α can be given as follows

α = inf x∈S V W (x) . (4)
Note that the degree of homogeneity of W is strictly positive by restricting the degree of V to m > -µ. An interesting consequence of (3) is the estimation of the decreasing rate of V along the solutions of (1). Lemma 2. (See, e.g. [START_REF] Haimo | Finite time controllers[END_REF][START_REF] Hong | On an Output Feedback Finite-Time Stabilisation Problem[END_REF]).

Let (1) be r-homogeneous of degree µ, with a differentiable strict Lyapunov function V : R n → R which is r-homogeneous of degree m. For all x(0) ∈ R n and all t ∈ R ≥0 , the following holds (with α as given in (3), and

V 0 := V (x(0))):
(1) if µ > 0, then the converge rate is in practical fixedtime, i.e.

V (x(t)) ≤ V 0 1 + µ m αV µ m 0 t -m µ ; (5) (2) if µ = 0, then the converge rate is exponential, i.e. V (x(t)) ≤ V 0 exp(-αt) ; (6) (3) if µ < 0, then the converge rate is in finite-time, i.e. V (x(t)) ≤ V (x(t)), where V (x(t)) =      V -µ m 0 --µα m t m -µ , t < mV -µ m 0 -µα , 0, t ≥ mV -µ m 0 -µα . (7)

Problem statement

Let us begin this section with two simple examples on the Euler discretisation of scalar homogeneous systems. Example 3. Consider the following scalar system ẋ = -a x 3 4 , a ∈ R >0 .

(8) Observe that, (8) is r-homogeneous of degree µ = -1 with r = 4. Now, for any initial condition x(0) ∈ R, the solution of ( 8) is given by

x(t) = x(0) 0 (|x(0)| 1 4 -at/4) 4 , at < 4|x(0)| 1 4 , 0, at ≥ 4|x(0)| 1 4
. Note that x = 0 is an asymptotically stable equilibrium of (8) for any a ∈ R >0 , moreover, it is finite-time stable. The explicit Euler discretisation of ( 8) is given by

x(k + 1) = x(k) -ha x(k) 3 4 , k ∈ Z ≥0 , (9)
where h ∈ R >0 is the discretisation step. In Fig. 1 we can see the trajectory generated by ( 9) with the parameters: a = 5, h = 0.05, and the initial condition x(0) = 1. Observe that, the continuous-time solution converges to zero in t = 0.8. However, the discrete-time approximation remains oscillating in the steady-state. Hence, we see that (9) is not a consistent discretisation of (8) since the finitetime stability is not preserved. Example 4. Consider the following scalar system ẋ = -ax 3 , a ∈ R >0 .

(10) Observe that (8) is r-homogeneous of degree µ = 2 with r = 1. Now, for any initial condition x(0) ∈ R, the solution of ( 8) is given by

x(t) = x(0)[1 + 2ax 2 (0)t] -1 2 .
(11) Note that x = 0 is an asymptotically stable equilibrium of (8) for any a ∈ R >0 , moreover, it is practically fixed-time 2ac 2 and all x(0) ∈ R (this bound is independent of the initial condition). The explicit Euler discretisation of (10) is given by

x(k + 1) = x(k) -hax 3 (k), k ∈ Z ≥0 . (12)
For the parameters a = 1 and h = 0.001, we compute ten iterations of ( 12) with the different initial conditions x(0) = 1, 2, . . . , 50. For each simulation we calculate the maximum error (shown in Fig. 2) between (12) (the Euler discretisation of (10)) and ( 11) (the exact solution of ( 10)), i.e. max k=1,...,10 |x(kh) -x(k)|. Observe that this error grows with the initial condition, and it is unbounded for initial conditions x(0) ≥ 45. Thus, ( 12) is not a consistent discretisation of (10) since there exist initial conditions such that the trajectories of ( 12) diverge.

As we mentioned in the introduction, we cannot have (in general) an exact discretisation for a nonlinear system. However, it is expected that a suitable discretisation preserves important properties of the solutions, for example, the convergence rates described in Lemma 2. It is also expected that if a Lyapunov function is available, then it can be used to improve the discretisation scheme. For the case of homogeneous systems, a homogeneous Lyapunov function provides the information of stability and conver-gence rates, as stated in Lemma 2. Hence, the problem to be solved in this paper can be described as follows:

For continuous r-homogeneous systems whose origin is asymptotically stable, to provide a methodology to construct discretisation schemes such that the obtained discrete-time system preserves the converge rate of the continuous-time system. This problem is solved by exploiting the information provided by the homogeneous Lyapunov function of the system, and by considering the system's dynamics projected on the unitary level set of the Lyapunov function. Such a projection is described in the following section.

PROJECTED DYNAMICS

In this section we develop the fundamentals for the discretisation scheme proposed in Section 4.

Let (1) be r-homogeneous of degree µ, and V : R n → R be a continuously differentiable function, which is positive definite and r-homogeneous of degree m. Define the following auxiliary variable

y = Λ r V -1 m (x) x , ∀x ∈ R n \ {0} . ( 13
)
Note that ( 13) is the homogeneous projection of x over the unitary level set of V , thus, y ∈ S V for all x ∈ R n \ {0}.

Let us compute the dynamics of the auxiliary variable y. By differentiating ( 13) along ( 1) we obtain

ẏ = Λ r V -1 m (x) I -1 m V -1 (x)Gx∇V (x) f (x) , (14)
where I is the identity matrix, and G := diag(r 1 , . . . , r n ).

From ( 13) we see that

x = Λ r V 1 m (x)
y, thus, we rewrite ( 14)

as follows ẏ = Λ r V -1 m (x) f (x) -1 m V -1 (x)Λ r V -1 m (x) GΛ r V 1 m (x) × ×y∇V (x)f (x) , = V µ m (x)f (y) + 1 m W (x) V (x) Gy , = V µ m (x)f (y) + 1 m V m+µ m (x)W (y) V (x) Gy , therefore, ẏ = V µ m (x) f (y) + 1 m W (y)Gy .
(15) Equation ( 15) describes the dynamics (1) projected on S V . However, observe that we cannot recover the trajectory of (1) directly from the trajectory of ( 15) because ( 13) is not bijective. To overcome this problem, we proceed to study the dynamics of V , i.e. the derivative of V along (1). Thus, from (2) and ( 13), we obtain

V = -W Λ r V 1 m (x) y = -V m+µ m (x)W (y) . (16)
Now, define the function v : R ≥0 → R ≥0 such that it satisfies the scalar differential equation (cf. ( 16))

v(t) = -v m+µ m (t)W (z(t)) , (17)
with the function z : R ≥0 → R n being the solution of the system (cf. (15))

ż(t) = v µ m (t) f (z(t)) + 1 m W (z(t))Gz(t) .
(18) It can be verified that S V is a positively invariant set for the trajectories of (18).

From these developments we are ready to state the main results of this section. First, we give a useful result about a solution representation of (17).

Lemma 5. (1) For all t ∈ R ≥0 , and any initial condition

v 0 := v(0) ∈ R ≥0 , the function v : R ≥0 → R ≥0 , v(t) = v 0 exp -Ŵ0 (t) , for µ = 0 , (19a) v(t) = v 0 1 + µ m v µ m 0 Ŵ0 (t) -m µ , for µ > 0 , (19b)
with Ŵ0 (t) := t 0 W (z(τ )) dτ , satisfies ( 17). (2) For µ < 0 and any initial condition v 0 ∈ R >0 , there exists Θ(v 0 ) ∈ R >0 such that the function v : R ≥0 → R ≥0 given by

v(t) =    v -µ m 0 --µ Ŵ0(t) m m -µ , -µ Ŵ0(t) m < v -µ m 0 0, -µ Ŵ0(t) m ≥ v -µ m 0 , (19c)
satisfies ( 17) for all t ∈ 0, Θ(v 0 ) , and v(t) → 0 as t → Θ(v 0 ).

We finalise this section with the statement of the fundamental results for the discretisation method described in Section 4. Theorem 6. Let (1) be r-homogeneous of degree µ, and

V : R n → R be a continuously differentiable Lyapunov function which is r-homogeneous of degree m. Define ζ = [v, z ] ∈ Z, Z = R >0 × S V . Consider (1) on R n \ {0}
and ( 17)-( 18) on Z. The solutions of ( 1) and the solutions of ( 17)-( 18) are equivalent with the homeomorphism Φ :

R n \ {0} → Z given by Φ(x) = V (x) Λ r V -1 m (x)
x .

(20)

Corollary 7. Let (1) be r-homogeneous of degree µ, and V : R n → R be a continuously differentiable Lyapunov function which is r-homogeneous of degree m. Let v and z be solutions of ( 17), ( 18), respectively, with initial conditions v(0

) = V (x(0)), z(0) = Λ r v -1 m (0)
x(0), for any

x(0) ∈ R n \ {0}.
(1) If µ ≥ 0, then the function x : R ≥0 → R n given by

x(t) = Λ r v 1 m (t) z(t) , (21a)
is solution of (1) for all t ∈ R ≥0 .

(2) If µ < 0, the function x : R ≥0 → R n given by

x(t) = Λ r v 1 m (t) z(t), t < Θ(v(0)), 0, t ≥ Θ(v(0)), (21b)
(with Θ as given in Lemma 5) is solution of (1) for all t ∈ R ≥0 .

DISCRETISATION SCHEME

The main idea of the discretisation scheme comes from the developments of Section 3, and it can be sketched as follows: since (18) describes the dynamics of (1) projected on S V , and v (as given in Lemma 5) describes the decreasing behaviour of V along the solutions of (1), the idea is to obtain a numerical solution2 of (18) on S V , and expand it to R n by using ( 21) and a discretisation of v. Remark 8. Although, a numerical solution of ( 18) can be obtained by different techniques, we restrict ourselves in this paper to the explicit (a.k.a. forward) Euler method.

z(0) = z(0), and for some h ∈ R >0 , z(k) approximates z(kh).

To construct the discretisation of v, we see from Lemma 5 that for any h ∈ R ≥0 ,

v(t + h) = v(t)exp -Ŵ (t) , for µ = 0 , (22a)
v(t + h) = v(t) 1 + µ m v µ m (t) Ŵ (t) -m µ , for µ > 0, (22b) and for µ < 0, v(t + h) =    v -µ m (t) --µ Ŵ (t) m m -µ , -µ Ŵ (t) m < v -µ m (t), 0, -µ Ŵ (t) m ≥ v -µ m (t), , (22c)
where Ŵ (t) := t+h t W (z(τ )) dτ . Thus, for the discretisation of v we only have to define a discrete-time approximation of Ŵ (t). For example, by using the forward Euler method with an integration step h, the discrete-time approximation v :

Z ≥0 → R of v is given by v+ = v exp -W (z)h , for µ = 0 , (23a)
v+ = v 1 + µ m v µ m W (z)h -m µ , for µ > 0 , (23b)
and for µ < 0,

v+ =    v -µ m --µhW (z) m m -µ , -µhW (z) m < v -µ m , 0, -µhW (z) m ≥ v -µ m , , (23c)
where we denote v = v(k), v+ = v(k + 1), and z = z(k). The function z :

Z ≥0 → R n is the discrete-time approximation of z given by z+ = Λ r V -1 m (z + ) z+ , (24) z+
= z + hv µ m f (z) + 1 m W (z)Gz , v+ > 0, z, v+ = 0.
Note that (24) can be seen as the explicit Euler discretisation of (18), including a scaling factor given by Λ r

V -1 m (z + )
.

This scaling is necessary to guarantee that z(k) ∈ S V for all k ∈ Z ≥0 , nonetheless, it is also necessary that z+ = 0, thus we require the following assumption. Assumption 9. For all z ∈ S V and all τ ∈ R >0 ,

-z = τ f (z) + 1 m W (z)Gz .
Observe that, if there exist τ ∈ R >0 and z ∈ S V such that z + τ F (z) = 0, with F (z) := f (z) + 1 m W (z)Gz, then the vector F (z) is necessarily collinear to z. This holds if and only if there exists w ∈ R n \ {0} which is orthogonal to both F (z) and z. Note that ∇V (z) is orthogonal to F (z), thus it is also orthogonal to z if and only if ∇V (z) z = 0. Hence, we conclude that a sufficient condition to guarantee that Assumption 9 holds is ∇V (z) z = 0 for all z ∈ S V . In simple words, there is no tangent vector to S V pointing towards the origin. Now, we can state the main result of this section. Theorem 10. Let (1) be r-homogeneous of degree µ with a strict Lyapunov function V : R n → R, which is continuously differentiable and r-homogeneous of degree m. Suppose that Assumption 9 holds. Consider the discretetime approximation x(k) of x(kh), given by x

(k + 1) = Λ r v 1 m (k+1) z(k + 1) , k ∈ Z ≥0 , (25)
where v and z are given by ( 23) and ( 24), respectively, with the initial conditions v(0

) = V (x(0)), z(0) = Λ r V -1 m (x(0))
x(0). Then, for all h ∈ R >0 and all x(0) ∈ R n \ {0}, x(k) → 0 as k → ∞. Moreover (with α as in (4)):

(1) if µ = 0, then the convergence rate is exponential, i.e. V (x(k)) ≤ V (x(0)) exp(-αhk) ; (2) if µ > 0, then the convergence rate is practically in fixed-time, i.e.

V (x(k)) ≤ V (x(0)) 1 + µ m V µ m (x(0))αhk -m µ ;
(3) if µ < 0, then the convergence rate is in finite-time, i.e. V (x(k)) ≤ V (k), where

V (k) =      V -µ m (x(0)) --µαh m k m -µ , k < mV -µ m (x(0)) -µαh , 0, k ≥ mV -µ m (x(0)) -µαh
.

Theorem 10 requires the availability of a homogeneous Lyapunov function. Unfortunately there is no universal technique to compute it, nevertheless the existence of such a function is guaranteed by some converse Lyapunov theorems [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF][START_REF] Nakamura | Smooth Lyapunov functions for Homogeneous Differential Inclusions[END_REF], and there are some methods to design homogeneous Lyapunov functions for particular classes of homogeneous systems, e.g. [START_REF] Polyakov | Unified Lyapunov function for a finite-time stability analysis of relay second-order sliding mode control systems[END_REF][START_REF] Sanchez | Design of Lyapunov functions for a class of homogeneous systems: Generalized forms approach[END_REF][START_REF] Efimov | Homogeneous Lyapunov Functions: From Converse Design to Numerical Implementation[END_REF]. Note that the conclusions of Theorem 10 are valid for any discretisation step h ∈ R >0 provided that Assumption 9 holds.

EXAMPLES

In this section we illustrate the proposed discretisation method with two different systems. One of them with the property of finite-time stability and the another one with the property of practical fixed-time stability.

Finite-time stability

In this section we resume the system of Example 3. We have seen that the origin of ( 8) is finite-time stable. To apply our proposed discretisation scheme, we consider the function V : R → R given by

V (x) = 4 5 |x| 5
4 . This function is r-homogeneous of degree m = 5, and a Lyapunov function for (8) for any a ∈ R >0 , with V = -W (x), W (x) = a|x|. Hence, the discrete-time approximation is given by ( 23), ( 24), ( 25 -4

5 z+ = ( 5 4 ) 4 5 sign(z + ), v+ = v -µ m -1 5 ha|z| 5 , ha|z| < 5v 1 5 , 0, ha|z| ≥ 5v 1 5 , z+ = z + ahv -1 5 z 3 4 + 4 5 z 2 , v+ > 0, z, v+ = 0.
For the simulation we use the parameters: a = 5 and h = 0.05. We simulate the system 1.5 seconds for the initial condition x(0) = 2. In Fig. 3 we show the error signal x(k) -x(kh) generated by both the standard Euler discretisation scheme and our proposed method. We can see that, although the errors are comparable along the transient period, the standard Euler scheme shows a permanent deviation in the steady-state. However, the proposed Lyapunov-based scheme provides zero deviation in the steady-state.

Practical fixed-time stability

For this example we consider the following system ẋ1 = -k 1 x 1 3 2 + x 2 , ẋ2 = -k 2 x 1 2 , (26) which is r-homogeneous of degree µ = 1 with r = [2, 3] . Consider the function V : R 2 → R given by

V (x) = 2 5 k 1 |x 1 | 5 2 -x 1 x 2 + 3 5 α|x 2 | 5 3
. This function is r-homogeneous of degree m = 5, and it can be proven that for any k 1 ∈ R >0 there exist α, k 2 ∈ R >0 such that V is a Lyapunov function for (26). For the simulation we consider the parameters: k 1 = 2, k 2 = 1, and α = 2. We simulate the system 1.2 seconds by using 12000 steps, i.e. with a step length h = 0.0001, for the different initial conditions x 2 (0) = 0 and x 1 (0) = 10 q with q = 3, . . . , 9. The norm of the system's states is shown in the logarithmic plot of Fig. 4, there we can appreciate the fixed-time convergence to the set {x ∈ R 2 : |x| ≤ 100}.

CONCLUSION

For continuous homogeneous systems with asymptotically stable origin, we developed a methodology to formulate discretisation schemes that preserve the convergence rate of the continuous-time system. The key ingredient of the method is the exploitation of the information provided by the Lyapunov function. It is important to mention that the proposed methodology does not restrict the discretisation of the projected dynamics to the explicit Euler method. Hence, a different technique could be used in such a process to obtain a different discretisation scheme, but preserving the main properties shown in this paper. Some future developments: the extension of the methodology to systems with discontinuities; study of different schemes obtained by modifying the discretisation for the projected dynamics; application of the proposed discretisation method to design sampled data controllers.

Fig. 1 .

 1 Fig.1. Solution of (8) and its discrete-time approximation generated by the explicit Euler method.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Discretisation error: dotted-line for standard Euler method; solid-line for proposed method.

We mean by numerical solution a sequence {z(k)} k∈Z ≥0 such that


Konstantin Zimenko
email: kostyazimenko@gmail.com

Andrey Polyakov
email: andrey.polyakov@inria.fr

Denis Efimov
email: denis.efimov@inria.fr

Robust Stabilization of Control Affine Systems with Homogeneous Functions

Keywords: Nonlinear control, control affine systems, robust stabilization, homogeneous systems

The stabilization problem of the affine control system ẋ = f 0 (x) + m i=1 u i f i (x) with homogeneous functions f 0 , f i is studied. This class of systems is of interest due to the robust properties of homogeneity and the fact that many affine systems can be approximated by or transformed to the class under consideration. An advantage of the introduced design method is that the tuning rules are presented in the form of linear matrix inequalities. Performance of the approach is illustrated by a numerical example.

INTRODUCTION

Homogeneous dynamical systems have a number of very useful properties for system analysis, control design and estimation. In particular, the local and global behaviours are the same; homogeneous systems have certain intrinsic robustness properties with respect to external perturbations, measurement noises and time delays; convergence rate of the system can be assessed by its homogeneity degree, etc. (see, for example, [START_REF] Bacciotti | Liapunov Functions and Stability in Control Theory[END_REF][START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF][START_REF] Ryan | Universal stabilization of a class of nonlinear systems with homogeneous vector fields[END_REF][START_REF] Hong | H ∞ control, stabilization, and inputoutput stability of nonlinear systems with homogeneous properties[END_REF][START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF][START_REF] Zimenko | A note on delay robustness for homogeneous systems with negative degree[END_REF]Efimov et al., 2016).

For the stability analysis, a Lyapunov function of a homogeneous system can also be chosen homogeneous (see, e.g. [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF][START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF][START_REF] Bacciotti | Liapunov Functions and Stability in Control Theory[END_REF], and its negativeness can be checked only on the unit sphere. The combination of this notion with the implicit Lyapunov function method in some cases allows formalizing control/observer tuning algorithms (even in the case of essentially nonlinear feedbacks) in the form of linear matrix inequalities (LMI). However, in most cases, this effect is achieved only for linear (almost linear) system models (for example, [START_REF] Zimenko | Robust feedback stabilization of linear MIMO systems using generalized homogenization[END_REF][START_REF] Rios | Robust output-control for uncertain linear systems: Homogeneous differentiator-based observer approach[END_REF].

The present paper addresses the control design problem for the system

ẋ = f 0 (x) + m i=1 u i f i (x), (1)
where x ∈ R n are states, u = (u 1 , ..., u m ) T ∈ R m are controls, and f 0 , f i are homogeneous functions. In addition

This work was partially supported by the Government of Russian Federation (Grant 08-08) and by the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898.

to the useful properties of homogeneity, interest in this class of systems is based on the following reasons:

• the system (1) is of wider class than linear control systems; • if there is a nonhomogeneous function f 0 or f i , it can be homogenized by an appropriate control; • nonhomogeneous functions may be approximated by the homogeneous ones (see, for example, [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF][START_REF] Hermes | Nilpotent and high-order approximations of vector field systems[END_REF][START_REF] Ménard | Homogeneous approximations and local observer design[END_REF]Efimov and Perruquetti, 2016).

The proposed homogeneous controller for the system (1) is based on 'universal' control design scheme given in [START_REF] Sontag | A 'universal' construction of artstein's theorem on nonlinear stabilization[END_REF]. Despite a significant nonlinearity of the system, the applicability condition of the presented control is in the form of LMI. This condition is obtained following homogeneous function representation in a certain canonical form. The representation procedure substantially repeats the mechanism of embedding nonlinear systems into linear differential inclusions.

The paper is organized in the following way. Notation used in the paper is given in Section 2. Section 3 presents some preliminaries used in the paper. Section 4 introduces the main result on controller design for the system (1). Simulations are shown in Section 5. Finally, concluding remarks are given in Section 6.

NOTATION

Through the paper the following notation will be used:

• N is the set of natural numbers;

• R + = {x ∈ R : x > 0}, where R is the field of real numbers; • The symbol 1, m is used to denote a sequence of integers 1, ..., m;

• | • | is used to signify the Euclidean norm in R n , • denotes a weighted Euclidean norm;

• S = {x ∈ R n : x = 1} is the unit sphere in R n ; • A = sup x∈R n Ax x if A ∈ R n×n ; • I n ∈ R n×n is the identity matrix;
• the inequality P > 0 (P ≥ 0) means that a symmetric matrix P = P T ∈ R n×n is positive definite (positive semi-definite); • the eigenvalues of a matrix G ∈ R n×n are denoted by λ i (G), i = 1, ..., n, λ min (G) = min i=1,n λ i (G); • C n (X, Y ) is the set of continuously differentiable (at least up to the order n) maps X → Y , where X and Y are open subsets of finite-dimensional spaces;

• ∂ ∂x = ∂ ∂x1 , ∂ ∂x2 , • • • , ∂ ∂xn ; • R(λ)
denotes the real part of a complex number λ.

PRELIMINARIES

Stability Notions

Consider the following system

ẋ(t) = f (x(t)), x(0) = x 0 , t ≥ 0, (2)
where

x(t) ∈ R n is the state vector, f ∈ R n → R n is continuous, f (0) = 0.
Definition 1 [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF][START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF] The origin of ( 2) is said to be globally finite-time stable if it is globally asymptotically stable and any solution x(t, x 0 ) of the system (2) reaches the equilibrium point at some finite time moment, i.e. x(t, x 0 ) = 0 ∀t ≥ T (x 0 ) and x(t, x 0 ) = 0 ∀t ∈ [0, T (x 0 )), x 0 = 0, where T : R n → R + ∪ {0}, T (0) = 0 is the settling-time function.

Definition 2 [START_REF] Polyakov | Nonlinear feedback design for fixedtime stabilization of linear control systems[END_REF] A set M ⊂ R n is said to be globally finite-time attractive for (2) if any solution x(t, x 0 ) of (2) reaches M in some finite time moment t = T (x 0 ) and remains there ∀t ≥ T (x 0 ), T : R n → R + ∪ {0} is the settling-time function. It is fixed-time attractive if in addition the settling-time function T (x 0 ) is globally bounded by some number T max > 0.

Theorem 1. [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF]. Suppose there exist a positive definite C 1 function V defined on an open neighborhood of the origin D ⊂ R n and real numbers C > 0 and σ ≥ 0, such that the following condition is true for the system (2)

V (x) ≤ -CV σ (x), x ∈ D \ {0}.
Then depending on the value σ the origin is stable with different types of convergence:

• if σ = 1, the origin is exponentially stable; • if 0 ≤ σ < 1, the origin is finite-time stable and T (x 0 ) ≤ 1 C(1-σ) V 1-σ 0
, where V 0 = V (x 0 ); • if σ > 1 the origin is asymptotically stable and, for every

ε ∈ R + , the set B = {x ∈ D : V (x) < ε} is fixed- time (independent on the initial values) attractive with T max = 1 C(σ-1)ε σ-1 . If D = R n
and function V is radially unbounded, then the system (2) admits these properties globally.

'Universal' Stabilizer for Control Affine Systems

Consider a system in the form (1), where f 0 , f i ∈ C ∞ , and f 0 (0) = 0. Let this system admits a control-Lyapunov function V that is, a smooth, proper, and positive definite function

R n → R + ∪ {0} so that inf u∈R m ∂V (x) ∂x f 0 (x) + ∂V (x) ∂x m i=1 u i f i (x) < 0 (3)
for each x = 0.

Definition 3 [START_REF] Sontag | A 'universal' construction of artstein's theorem on nonlinear stabilization[END_REF] The control-Lyapunov function V satisfies the small control property if for each ε > 0 there is a δ > 0 such that, if x = 0 satisfies x < δ, then there is some u with u < ε such that

∂V (x) ∂x f 0 (x) + ∂V (x) ∂x m i=1 u i f i (x) < 0.
Definition 4 [START_REF] Sontag | A 'universal' construction of artstein's theorem on nonlinear stabilization[END_REF] Let k : R n → R m be a mapping, smooth on R n \ {0} and with k(0) = 0. This is a smooth feedback stabilizer provided that, with k = (k 1 , ..., k m ) T the closed-loop system

ẋ = f 0 (x) + m i=1 k i (x)f i (x)
is globally asymptotically stable. Theorem 2. [START_REF] Sontag | A 'universal' construction of artstein's theorem on nonlinear stabilization[END_REF]. There is a smooth feedback stabilizer k iff there is a smooth control-Lyapunov function V , and k can be chosen continuous at the origin if V satisfied the small control property. Moreover, such k can be chosen in the form

k(x) =    a(x) + a 2 (x) + |B(x)| 4 |B(x)| 2 B(x), if |B(x)| = 0, 0, if |B(x)| = 0, where a(x) = ∂V (x) ∂x f 0 (x), B(x) = (b 1 , ..., b m ) T , b i (x) = ∂V (x)
∂x f i (x) for i = 1, m.

Generalized Homogeneity

The homogeneity is a dilation symmetry property. For example, if a mathematical object f (e.g., function, vector field, and operator) remains invariant with respect to scaling (dilation operation) of its argument f (e s x) = e νs f (x), s ∈ R, x ∈ R n then it is called homogeneous, where ν ∈ R is a constant called the degree of homogeneity.

In the general case, we can consider a non-uniform scaling x → d(s)x, where an operator d(s) : R n → R n is called dilation in the space R n if it satisfies:

• group property: d(0) = I n and d(t + s) = d(t)d(s) = d(s)d(t) for t, s ∈ R; • continuity property: d is a continuous map; • limit property:
lim s→-∞ d(s)x = 0 and lim s→+∞ d(s)x = +∞ uniformly on the unit sphere S.

In [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF][START_REF] Rosier | Etude de Quelques Problemes de Stabilization[END_REF][START_REF] Khomenuk | On systems of ordinary differential equations with generalized homogeneous right-hand sides[END_REF], the dilation d is suggested to be generated as a flow of C 1 vector field. Such a dilation is known as geometric dilation.

In this paper we deal with generalized homogeneity based on groups of linear dilations, that also can be called linear geometric homogeneity. For this type of homogeneity there exists a generator matrix G d = lim s→0 d(s)-In s that satisfies the following properties [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF]:

∂ ∂s d(s) = G d d(s) = d(s)G d , d(s) = e G d s = +∞ i=0 s i G i d i! , s ∈ R.
Definition 5 [START_REF] Polyakov | On homogeneous distributed parameter systems[END_REF] The dilation d is said to be monotone if d(s) < 1 as s < 0. It is said to be strictly monotone if ∃β > 0 : d(s) < e βs for s ≤ 0.

Thus, monotonicity means that d(s) is a strong contraction for s < 0 (strong expansion for s > 0) and implies that for any x ∈ R\{0} there exists a unique pair (s 0 , x 0 ) ∈ R× S such that x = d(s 0 )x 0 . Note, that monotonicity property may depend on a norm • . Theorem 3. [START_REF] Polyakov | Sliding mode control design using canonical homogeneous norm[END_REF]. If d is a dilation in R n , then

• the generator matrix G d is anti-Hurwitz (i.e. R(λ i (G d )) > 0, i = 1, n)
and there exists a matrix P ∈ R n×n such that

P G d + G T d P > 0, P > 0.
(4) • the dilation d is strictly monotone with respect to the norm x = √ x T P x for x ∈ R n and P satisfying ( 4):

e αs ≤ d(s) ≤ e βs if s ≤ 0, e βs ≤ d(s) ≤ e αs if s ≥ 0, where α = 1 2 λ max P 1 2 G d P -1 2 + P -1 2 G T d P 1 2 , β = 1 2 λ min P 1 2 G d P -1 2 + P -1 2 G T d P 1 2
.

The dilation d introduces a sort of norm topology in R n by means of the so-called homogeneous norm [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF].

Definition 6 (Polyakov, 2019) A continuous function p : R n → R + ∪ {0} is said to be a d-homogeneous norm if p(x) → 0 as x → 0 and p(d(s)x) = e s p(x) > 0 for x ∈ R n \ {0} and s ∈ R.

For monotone dilations we define the canonical homogeneous norm • d : R n → R + ∪{0} as x d = e sx for x = 0, where s x ∈ R such that d(-s x )x = 1 and, by continuity, we assign 0 d = 0. Note that d(s)x d = e s x d and d(ln x d )x = 1.

(5)

Definition 7 [START_REF] Polyakov | On homogeneous distributed parameter systems[END_REF] A vector field

f : R n → R n (a function h : R n → R) is said to be d- homogeneous of degree ν ∈ R if f (d(s)x) = e νs d(s)f (x), ∀x ∈ R n , ∀s ∈ R. (resp. h(d(s)x) = e νs h(x), ∀x ∈ R n , ∀s ∈ R.) (6)
Let F d (R n ) be the set of vector fields R n → R n satisfying the identity (6), which are continuous on

R n \ {0}. Let deg F d (f ) denote the homogeneity degree of f ∈ F d (R n ).
Lemma 4. [START_REF] Polyakov | Sliding mode control design using canonical homogeneous norm[END_REF]. The vector field f ∈ F d (R n ) is Lipschitz continuous (smooth) on R n \ {0} if and only if it satisfies a Lipschitz condition (it is smooth) on the unit sphere S, provided that d is strictly monotone on R n equipped with a (smooth on R n \ {0}) norm • .

If a function (or a vector field) is smooth, then homogeneity is inherited by its derivatives in a certain way.

Claim 5. If f ∈ F d (R n ) ∩ C 1 (R n \ {0}, R n ), then e deg F d (f )s d(s) ∂f (x) ∂x = ∂f (z) ∂z z=d(s)x d(s) (7) for x ∈ R n \ {0} and s ∈ R.
The rate of convergence of homogeneous systems can be assessed by its homogeneity degree [START_REF] Nakamura | Smooth Lyapunov functions for homogeneous differential inclusions[END_REF][START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF][START_REF] Polyakov | Sliding mode control design using canonical homogeneous norm[END_REF]. Theorem 6. An asymptotically stable d-homogeneous system ẋ = f (x), f : R n → R n is uniformly finite-time stable if and only if deg

F d (f ) < 0.
Analogously, if deg F d (f ) > 0 and the system is stable, then any compact set containing the origin is fixed-time attractive.

The homogeneity theory provides many other advantages to analysis and design of nonlinear control system. For instance, some results about ISS of homogeneous systems can be found in [START_REF] Bernuau | Robustness of homogeneous and homogeneizable differential inclusions[END_REF][START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF]; [START_REF] Ryan | Universal stabilization of a class of nonlinear systems with homogeneous vector fields[END_REF].

Of particular interest is the use of homogenising control algorithms due to delay robustness properties of homogeneous systems (see, for example, Efimov et al., 2016;[START_REF] Zimenko | A note on delay robustness for homogeneous systems with negative degree[END_REF][START_REF] Zimenko | Independent of delay stabilization using implicit lyapunov function method[END_REF].

MAIN RESULTS

In this paper we consider the system (1), where f i (0) = 0 for i = 0, m, and

f i ∈ F d (R n ) ∩ C 1 (R n , R n ) with strictly monotone dilation d equipped with the norm x = √ x T P x, P = P T > 0.
The main goal of the paper is to propose a constructive (i.e. equipped with reliable tuning rules and robustness abilities) stabilizing control algorithm.

Canonical Representation of Homogeneous Systems

Let us consider the system in the form ẋ = f (x), ( 8)

where f ∈ F d (R n ) ∩ C 1 (R n , R n ), f (
0) = 0. For this system let us define the following bounds

g ij = sup y∈R n : y <1 ∂f i (z) ∂z j z=y , g ij = inf y∈R n : y <1 ∂f i (z) ∂z j z=y , that are always exist due to f ∈ C 1 (R n , R n ). Denote the bounded convex domain Ω = θ = (θ 11 , ..., θ 1n , ..., θ nn ) : g jl ≤ θ jl ≤ g jl , j, l = 1, n
with the set of vertices defined by V = α = (α 11 , ..., α 1n , ..., α nn ) : α jl ∈ {g jl , g jl }, j, l = 1, n .

Proposition 7. Every trajectory of ( 8) is also a trajectory of

ẋ = N i=1 α i (x) x deg F d (f ) d d(ln x d )A i d(-ln x d )x, (9)
where

A i ∈ V, N i=1 α i (x) = 1, 0 ≤ α i (x) ≤ 1, and N ∈ N.
Remark 1 Proposition 7 is mostly based on the mechanism of representing a nonlinear systems in the form of linear differential inclusions and convex embedding (see, for example, [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF][START_REF] Zemouche | Observers for a class of lipschitz systems with extension to H ∞ performance analysis[END_REF][START_REF] Pyatnitskiy | Criteria of asymptotic stability of differential inclusions and periodic motions of time-varying nonlinear control systems[END_REF].

Note that this approach can be used for the case f (0) = 0 also, where

f (x) = N i=1 α i (x) x deg F d (f ) d d(ln x d )A i d(-ln x d )x+f (0).
The choice of the form

N i=1 α i (x) x deg F d (f ) d d(ln x d )A i d(-ln x d )x
for homogeneous functions representation is based on the fact that this form may be effective in control design with a simple tuning in the form of LMI as we will demonstrate below.

It should be noted that the condition f ∈ C 1 (R n , R n ) of Proposition 7 sufficiently restricts the class of homogeneous functions for representation in the form (9). One of the main directions for the future research is development of such method, that will not require such restrictions on the function f .

Homogeneous Control Design

Using the result of Proposition 7 the control for (1) can be obtained based on the scheme of a 'universal' stabilizing control design given in the work of [START_REF] Sontag | A 'universal' construction of artstein's theorem on nonlinear stabilization[END_REF].

Before stating the main contribution of this paper, accordingly to Proposition 7 let us define the matrices A i,j , i = 0, m, j = 1, N , and α i,j (x), such that f i in (1) can be represented as

f i (x) = N j=1 α i,j (x) x deg F d (fi) d d(ln x d )A i,j d(-ln x d )x. (10) Theorem 8. Let deg F d (f 0 ) ≥ -1, and deg F d (f 0 ) ≥ deg F d (f i ) for i = 1, m. If the system of LMIs P A 0,j + A T 0,j P < m i=1 τ i (P A i,l + A T i,l P ), j, l = 1, N (11)
is feasible for some τ i ∈ R, then the control in the form

u i (x) =        -x νi d b i (x) a(x) + a 2 (x) + |B(x)| 4 |B(x)| 2 , if |B(x)| = 0, 0, if |B(x)| = 0, ( 12
)
where i = 1, m, ν i = deg F d (f 0 ) -deg F d (f i ), B(•) = (b 1 (•), ..., b m (•)) T , and
a(x) = 2x T d T (-ln x d )P f 0 (d(-ln x d )x) x T d T (-ln x d )(P G d + G T d P )d(-ln x d )x , b i (x) = 2x T d T (-ln x d )P f i (d(-ln x d )x) x T d T (-ln x d )(P G d + G T d P )d(-ln x d )x
, globally asymptotically stabilizes the system (1) at the origin.

Remark 2

The system of LMIs (11) guarantees that the condition

|B(x)| = 0 ⇒ a(x) < 0 (13)
is satisfied, which according to (3) follows from the assumption that V = x d is a control-Lyapunov function.

Remark 3 If in Theorem 8 the strict inequality deg

F d (f 0 )> deg F d (f i
) is satisfied, then the control-Lyapunov function satisfies the small control property and the control ( 12) is continuous at 0.

The controller (12) homogenizes the closed-loop system with homogeneity degree deg F d (f 0 ). Thus, if deg F d (f 0 ) > 0, then any compact set containing the origin is fixedtime attractive. If deg F d (f 0 ) < 0 the system is finitetime stable. It should be noted that in the case of the use of Proposition 7 for system representation, the condition f ∈ C 1 (R n , R n ) restricts an application of control ( 12), especially for the case deg F d (f 0 ) < 0. However, as it shown in the following remark, the condition f ∈ C 1 (R n , R n ) can be neglected if instead of Proposition 7 we would use the numerical calculation of a(x), B(x) on a proper grid.

Remark 4

The system of LMI ( 11) is only a sufficient condition to satisfy (13). In general, since

a(x) = 2y T P f 0 (y) y T (P G d + G T d P )y , b i (x) = 2y T P f i (y) y T (P G d + G T d P )y , y = d(-ln x d )x ∈ S, i = 1, m,
the condition (13) can be numerically checked using a numerical grid on a sphere. Explicitly calculating a(x) and B(x) we can obtain that f0) ,

∂V (x) ∂x f 0 (x) + ∂V (x) ∂x m i=1 u i f i (x) ≤ -CV 1+deg F d (
where C = min y∈S a 2 (y) + |B(y)| 4 . Then, utilizing Theorem 1 we can get estimations for settling-time functions. Moreover, in this case we do not apply the representation (10), and the condition f ∈ C 1 (R n , R n ) can be neglected.

EXAMPLE

Consider the system (1) with x ∈ R 3 , m = 3, where

f 0 = -x 1 x 3 -x 1 x 2 , f 1 = x 2 -x 3 0 , f 2 =   x 3 3 0 x 3   and f 3 = x 1 x 2 0 .
In this system f 0 , f 1 , f 2 and f 3 are d-homogeneous of degrees 1, -1, 0 and 0, respectively, where d is strictly monotone dilation equipped with the norm x = The paper presents stabilizing control algorithm for affine control system (1) with homogeneous functions f i , i = 0, m. Due to homogeneity the closed-loop system has a number of robust properties necessary in practice (e.g. ISS, robustness with respect to delays, etc.). Despite nonlinearity the main condition the system has to satisfy is presented in the form of LMI. This condition is obtained with the use of homogeneous function representation in the form (10).

There are two main directions for future research. In particular, it is expedient to develop a convex embedding method, that allows representing of homogeneous systems in the form (10) for the wider class than in Proposition 7.

The second direction is related to transformation and approximation techniques to represent nonhomogeneous systems in the form (1) with homogeneous f i .

  Fig. 1. Evolution of the state of the closed-loop system