Philippe Jarne
email: philippe.jarne@cefe.cnrs.fr

Keywords: biodiversity dynamics, eco-evolutionary loops, ecology-evolution linkage

At the border between ecology and evolution

This paper is a translation by P. Jarne of an opinion paper published in the 'Regards sur la biodiversité" (Views on biodiversity) of the Web site of the

At the frontier between ecology and evolution" -everything seems curious in this title. Ecology and evolution, biodiversity sciences1 indeed, but ecology is a science or a set of sciences, and evolution, a theory or a historical process. Why associate a science and a theory? And is there a borderline between the two, even though they are so often associated, if only (recently) in the name SFE2 (French society for ecology and evolution), or in the names of training courses, laboratories or university departments? Such a question would probably have seemed strange to Charles Darwin, an evolutionist and ecologist if ever there was one. But in fact, ecology and evolution have slowly separated since the end of the 19 th century to form distinct fields of study, which are globally little connected. However, in the last twenty years or so, explicit attempts have been made to bring them together, and terms such as "eco-evolutionary dynamics" or "ecoevolutionary loop" have flourished [START_REF] Vellend | The theory of ecological communities[END_REF]Hendry, 2017), to the point where we have come to wonder whether any study of ecology or evolution is not eco-evolutionary (i.e. when an ecological change leads to an evolutionary change that will, in turn, affect the ecological interactions). I will defend here the idea that this is not the case, but that we can indeed affirm that some studies are really eco-evolutionary, and that it is necessary to widen this zone of hybridization in order to understand the dynamics of biodiversity in general, and to predict its future trajectories in particular.

A (small) bit of history and definitions

The separation, or distinction, between ecology and evolution has been a subject of reflection for a century. The structuring of the "new evolutionary synthesis" in the years 1930-1940, under the leadership of theorists such as R.A. Fisher or S. Wright, and empiricists such as T. Dobzhansky or E. Mayr (see e.g. Gayon, 1992), is probably not unrelated to this, any more than is the development of conceptual bases for ecology at the same time, under the impetus of personalities such as H. Gleason or F.E. Clements (whose positions are quite opposed). This separatist dynamic [possibly characteristic of sciences under construction or consolidation] was not lost on the protagonists of both camps, and gave rise to recurrent attempts at rapprochement (Futuyma, 1986;Huneman, 2019). For example, one can cite the classic book of Allee et al. (1949), which devotes a consistent section to this new synthesis, but it is more of an attempt to "acclimatize" the new synthesis, to use Huneman's (2019) expression, strongly marked by the proximity between W.C. Allee and S. Wright (both professors at the University of Chicago), than a real attempt to consider ecology and evolution in a fusional vision. As a reverse pathway, from evolution to ecology, we can consider ecological genetics, whose most famous proponent is E.B. Ford, or evolutionary ecology, which appeared in the 1950s and emphasizes life-history traits and their link with population demography (Huneman, 2017). However, in both cases, it is questionable whether the environment is little more than a "tapestry" (or background) whose structure and functioning are not of fundamental importance.

Prominent figures of the 1960s (e.g., R.H. McArthur, R. Levins, R.C. Lewontin or E.O. Wilson) also made attempts to think synthetically and to construct a conceptual framework in which simple models can play a supporting role in understanding eco-evolutionary processes, from individuals to biological communities. In fact, these attempts did not change the work programs in ecology and evolution, and the scientific communities remained very compartmentalized in the period 1970 to 2000. There are, however, some attempts and successes, such as the explicit consideration of demography in evolutionary biology (e.g., Charlesworth, 1980), or models that simultaneously consider species and material dynamics and their evolutionary consequences (e.g., Loreau, 2010). An explicit evolutionary framework for explaining the complexity of biodiversity can also be noted in the writings of prominent ecologists (Levin, 2000).

Population biology / genetics

Community ecology

Ecosystem ecology

I have talked about ecology and evolution so far without really defining them, except that they are at the heart of biodiversity sciences. Nor have the "border" been drawn. Let us start with a central point: ecology and evolution are interested in the dynamics of biodiversity, including all living organisms, in the forces and processes that drive it, in the reconstruction of its history and in the projection of its future (see Glossary). However, I would like to make a distinction here, admittedly somewhat artificial, in three vast fields of work in a partly hierarchical vision of living organisms: population biology (including genetics), community ecology and ecosystem ecology (Figure 1). Broadly speaking, we can say that population biology is concerned with evolutionary and ecological processes, from the individual level (genotype and phenotype) to the species level. Community ecology focuses rather on the assembly and dynamics of communities, and species play the central role. Ecosystem ecology is interested in the relationship between species and the environment, and perhaps even more so in environmental dynamics, in the flow of matter and energy in ecosystems. From population biology to ecosystem ecology, we thus move from a perspective where evolutionary thinking (and intraspecific diversity) plays a major role to one where ecology dominates, with a biological entry point centered on the species, or even the community (Figure 1). I insist on the fact that this is a caricatured vision, and one can note that demography has not been mentioned in these definitions. We will try to have a more nuanced approach in what follows, by exposing what separates and what brings together, before identifying what is truly eco-evolutionary.

Glossary

• Ecology: scientific ecology is a science (or set of sciences) that studies the interactions of living things with each other and with their environment. See e.g. Ricklefs & Relyea (2019) or https://en.wikipedia.org/wiki/Ecology. • Epigenetics: the set of mechanisms or processes that modify gene expression without altering the nucleotide sequence, through reversible and transmissible (within or between generations) marking. • Evolution: a gradual process of adaptation to the environment via (epi)genetic variation.

Evolution is a theory that gave birth to the evolutionary sciences, such as evolutionary genetics or paleontology. For a detailed presentation, one can consult numerous books (e.g., David & Samadi, 2021) or an online encyclopedia (e.g., https:// https://en.wikipedia.org/wiki/Evolution). • Ecological and evolutionary stability: we should talk about quasi-stability since no living system is stable (by definition). A system is stable if it can return to the same state (or approximately) after an environmental disturbance. This characteristic can be described by the term "resilience". Ecological equilibrium and evolutionary equilibrium are not necessarily the same thing, because evolution can lead to the emergence of new interactions between the elements of a system via mutation, for example, leading to new equilibria. • Trait: we retain a broad meaning of phenotypic trait resulting from genetic or epigenetic expression. These traits can be molecular, individual (e.g. morphological) or more integrative such as life traits (e.g., age at first reproduction, longevity). They may also be functional, related to response to environmental factors (e.g., [START_REF] Violle | Let the concept of trait be functional![END_REF]Garnier & Navas, 2013).

What separates, what brings together

Clearly highlighting what separates and brings together ecology and evolution in terms of concepts, ideas, approaches ... is not always easy because, as is often the case when dealing with sisters and brothers, what makes them diverge can also be a common point, all the more so as divergences and overlaps have evolved over time. I will limit myself to a few aspects that seem to me to be major, thus proposing a partial vision.

The nature of diversity. As mentioned above, ecology and evolution are interested in the same object, biodiversity; this means that the notion of variation, from the genotype to the ecosystem, plays a central role, because it is the fuel of the dynamics of ecological systemsecological variation, evolutionary variation. Such variation plays a much lesser role in other areas of biology, and even less so in physics, which gives a special epistemological status to the biodiversity sciences. Of course, the emphasis is not on the same diversity, from the genotype for the geneticist to the trophic chain for the ecosystem scientist; the former, by focusing on a species, will readily dispense with interspecific relationships, whereas intraspecific diversity is not central to community or ecosystem ecology. However, over the last twenty years, there has been a strong desire for rapprochement, which takes different forms.

If we look at the evolutionary side, a growing number of research programs are developing under natural conditions to fully account for environmental variation, while still often remaining focused on one species (Hendry, 2017). On the ecology side, the development of work aimed at understanding the relationship between specific diversity and ecosystem functioning now also takes intraspecific diversity into account, strongly driven by theoretical and conceptual developments [START_REF] Violle | The return of the variance: intraspecific variability in community ecology[END_REF][START_REF] Vellend | The theory of ecological communities[END_REF]. Attempts to link genetic and specific diversities to explain community functioning, via the environment, is another convergence point between ecology and evolution (Lamy et al., 2017).

Selective value vs. material and energy flows.

A comprehensive understanding of biodiversity dynamics can hardly ignore a common "currency". Population biology considers this to be the selective value ("fitness"), i.e. the number of offspring transmitted to the next generation by a gene, a genotype or an individual; a similar idea is found in community ecology, via the number of individuals produced by a species. Even if the unit considered is not the same, the dynamics is mediated by the number of units at generation N+1 produced by a unit at generation N (thus via individual dynamics), and this is what will measure the effect of natural selection or biotic interactions.

At the other end of the spectrum, ecosystem ecology has found its exchange value in flows, of matter or energy, thus much closer to physico-chemical approaches, and in particular to thermodynamic notions like entropy (and information). In the ecological psyche, this perspective is inevitably associated with the work of the Odum brothers, but also with generic hypotheses of the functioning of the earth such as Gaia (Lovelock, 1972). There will also appear here a thinking based on "complex systems" (Holling, 2001) which incorporates notions such as resilience, self-organization, complex adaptive systems, tipping point... all notions little used until recently in evolutionary biology, except in certain very theoretical approaches, and a little more in population dynamics (but see [START_REF] Nosil | Biodiversity, resilience and the stability of evolutionary systems[END_REF].

It seems difficult to directly transform a selective value into a flow, or the opposite, which do not concern the same entry into biodiversity -one rather on the biotic side and an emphasis on adaptation to the environment, the other on abiotic dynamics. However, selective value feeds on matter and energy, and the analysis of flows only makes sense for the understanding of biodiversity if these flows are embodied in units of selective value -this is even a foundation of functional ecology (Calow, 1987). One possible integration is through the simultaneous and coupled consideration of biotic and abiotic dynamics, for example by analyzing the link between physiology and selective value or via simple predator-prey models in which environmental dynamics (e.g., litter) depend on, and influence, the dynamics of predators and prey (Loreau, 2010). This type of models also allows us to understand the evolutionary dynamics of such systems.

Transmission and temporality.

A key point of evolutionary theories is the transmission of genetic information (or epigenetics) from generation to generation, which depends on many processes, such as the reproductive system. These processes are themselves subject to natural selection, and therefore depend on environmental dynamics. For example, an environmental modification that reduces the density of a population, or in plants that of pollinators, can lead to a uniparental selection regime, since it becomes more difficult to find a partner to reproduce. We can thus see that environmental temporality interacts with evolutionary dynamics. However, for a long time, ecology was considered as the science of short timescales, and evolution as the science of long timescales -simply because it would take time for an evolutionary process to take place, and for a trait to become fixed. Of course, it took tens of millions of years (and many trials) for a fish to become a mammal or a bird, or for flowering plants to diversify. Of course, environmental conditions can change drastically over a very short period of time -recent climatic changes are a strong reminder of this -and can rapidly lead to variations in species occurrence in an ecosystem, or even to their disappearance.

However, work carried out over the last few decades has shown that ecological time and evolutionary time can overlap. For example, experimental evolutionary work (Box 1) by R.

Lenski and his colleagues on the laboratory star Escherichia coli indicates a very rapid adaptation to environmental changes (see "For more information"). One could argue that these are organisms with a rapid life cycle (7-8 generations/day in this experiment). So let us consider organisms with slower life cycles: mosquitoes. The unfortunate "experiments" that we humans have conducted to eradicate them have systematically led to evolutionary reactions through the selection of resistance genes. Here, another argument could be developed: the selection pressure is considerable, both by the pesticide dose and by the geographical extent of the treatments. But rapid evolutionary processes have also been observed in natural populations of species with slower life cycles experiencing an environmental stress (e.g., Darwin's finches in the Galapagos; Grant & Grant, 2003), predation (e.g., guppies in Trinidad; [START_REF] Reznick | Experimental studies of evolution and eco-evo dynamics in guppies (Poecilia reticulata)[END_REF] or hunting pressure (e.g., elephants; Allendorf & Hard, 2009). Of course, organisms with long life spans and limited fecundity will have difficulties escaping environmental variation via rapid genetic evolution, but they may have individual plasticity that can be mobilized for a short-term response. However, it should be remembered that ecology and evolution can therefore play on the same temporal scale, and that this can serve as a basis for the definition of eco-evolutionary processes.

Forces and processes. Population biology, in particular population genetics, has long formalized population functioning under the impact of forces or processes whose relative impact it seeks to understand. We can mention mutation, which generates variation, population size which directly impacts the respective effect of genetic drift and natural selection, or migration. This framework is found in models from the 1930s onwards, with a more or less great influence given to the spatial structuring of populations (Fisher, 1930;[START_REF] Wright | Evolution in Mendelian populations[END_REF].

Such a frame appears in ecology only in the 1960s [START_REF] Mcarthur | The theory of island biogeography[END_REF], and has been revived more recently (e.g., Hubbell, 2001;[START_REF] Vellend | The theory of ecological communities[END_REF]; here we speak of speciation that brings new species into existence, ecological drift, interspecific competition, or migration. Thus, "natural" analogies can be detected between the levels of populations in species and species in biological communities. These kinds of ideas have also been generalized to ecosystems, with the notion of meta-ecosystems, including not only species migration, but also matter and energy flows (Loreau, 2010).

In fact, these approaches led authors to propose using these analogies to understand the dynamics of biodiversity jointly at different levels of integration of living organisms according to environmental conditions that will, for example, act directly on the population dynamics of all species [START_REF] Vellend | The theory of ecological communities[END_REF]Hendry, 2017;[START_REF] Mcpeek | Evolutionary community ecology[END_REF]. Consider site size as an environmental parameter: a larger site will be able to support larger populations, within species, as well as more species, because it may offer more resources or different microenvironments, or is more stable over time and/or more easily reached by migrants. With larger populations, there will be, on average, a higher evolutionary potential, and thus a possibility of maintaining ecological and evolutionary stability in the face of environmental change. Every analogy has its limits -here, natural selection at the population level (interactions between alleles) cannot take the same forms as the variety of interspecific interactions (mutualism, parasitism, competition, predation ...). But it remains that we have a comparative entry between evolution and ecology, via forces and processes, which act at different levels of integration.

Box 1. Approaches and tools for a dialogue

Ecology and evolution have been separated since the end of the 19th century in terms of approaches and tools, for example with a much greater emphasis on the study of biodiversity in the "natural environment" and on physico-chemical analyses for ecology and on experimental approaches in the laboratory or molecular biology for evolution. Considerable convergence has occurred in recent years, in connection with conceptual and technological developments, which seem to us to open up areas of dialogue on common objects of study. Indeed, these approaches and tools require a high level of technical competence given their increasing sophistication, and therefore the creation of places (sensu lato) for transmission and exchange (e.g., work groups, dedicated training courses, etc.). They also require more and more team work and projects integrating multiple approaches, or even exchanges between disciplines (inter-, trans-, multi-...). We can think of molecular and chemical methods, sensors, statistical methods (including artificial intelligence) or data mining. I will develop here the case of experimentation. A fundamental question, when one is interested in the functioning and dynamics of biodiversity, is to know if the controlled experimental approach is sufficient or if it is necessary to work "in nature". On the one hand, we consider the real environment undergone by the organisms, on the other hand we wish to have control over the environment. This old debate is far from over (see e.g. Hendry, 2019), echoing reductionist vs. holistic perspectives, but also the possibility of a hypothetical-deductive approach, and there is certainly no single answer. Moreover, research programs are increasingly jointly incorporating a range of approaches from microcosms under controlled laboratory conditions, especially for organisms with short generation times, to ensembles of field sites undergoing ambient environmental conditions (including the human environment), through instrumented experimental devices such as the Ecotron or the use of sensors to monitor biotic or abiotic parameters (Le Galliard & Montoya, 2017). In this perspective, rather than opposing laboratory and field, the trend is to develop long-term approaches, known as evolution or experimental ecology. While they do not exhaust the reductionist / holistic debate, they do allow for a more integrative understanding of the processes at play in a temporal perspective and an explicit consideration of the different levels of variation, for example intra-individual vs. interspecific, and of the appearance of new variation through mutation (or even speciation if we can wait long enough). They also make it possible to integrate various environmental forcing into ecology and evolution studies, such as climate change or the arrival of new variants or species, or the spatial structuring of biodiversity in a more or less fragmented form.

I have only touched on an (important) subset of what could integrate, or unlink, ecology and evolution. Other aspects that play a significant and growing role in this integration could have been mentioned. In the first place, natural selection and niche are strongly linked, natural selection emerging from ecological interactions, and I also did not insist much on the notion of space. One can also think of the notion of trait -in its sense of phenotypic trait resulting from genetic or epigenetic expression. Subject to selection and expressing plasticity, it can respond to environmental variation, from the individual to the ecosystem level, and regulate intra-and interspecific competitive relationships (see for example Garnier & Navas, 2013 in ecology; Roff, 1992 in evolutionary biology). Moreover, a trait can also be directly related to selective value or material and energy flows [START_REF] Violle | Let the concept of trait be functional![END_REF], providing a common entry point into the "population biology / community ecology / ecosystem ecology" triptych mentioned above. From a more pragmatic point of view, common tools and approaches have a role to play in the eco-evolutionary dialogue (Box 1). Phylogenetics could fall into this category; used for more than half a century to reconstruct kinship relationships between populations or species (Felsenstein, 2004), ecophylogenetics has developed with the idea of detecting ecological signals in the phylogenetic patterns of species communities, as a result of both evolutionary history and ongoing ecological interactions (Cavender-Bares et al., 2009;Mouquet et al., 2012). The results are quite disappointing, however, as these signals are weak.

Eco-evolution, really?

If the above explains why and how ecology and evolution are interested in the dynamics of biodiversity, when is it important to develop an eco-evolutionary thought and framework? As a first approach, we can categorize current approaches into (i) evolutionary studies, often "species-centric", focusing on genetic variation and introducing a dose of environment into their approaches; (ii) ecological studies that consider evolutionary impact (or mechanisms), focusing on biological communities or ecosystems; (iii) (probably rarer) integrative work. All three approaches are valid, both from a basic research perspective and when it comes to managing biodiversity [START_REF] Sarrazin | Evolution in the Anthropocene[END_REF]. In fact, any study of biodiversity is in the absolute sense eco-evolutionary since its dynamics and structure depend on ecological and evolutionary processes.

However, do we really need evolutionary thinking to explain species-environment relationships here and now (small spatial scale and short timescale)? Symmetrically, does the understanding of long-term evolutionary processes require a detailed ecological knowledge? Can we not refer evolution to long time and ecology to short time? In a way, decouple the time scales, as theorists sometimes do, by assuming a weak selection regime, such that population dynamics have time to equilibrate before a new mutation appears and potentially disrupts this equilibrium (e.g., Lion 2018). What place and need then for eco-evolutionary studies? Following Bassar et al.'s (2021) proposal, we will consider a dynamic to be truly ecoevolutionary when ecological and evolutionary processes are contemporaneous, i.e. when they are positioned in the same temporality or unfold at the same pace. This emphasizes reciprocal interactions between ecology and evolution on the same time scale, while noting that it is not always easy to appreciate this contemporaneity.

A growing number of studies can be considered eco-evolutionary, and we will take the example of guppies (Poecilia reticulata) in Trinitad (West Indies), a study conducted over the past 40 years by D. Reznick and colleagues [START_REF] Reznick | Experimental studies of evolution and eco-evo dynamics in guppies (Poecilia reticulata)[END_REF] based on a combination of field work (Fig. 2), mesocosm and molecular approaches. Guppies belong to fish communities in which the predation they experience, e.g. by cichlids, is low or high, both upstream and downstream of rivers; marked differences were noted, in response to predation for a range of traits (reproduction, morphology ...). The selection pressure is in fact due to the difference in density in the populations; the density is indeed much higher when predation is low. The high density in populations with low predation reduces food availability, and vice versa, leading to an existence adapted to these food conditions (e.g. smaller size). Under these conditions, primary productivity is also affected, as are the dynamics of other fishes that compete with guppies (Rivulus hartii). We can thus clearly see here the eco-evolutionary loop that operates on the scale of a few tens of generations, involving the different levels of biodiversity, mentioned at the beginning of the article, populations, communities and ecosystems.

Figure 2. A waterfall on the El Cedro River in Trinidad (West Indies) that separates fish communities where predation on guppies is high (downstream) from communities where predation is more limited. Predation not only leads to trait changes in guppies, but also to changes in ecosystem functioning. This is one of the sites where D. Reznick introduced guppies (upstream) in 1981 to study their evolution in the absence of predation. Photo credit: Joshua Goldberg.

We can also found examples where the starting point is not populations and evolutionary biology but rather the relationship between biodiversity and ecosystem functioning (BEF; e.g., are species-richer ecosystems more productive?) to integrate the evolutionary process. This is the case of a study conducted in the long-term grassland plant experiment ("the Jena experiment"; see "Read more"). The treatments included a variable diversity of species per experimental plot, in particular monocultures versus polycultures. The BEF relationship was shown to strengthen over generations, which may result from complementarity between species in resource use in polycultures. The study by [START_REF] Zuppinger-Dingley | Selection for niche differentiation in plant communities increases biodiversity effects[END_REF] suggests that this strengthening could also result from selection for niche differentiation between species via phenotype differentiation (e.g., leaf size in plants) -referred to as "character displacement" in the literature. Here, selection was based on the (epi)genetic variation of the plants that were selected to constitute the first generation of the experiment, leading to a reduction in interspecific competition. In this example, the eco-evolutionary loop is thus based on selection that leads to differential resource use.

As a conclusion

Ecology and evolution offer a necessary framework for understanding the structure and dynamics of biodiversity, a complex subject if ever there was one. Probably not a theory at all (do we really need one, despite Lawton (1999)?), but much more than a set of case studies ... via a spatio-temporal perspective, theories of adaptation and interactions of species, among themselves and with their environment. It seems important to keep a distinction between ecology and evolution, to let them develop in their respective niches, to refine and to increase their complexity in order to analyze and understand biodiversity. To use the metaphor of the brother and sister again, ecology and evolution, as they have grown older and matured, have come closer together, understanding each other better, as is often the case (one hopes) with siblings, even if they should interact more strongly. This rapprochement has taken shape over the last two decades on a theoretical and empirical basis, via data mining and the development of common tools; and the term eco-evolutionary is used in a common way, so common that it has come to lose a real meaning, perhaps to the point of tending to make us believe that in ecology and evolution, everything is in everything. It is proposed here to stick to a stricter definition of what is eco-evolutionary, based on a common temporality, while amplifying a dialogue through the development of common concepts, integrated (integrative) approaches jointly examining the complexities of ecological systems through collective work. Neither holism nor reductionism ... the right amount of parsimony to deconstruct these complexities and their dynamics.

If ecology and evolution are necessary to understand the dynamics of biodiversity (this is the focus of this article), they are not sufficient. In particular, I have not addressed the contribution of the humanities and social sciences, for example through the notion of socioecosystems or the rise of the "environmental humanities". Eco-evolutionary integration also seems to me to be an interesting way of addressing key points of the relationship of humans to the world. Evolution has taught us where we came from, how we fit into the history of life on Earth, or how we relate to the world (for example, through various value systems). Ecology forces us to think about our relationship to biodiversity, questions us about which ecosystem engineers we are, for better and (I fear) for worse. Eco-evolution can help us to project our futures, and to get out of the vortex of environmental destruction in which we have put ourselves -I believe in it in my most optimistic phases.

Figure 1 :

 1 Figure1: Simplified diagram of the decreasing impact of evolutionary theory (red triangle) in studies considering increasing levels of integration of biodiversity (blue triangle), moving from genes (population genetics) to ecosystems (ecosystem ecology). The blue triangle could also be seen as an indicator of space, with increasingly larger spatial scales (in general) toward the figure bottom.

I mean biodiversity in the current sense of the term, incorporating living entities and their environments.

Acknowledgements

I thank I. Arpin, C. Violle, E Fronhofer, P. Huneman and A. Teyssèdre for stimulating comments on the text, and D. Reznick for kindly exchanging over the evolution of guppies in Trinidad.

For more information Ecotron websites: https://www.anaee-france.fr/service/ecotrons On the experiment of R. Lenski and collaborators: https://en.wikipedia.org/wiki/E._coli_long-term_evolution_experiment On the Jena experiment: http://the-jena-experiment.de


Tarek Ahmed-Ali

Fouad Giri
email: fouad.giri@unicaen.fr

Miroslav Krstic
email: krstic@ucsd.edu

Laurent Burlion
email: laurent.burlion@onera.fr

Françoise Lamnabhi-Lagarrigue
email: francoise.lamnabhi-lagarrigue@lss.supelec.fr

Adaptive observer design with heat PDE sensor

The problem of state and parameter estimation is addressed for systems with cascade structure including finite-dimensional dynamics followed in series with infinite-dimensional dynamics. The formers are captured through an ODE that is state-and parameter-affine. The latter, referred to as sensor dynamics, are represented by a diffusion parabolic PDE. Both equations are subject to parameter uncertainty. Furthermore, the connection point between the ODE and the PDE blocs is not accessible to measurements. The aim is to get online estimates of all inaccessible states and unknown parameters of both the ODE and the PDE subsystems. This observation problem is dealt with by combining the backstepping design method and the extended Kalman observer approach. The obtained adaptive observer is shown to be exponentially convergent under an ad-hoc persistent excitation condition.

Introduction

An intensive research activity has been devoted to observer design over the past three decades. This activity has mainly been focused on (finite-dimensional) nonlinear systems described by ordinary differential equations (ODEs) leading to numerous observer design techniques and related stability and convergence results. So far, much less research effort has been devoted to the problem of state observation for systems involving partial differential equations (PDEs). In this respect, the emphasis has been put on the problem of designing exponential boundary observers for various types of systems including linear wave equation [START_REF] Guo | The stabilization of a one-dimensional wave equation by boundary feedback with noncollocated observation[END_REF], semilinear diffusion equation [START_REF] Fridman | Robust sampled-data control of a class of semilinear parabolic systems[END_REF]. Attention has also been paid to designing finitetime convergent observers, see e.g. [START_REF] Miranda | Observer design for a class of hyperbolic PDE equation based on a distributed super twisting algorithm[END_REF] and sampled-data exponentially convergent observers for parabolic-type PDEs (Ahmed-Ali, [START_REF] Ahmed-Ali | A new approach to enlarging sampling intervals for some sampled-data systems and observers[END_REF], 2016;Ahmed-Ali, Giri, Krstic, & Lamnabhi-Lagarrigue, 2016).

In [START_REF] Smyshlyaev | Backstepping observers for a class of parabolic PDEs[END_REF], a quite different approach to boundary observer design has been developed for a class of parabolic partial integro-differential equations. The resulting observers involve gain kernels that are constructed using the continuum version of the backstepping design method. This approach has been applied to observer design for various systems described by PDEs e.g. magnetohydrodynamic system [START_REF] Vazquez | Magnetohydrodynamic state estimation with boundary sensors[END_REF], linear hyperbolic system [START_REF] Vazquez | Backstepping boundary stabilization and state estimation of a 2-2 linear hyperbolic system[END_REF], and drilling system [START_REF] Hauge | Application of an infinitedimensional observer for drilling systems incorporating kick and loss detection[END_REF]. Later on, the backstepping-like approach has proved to be applicable to various other classes of systems including cascades of ODEs and PDEs. For these cascade systems, the aim is to recover the (finite-dimensional) state of the system ODE part and the (infinitedimensional) state of the PDE part. A major difficulty lies in the fact that the connection point between the two parts is not accessible to measurements. In [START_REF] Krstic | Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays[END_REF], the observation problem has been addressed for linear ODE-PDE cascade systems where the PDE part can be seen as representing sensor dynamics. The latter have been captured by a (purely convection) firstorder hyperbolic PDE which models transport delays. Using the backstepping design approach, new state transformations and Lyapunov-Krasovskii functional have been employed in observer design resulting in a new family of boundary predictor observers. A similar design approach has been developed in [START_REF] Krstic | Compensating actuator and sensor dynamics governed by diffusion PDEs[END_REF] to get state observers for ODE-PDE cascades where the PDE part is a (diffusion) parabolic equation (describing e.g. heat diffusion phenomena). Like the system, the observer structure is an ODE-PDE cascade augmented by innovation terms with observer gains. The design PDEs yielding observer gains, which were hyperbolic of first order in the delay problem considered in [START_REF] Krstic | Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays[END_REF], have turned out to be hyperbolic of second-order in the diffusion problem studied in [START_REF] Krstic | Compensating actuator and sensor dynamics governed by diffusion PDEs[END_REF]. Extensions of this observer design to sampled-output ODE-PDE systems have been proposed in Ahmed-Ali, [START_REF] Karafyllis | Stability of sampled-data PDE-ODE systems and application to observer design and output-feedback stabilization[END_REF] and [START_REF] Karafyllis | Stability of sampled-data PDE-ODE systems and application to observer design and output-feedback stabilization[END_REF].

In this paper, the problem of state observation for ODE-PDE systems, where the PDE is a diffusion parabolic equation, is addressed in presence of parametric uncertainty. The unknown parameter vectors come linearly in the finite-and infinitedimensional subsystem, but the associated regression vectors are allowed to be nonlinear in the system output adding thus extra nonlinear dynamics. Clearly, this class of systems is much wider than that of [START_REF] Krstic | Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays[END_REF] which was limited to parameter-uncertainty-free linear ODE-PDE systems. We seek an observer that provides online estimates of the ODE and PDE states and their unknown parameters. This observation problem is dealt with by designing an adaptive observer making use of the (infinite-dimensional) backstepping design technique and the (finite-dimensional) Kalman observer approach. Interestingly, the resulting observer turns out to be a nonlinear adaptive version of the generalized-predictor observer of [START_REF] Krstic | Compensating actuator and sensor dynamics governed by diffusion PDEs[END_REF]. It involves design PDEs yielding observer gains similar to those of their nonadaptive counterparts. Besides, the adaptive observer includes additional filters, defined by ODEs and PDEs, that define the (timevarying) direction along which the parameter adaptive laws evolve in the parameter space.

Compared to Ahmed-Ali, Giri, [START_REF] Ahmed-Ali | Adaptive observer for parabolic PDEs with uncertain parameter in the boundary condition[END_REF], Ahmed-Ali, Giri, Krstic, and Lamnabhi-Lagarrigue (2015) and Ahmed-Ali, Giri, Krstic, Burlion, and Lamnabhi-Lagarrigue (2016), the observation problem that is presently studied is quite different. Indeed, the class of systems considered in Ahmed-Ali, Giri, Krstic, Burlion, andLamnabhi-Lagarrigue (2016, 2015) is described using solely a semilinear heat PDE, i.e. no ODE subsystem was involved in those systems. Furthermore, the adaptive observers in Ahmed-Ali, Giri, [START_REF] Ahmed-Ali | Adaptive observer for parabolic PDEs with uncertain parameter in the boundary condition[END_REF] and Ahmed-Ali, Giri, Krstic, Lamnabhi-Lagarrigue, and Burlion (2016) necessitated as many sensors as the number of unknown parameters. In Ahmed-Ali, Giri, Krstic, and Lamnabhi-Lagarrigue (2015), an ODE-PDE system structure was considered but the system was not subject to parameter uncertainty and so the proposed observer was not adaptive. In Ahmed-Ali, [START_REF] Ahmed-Ali | Using exponential time-varying gains for sampled-data stabilization and estimation[END_REF]; Ahmed-Ali, Giri, Krstic, and Lamnabhi-Lagarrigue (2016), the problem of observer design is considered for delayed ODE systems subject to parameter uncertainty. The constant delay is described by first-order hyperbolic PDE leading to an ODE-PDE representation of the system. As the parameter uncertainty only enters in the ODE, the adaptive observer design problem proves to be much simpler than the present problem where parameter uncertainty also affects the PDE.

The paper is organized as follows: in Section 2, the observation problem under study is formulated and an adaptive observer is proposed; the observer convergence is analyzed in Section 3 and numerical simulation results are provided in Section 4; a conclusion and reference list end the paper. To alleviate the presentation, some technical proofs are appended. Notations. Throughout the paper, R n denotes the n dimensional real space and the corresponding Euclidean norm is denoted ∥•∥.

This also denotes the induced matrix norm in R n×m , the set of all n × m real matrices. Functions that are continuously differentiable with respect to all their arguments are denoted C 1 . L 2 [0, D] is the Hilbert space of square integrable functions and the corresponding

L 2 norm is denoted ∥•∥ 2 . Accordingly, ∥w∥ 2 =   D 0 w 2 (ς)dς  1/2 for all w ∈ L 2 [0, D]. H 1 (0, D) is the Sobolev space of absolutely continuous functions w : [0, D] → R; x → w(x) with dw/dx ∈ L 2 [0, D]. H 2 (0, D) is the Sobolev space of scalar functions w : [0, D] → R; x → w(x) with absolutely continuous dw/dx ∈ L 2 [0, D] and d 2 w/dx 2 ∈ L 2 [0, D].
For any w ∈ H 1 (0, D) such that w(0) = 0 or w(D) = 0, the following Wirtinger's inequalities hold:

 D 0 w 2 (x)dx ≤ 4D π 2  D 0 w 2 x (x)dx (1a) max 0≤x≤D w 2 (x) ≤ D  D 0 w 2 x (x)dx. (1b)
Finally, given a two-dimensional function w :

[0, D] × R + → R; (x, t) → w(x, t), the notation w[t] and w x [t] refer to the functions defined on 0 ≤ x ≤ D by (w[t])(x) = w(x, t) and (w x [t])(x) = ∂w(x, t)/∂x.

Problem formulation and adaptive observer statement

The system under study is composed of a finite-dimensional nonlinear subsystem connected in series with an infinitedimensional subsystem as depicted in Fig. 1. The former assumes the following state-space representation:

Ẋ(t) = AX (t) + φ 1 (t)θ 1 , t ≥ 0, (2) with X (0) = X 0 ∈ R n is arbitrary. The infinite-dimensional
subsystem is modeled by a parabolic PDE of the form,

u t (x, t) = u xx (x, t) + φ 2 (x, t)θ 2 , 0 ≤ x ≤ D (3a) u x (0, t) = 0, u(D, t) = CX (t) (3b)
where A ∈ R n×n and C ∈ R 1×n are known constant matrices and the pair (A, C ) is observable;

φ 1 ∈ C 1 ([0, ∞) : R n×m 1 ) and φ 2 ∈ C 1 ([0, D] × [0, ∞) : R 1×m
2 ) are known bounded functions; the domain length D > 0 is a known scalar. The state vector X (t) ∈ R n and the distributed state u(x, t) ∈ R are not accessible to measurement, except for the boundary state u(0, t) which stands as the output of the whole system. The system parameter vectors

θ 1 ∈ R m 1 and θ 2 ∈ R m 2 are not known but their dimensions (m 1 , m 2 ) are.
The aim is to design an observer able to provide accurate online estimates of the system state functions X (t) and u(x, t) (t > 0, ≤ x ≤ D), on the one hand, and the unknown parameter vectors θ 1 and θ 2 , on the other. The observer must only make use of the system output y(t) = u(0, t). The uncertain quantities in (2) and (3a) constitute a new feature of this study compared to [START_REF] Krstic | Compensating actuator and sensor dynamics governed by diffusion PDEs[END_REF] where φ 1 (t)θ 1 = φ 2 (x, t)θ 2 = 0. On the other hand, the functions φ 1 (t) and φ 2 (x, t) are presently allowed to be output dependent i.e. one might have φ 1 (t) = ψ 1 (t, y(t)) and φ 2 (x, t) = ψ 2 (x, t, y(t)) for some functions ψ 1 (•, •) and ψ 2 (•, •). That is, the present observer design is not limited to linear systems, unlike [START_REF] Krstic | Compensating actuator and sensor dynamics governed by diffusion PDEs[END_REF].

To cope with the observation problem at hand, the adaptive observer of Table 1 is proposed.

Ẋ(t) = A X(t) + φ 1 (t) θ1 (t) -K (û(0, t) -u(0, t)) + λ 1 (t) θ1 (t) + λ 2 (t) θ2 (t) (4a) ût (x, t) = ûxx (x, t) + φ 2 (x, t) θ2 (t) -CM(x) M -1 (D)K (û(0, t) -u(0, t)) +  λ 3 (x, t) + CM(x)M -1 (D)λ 1 (t)  θ1 (t) +  λ 4 (x, t) + CM(x)M -1 (D)λ 2 (t)  θ2 (t) (4b) ûx (0, t) = 0, û(D, t) = C X(t) (4c)
for all t ≥ 0 and all x ∈ [0, D], where

K ∈ R n is such that A -KCM -1 (D) is Hurwitz.
Parameter adaptive law

θ (t) = -ρR(t)Λ(t)ũ(0, t) (4d) Ṙ(t) = R(t) -R(t)Λ(t)Λ T (t)R(t) (4e) with θ (t) =  θ T 1 (t) θT 2 (t)  T ∈ R m , R(t) ∈ R m×m , m = m 1 + m 2 ,
and

Λ(t) =  CM -1 (D)λ 1 (t) + λ 3 (0, t) CM -1 (D)λ 2 (t) + λ 4 (0, t)  T ∈ R m , (4f)
where θ (0

) ∈ R m , R(0) = R T (0) > 0 and ρ > 0 are arbitrary. Filters λ1 (t) = (A -KCM -1 (D))λ 1 (t) + φ 1 (t) -K λ 3 (0, t) (4g) λ2 (t) = (A -KCM -1 (D))λ 2 (t) -K λ 4 (0, t) (4h) λ 3,t (x, t) = λ 3,xx (x, t) -CM(x)M -1 (D)φ 1 (t) (4i) λ 4,t (x, t) = λ 4,xx (x, t) + φ 2 (x, t) (4j) λ 1 (0) = 0 ∈ R n×m 1 , λ 3 (D, t) = 0 ∈ R 1×m 1 , λ 3,x (0, t) = 0, λ 2 (0) = 0 ∈ R n×m 2 (4k) λ 4 (D, t) = 0 ∈ R 1×m 2 , λ 4,x (0, t) = 0 (4l) M(x) = (I 0) exp  0 A I 0  x   I 0  ∈ R n×n (4m)
Remark 1. (a) Relevant properties of the matrix function M(x) defined by (4m) are described in Appendix B. Accordingly, one has

A = M(D)AM -1 (D) which implies A -KCM -1 (D) = M(D)AM -1 (D) -KCM -1 (D) = M(D)  A -M -1 (D)KC  M -1 (D).
(5)

This shows a similarity between the matrices A -KCM -1 (D) and A-M -1 (D)KC which then have identical eigenvalues. On the other hand, we know that the pair (A, C ) is observable (by assumption). Therefore, A -M -1 (D)KC can be made Hurwitz by appropriately choosing the gain K . The same conclusion applies to the similar matrix A-KCM -1 (D). It is thus demonstrated that the requirement on the gain K (to be selected so that A -KCM -1 (D) is Hurwitz) is not an issue. (b) A well posedness analysis of the system and the adaptive is outlined in Appendix A.

(c) Introduce the variable change

L = M -1 (D)K . Then, A - LC = M -1 (D)  A -KCM -1 (D)  M(D)
is clearly Hurwitz and the adaptive observer (4a)-(4c) rewrites in terms of L as follows:

Ẋ(t) = A X(t) + φ 1 (t) θ1 (t) -M(D)L ũ(0, t) + λ 1 (t) θ1 (t) + λ 2 (t) θ 2 (t) (6a) ût (x, t) = ûxx (x, t) -CM(x)Lũ(0, t) + φ 2 (x, t) θ2 (t) +  λ 3 (x, t) + CM(x)M -1 (D)λ 1 (t)  θ1 (t) +  λ 4 (x, t) + CM(x)M -1 (D)λ 2 (t)  θ2 (t) (6b) ûx (0, t) = 0, û(D, t) = C X(t). (6c)
Clearly, if φ 1 (t), φ 2 (x, t) and θ i (t) (i = 1, 2) are all set to zero then, the adaptive observer (6a)-(6c) boils down to the (nonadaptive) observer ( 84)-(87) of [START_REF] Krstic | Compensating actuator and sensor dynamics governed by diffusion PDEs[END_REF].

(d) The observer proposed in Ahmed-Ali, Giri, Krstic, and Lamnabhi-Lagarrigue (2015) concerned a class of ODE-PDE systems involving no parameter uncertainty. That non-adaptive high-gain type observer is quite different from the present adaptive observer of Table 1.

(e) In Ahmed-Ali, [START_REF] Ahmed-Ali | Using exponential time-varying gains for sampled-data stabilization and estimation[END_REF]; Ahmed-Ali, Giri, Krstic, and Lamnabhi-Lagarrigue (2016), ODEs with output delay and parameter uncertainty were considered. Following [START_REF] Krstic | Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays[END_REF], the observer design was performed on the basis of an ODE-PDE representation where the PDE, a firstorder hyperbolic subject to no parameter uncertainty, represents the constant well-known delay. The resulting adaptive observer turned out to be a much simpler version of the present adaptive observer, e.g. while the parameter adaptive law in the former involved a single filter, the present adaptive law involves four interconnected filters, i.e. (4g)-(4l). In this respect, note also that the present adaptive observer is not an extension of the one in Ahmed-Ali, Giri, Krstic, Burlion, and Lamnabhi-Lagarrigue (2016). Indeed, the former does not boil down to the latter in the simpler case where θ 2 = 0 because the PDEs are different in the two papers. That is, the adaptive observers in both papers are quite different from each other.

Adaptive observer analysis

Introduce the state and parameter estimation errors:

X = X -X , ũ = û -u, θ1 = θ1 -θ 1 , θ2 = θ2 -θ 2 . (7a)
Also, for convenience, the following notations are introduced:

θ =  θ T 1 θ T 2  T ∈ R m , θ = θ -θ. (7b)
Using ( 1)-( 2) and (3a)-(3b), it is readily seen that these errors undergo the following equations:

Ẋ = A X + φ 1 (t) θ1 -K ũ(0, t) + λ 1 (t) θ1 (t) + λ 2 (t) θ 2 (t) (8a) ũt (x, t) = ũxx (x, t) + φ 2 (x, t) θ2 (t) -CM(x) M -1 (D)K ũ(0, t) +  λ 3 (x, t) + CM(x)M -1 (D)λ 1 (t)  θ1 (t) +  λ 4 (x, t) + CM(x)M -1 (D)λ 2 (t)  θ2 (t) (8b) ũ(D, t) = C X(t) (8c) ũx (0, t) = 0 (8d) θ(t) = -ρR(t)Λ(t)ũ(0, t) (8e) Ṙ-1 = -R -1 + ΛΛ T (8f)
where the penultimate equation is obtained noticing that θ(t) = θ(t), θ i (t) = θi (t) (i = 1, 2) and the argument t is omitted (when clear) to alleviate expressions. In (8f), Ṙ-1 refers to the derivative dR -1 /dt. Eq. ( 8f) is equivalent to (4e) provided that R(t) is uniformly invertible. To meet this requirement, a persistent excitation (PE) property is needed. To this end, the following notation is introduced:

Λ(t) =  CM -1 (D)λ 1 (t) + λ 3 (0, t) CM -1 (D)λ 2 (t) + λ 4 (0, t)  T .
Then, the PE assumption is stated as follows:

PE Assumption. The vector signal Λ(t) is supposed to be persistently exciting in the sense that, ∃δ, ε 0 > 0, ∀t > 0 :

 t+δ t Λ(s)Λ T (s)ds > ε 0 I (9)
where I ∈ R m×m denotes the identity matrix.

Intuitively, the PE assumption means that the vector subset {Λ(s); t ≤ s ≤ t + δ} spans the parameter vector space R m , whatever t. It is readily seen from ( 4g)-(4j) that Λ(t) only depends on the known input signals φ 1 (t) and φ 2 (0, t) and the auxiliary function M(x) (i.e. Λ(t) does not depend on the state estimates generated by the observer). Therefore, it is quite possible to check whether the PE condition ( 9) is satisfied. Given that φ 1 (t) and φ 2 (x, t) linearly enter (4g)-(4j), the PE requirement is met if the power spectra of φ 1 (t) and φ 2 (0, t) are rich enough (e.g. Ioannou & Sun, 2006). Now, it is shown in many places (e.g. [START_REF] Ioannou | Robust adaptive control[END_REF][START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF] that if (9) holds then the time-varying matrix gain inverse R -1 (t) exists, is positive definite, and stays bounded away from 0. Specifically, one then has

r 0 ≤ R -1 (t) ≤ r 1 , for all t ≥ 0 (10)
for some couple (r 0 , r 1 ) of positive scalars. In the sequel, it is supposed that property (9) holds so that one can make use of ( 9)-(10).

Remark 2. Conditions ( 9)-(10) will prove to be crucial in making the parameter estimation error θ exponentially convergent to zero.

It is worth noting that the structure of the parameter adaptive law (4d)-( 4f) is quite similar to the so-called ''forgetting-factor leastsquares'' estimator (see e.g. Ioannou & Sun, 2006, p. 198). Except for the structural similarity, the adaptive law (4d)-( 4f) is novel due to the way the regressor Λ(t) is generated.

The exponential convergence of the adaptive observer is now stated in the following theorem:

Theorem 1. Consider the adaptive observer of Table 1 and let there the gain ρ of the parameter adaptive law be such that ρ > 1/2.

Then, when applied to the system (2)-(3), the observer is globally exponentially convergent in the sense that the errors, X(t), θ(t), and the norm  D 0 ũ2 (x, t)dx, are exponentially vanishing (as t → ∞), whatever the initial conditions are X(0

) ∈ R n , û[0] ∈ L 2 (0, D), θ(0) ∈ R m .
Proof. The proof of the theorem is divided in four parts. In the first part, the four auxiliary state vectors, λ 1 (t) to λ 4 (x, t), generated by (4g)-( 4l) are shown to be bounded. The second part of the proof is devoted to introducing a transformation of the error system (8a)-(8f). The transformed system is shown in the third part to be exponentially stable. Finally, the results of the theorem are established in the final part.

Part 1. Boundedness of λ 1 (t), λ 2 (t), λ 3 (x, t) and λ 4 (x, t).

It will be shown that there exist real numbers, λ i,max > 0 (i = 1, 2, 3, 4), such that the following properties hold:

∥λ i (t)∥ ≤ λ i,M (i = 1, 2) and ∥λ i (x, t)∥ ≤ λ 1,M (i = 3, 4), for all t ≥ 0 and all x ∈ [0, D]. (11)
To alleviate the paper body, the proof of this result is placed in Appendix C.

Part 2. Transformation of the error system (8a)-(8f).

Consider the following backstepping transformations, partly inspired by [START_REF] Krstic | Compensating actuator and sensor dynamics governed by diffusion PDEs[END_REF]:

Z (t) = X(t) -λ 1 (t) θ1 (t) -λ 2 (t) θ2 (t), (12a) ε(x, t) = ũ(x, t) -CM(x)M -1 (D) X(t) -λ 3 (x, t) θ1 (t) -λ 4 (x, t) θ2 (t). (12b)
Differentiating Z (t), one gets using (8a)-( 8b), ( 12a)-( 12b) and (4m):

Ż = Ẋ -λ1 θ1 -λ 1 θ1 -λ2 θ2 -λ 2 θ 2 = A X + φ 1 (t) θ1 -K ũ(0, t) -λ1 θ1 -λ2 θ2 (using (8a)) = AZ + Aλ 1 θ1 + Aλ 2 θ2 + φ 1 (t) θ1 -K ũ(0, t) -λ1 θ1 -λ2 θ2 (using (12a)) = AZ -K ε(0, t) -KCM -1 (D) X(t) -[K λ 3 (0, t) + λ1 -Aλ 1 -φ 1 (t)] θ1 -[K λ 4 (0, t) + λ2 -Aλ 2 ] θ2 (using (12b)) =  A -KCM -1 (D)  Z -K ε(0, t) -[K λ 3 (0, t) + λ1 -Aλ 1 -φ 1 (t) + KCM -1 (D)λ 1 ] θ1 -[K λ 4 (0, t) + λ2 -Aλ 2 + KCM -1 (D)λ 2 ] θ2
using (12a), where the arguments (x, t) have been omitted to alleviate expressions. Using (4g) and (4j), the last equality further simplifies to:

Ż =  A -KCM(0)M -1 (D)  Z -K ε(0, t). (13a)
Similarly, differentiating ε(x, t) with respect to t, one obtains using (8a)-( 8b) and (12b):

ε t = ũt -CM(x)M -1 (D) Ẋ -λ 3,t θ1 -λ 3 θ 1 -λ 4,t θ2 -λ 4 θ 2 = ũxx -CM(x)M -1 (D)A X -  CM(x)M -1 (D)φ 1 + λ 3,t  θ1 +  φ 2 -λ 4,t  θ2 (using (8a)-(8b)) = ε xx + C  d 2 M dx 2 (x)M -1 (D) -M(x)M -1 (D)A  X -  CM(x)M -1 (D)φ 1 + λ 3,t -λ 3,xx  θ1 +  φ 2 -λ 4,t + λ 4,xx  θ2 (using (12b)).
Using (4i)-(4j) and Appendix B (Part 2), the last equality boils down to:

ε t (x, t) = ε xx (x, t). ( 13b
)
This is completed by the following two boundary conditions:

ε(D, t) = 0 (13c) ε x (0, t) = 0 (13d)
where ( 13c) is obtained from (12b) using ( 8c)-(8d); Eq. ( 13d) is also obtained from (12b) using Appendix B (Part 1) and ( 4k)-( 4l).

In addition to (13a)-(13d), Eqs. (8e)-(8f) which govern θ must be retained but the term ũ(0, t) on the right side of (8e) has to be expressed in terms of ε(0, t), Z (t) and θ(t), using (12a)-( 12b). The resulting equation together with (13a)-(13d) constitutes the new error system expressed in terms of the new coordinates, Z and ε.

For convenience, this new system is recapitulated here:

Ż =  A -KCM -1 (D)  Z -K ε(0, t) (14a) ε t (x, t) = ε xx (x, t) (14b) ε(D, t) = 0 (14c) ε x (0, t) = 0 (14d) θ = -ρRΛΛ T θ -ρRΛ  ε(0, t) + CM -1 (D)Z  (14e) Ṙ-1 = -R -1 + ΛΛ T . ( 14f
)
Part 3. Stability analysis of the transformed system (14a)-(14f).

To analyze the stability of this new error system, consider the following Lyapunov functional candidate:

V = Z T PZ + a  D 0 ε 2 (x, t)dx + θ T R -1 θ (15)
with P any symmetric positive definite matrix satisfying the algebraic equation,

P(A -KCM -1 (D)) + (A -KCM -1 (D)) T P ≤ -µ I. (16)
The matrix P exists because (A -KCM -1 (D)) is Hurwitz. At this stage, the scalars a > 0 are arbitrary. Differentiating (15) yields, using ( 16) and ( 14a)-( 14f):

V = ŻT PZ + Z T P Ż + 2a  D 0 ε(x, t)ε t (x, t)dx + θ T Ṙ-1 θ + 2 θT R -1 θ =  [A -KCM -1 (D)]Z -K ε(0, t)  T PZ + Z T P  [A -KCM -1 (D)]Z -K ε(0, t)  + 2a  D 0 ε(x, t)ε xx (x, t)dx + θT [-R -1 + ΛΛ T ] θ + 2 θ T R -1  -ρRΛΛ T θ -ρRΛ[ε(0, t) + CM -1 (D)Z]  ≤ -µ    Z   2 -2 ZT PK ε(0, t) -2a  D 0 ε 2 x (x, t)dx -θ T R -1 θ + (Λ θ) 2 -2ρ(Λ T θ) 2 -2ρ θT Λε(0, t) -2ρ θ T ΛCM -1 (D)Z ( 17
)
where an integration by parts and the boundary conditions (14c-d) have been used in the last inequality. Applying Young's inequality to the cross products on the right side of ( 17), one gets:

V ≤ -µ ∥Z∥ 2 -2a  D 0 ε 2 x (x, t)dx -θ T R -1 θ -(2ρ -1)( θ T Λ) 2 + ξ ∥Z∥ 2 + ∥PK ∥ 2 ξ (ε(0, t)) 2 + ρ υ ε 2 (0, t) + ρυ( θT Λ) 2 + ρω ∥Z∥ 2 + ρ   CM -1 (D)   2 ω ( θ T Λ) 2 ≤ -(µ -ξ -ρω) ∥Z∥ 2 -θT R -1 θ -  2a - D ∥PK ∥ 2 ξ - Dρ υ   D 0 ε 2 x (x, t)dx -  2ρ -1 -ρ(υ +   CM -1 (D)   2 ω )  ( θT Λ) 2 (18)
where the last inequality is obtained applying Wirtinger's inequality (1b) and the scalars ξ > 0, υ > 0 and ω > 0 are arbitrary. Let these free parameters be selected so that:

µ -ξ -ρω > 0 (19a) 2a - D ∥PK ∥ 2 ξ - Dρ υ > 0 (19b) 2ρ -1 -ρ(υ +   CM -1 (D)   2 ω ) > 0. ( 19c
)
The last inequality is satisfied provided the pair (υ, ρ) is set such

that υ +   CM -1 (D)    2 ω < 2ρ-1
ρ . This is possible because ρ > 1/2. Furthermore, letting ξ = 1 and noting that a is independent on the rest of the parameters (especially D, K , P, ρ), it follows that (19b) is satisfied provided that a is sufficiently large. In turn,(19a) is satisfied by letting µ be sufficiently large. , it follows from ( 18) and ( 15) that:

V ≤ -(µ -1 -ρω) ∥Z∥ 2 -θT R -1 θ -  2a -D ∥PK ∥ 2 - Dρ υ  π 2 4D  D 0 ε 2 (x, t)dx ≤ -min  1, µ -1 -ρω λ max (P) , π 2 4D (2 - D ∥PK ∥ 2 a - Dρ aυ )  V (20)
with λ max (P) being the largest eigenvalue of P, where Wirtinger's inequality (1a) has been used to get the first inequality. Clearly, this implies that V is exponentially vanishing (as t → ∞). Due to (15), so are Z (t), θ(t) and  D 0 ε 2 (x, t)dx.

Part 4. Exponential convergence of the original errors.

Using the fact that M(x), λ 1 (t) and λ 3 (x, t) are bounded and Z(t), θ(t) are exponentially convergent, it follows from (12a)-(12b) that X(t) and ũ(x, t) are also exponentially vanishing (as t → ∞). Theorem 1 is proved.

Remark 3. The transformation defined by (12a)-( 12b) and (4g)-( 4l) is a key feature of the observer design and analysis. It is partly inspired by, but is also a generalization of, previous finite-or infinite-dimensional transformations. Indeed, the finite-dimensional transformation in [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF], which applies to adaptive observer design for ODEs, is obtained letting λ 2 (t), λ 3 (x, t) and λ 4 (x, t) be zero and deleting (12b). Also, the infinite- dimensional transformation in [START_REF] Krstic | Compensating actuator and sensor dynamics governed by diffusion PDEs[END_REF], that applies to nonadaptive observer design for ODE-PDEs, is obtained by letting all of λ 1 (t), λ 2 (t), λ 3 (x, t) and λ 4 (x, t) be zero and deleting (12a).

The infinite-dimensional transformation in Ahmed-Ali, Giri, [START_REF] Ahmed-Ali | Adaptive observer for parabolic PDEs with uncertain parameter in the boundary condition[END_REF] and Ahmed-Ali, Giri, Krstic, Lamnabhi-Lagarrigue, and Burlion (2016), used in adaptive observer design for parabolic PDEs, is obtained letting λ 1 (t), λ 2 (t), and λ 3 (x, t) be zero and deleting (12a). Leaving aside the difference between the PDEs in this paper and in Ahmed-Ali, [START_REF] Ahmed-Ali | Using exponential time-varying gains for sampled-data stabilization and estimation[END_REF]; Ahmed-Ali, Giri, Krstic, and Lamnabhi-Lagarrigue (2016), the transformation in the latter can be viewed as a particular case of (12a)-(12b), obtained letting λ 2 (t) and λ 4 (x, t) be zero. Finally, note that only the new transformation (12a)-(12b) features a coupling between the (finite-dimensional) transformation, defined by (12a) and ( 4g)-(4h), on one hand, and the transformation (12b)

Simulation

To illustrate the performances of the adaptive observer of Table 1, a system of the form (2)-( 3) is considered with the following parameters and functions:

A =  0 1 -1 0  ; C =  1 0  ; D = 1; θ 1 = 0.5; θ 2 = 1.2 φ 1 (t) =  0 1 + (sin(3t)) 2  ; φ 2 (x, t) = e 0.2x  5 + (sin(3t)) 2  ;
with the initial conditions X (0) =  1 -1  T and u(x, 0) = 0 (0 ≤ x < D). The parameters of the adaptive observer of Table 1 are set to K = [17.1 11.8] T (leading to the eigenvalues (-3, -6) of the matrix A -KCM -1 (D)), ρ = 1, and R(0) = 5I. The observer is run with the initial conditions X(0) = 

0 0  T , û(x, 0) = 2 (0 ≤ x < D), θ1 (0) = 0, θ2 (

Conclusion

The problem of state observation is addressed for ODE-PDE cascades modeled by ( 2)-( 3). The aim is to get online estimates of the states X (t) and u(x, t) (0 ≤ x ≤ D), on the one hand, and of the parameter vectors θ 1 and θ 2 , on the other.

To this end, the adaptive observer of Table 1 is designed and shown to be exponentially convergent. The gain matrix M(x), the filters (4g)-(4l), and the forgetting-factor least-squares estimators (4d)-( 4f) are instrumental components of this observer. Also, the presence of the parameter estimate speed θ in the state observer (4a)-( 4b) is an appealing feature. The observer analysis shows that the backstepping transformation defined by (12a)-(12b) plays a crucial role in the proof of exponential convergence.

(X, p) as follows:

Ẋ(t) = AX (t) + φ 1 (t)θ 1 , t ≥ 0, (A.2) p t (x, t) = p xx (x, t) + φ 2 (x, t)θ 2 -CM(x)M -1 (D)φ 1 (t)θ 1 (A.3) p x (0, t) = p(D, t) = 0, for all t ≥ 0 (A.4) u(x, t) = p(x, t) + CM(x)M -1 (D)X(t). (A.5)
One feature of the new system representation is that the infinitedimensional subsystem, here defined by (A.3)-(A.4), is decoupled from the finite-dimensional subsystem described by (A.2) (while a coupling existed in the initial model ( 2)-( 3)). Then, the existence of solutions of each parts can be analyzed separately. The existence and uniqueness of the solution X ∈ C 1 ([0, ∞) : R n ) of (A.2) is not an issue, due to the usual existence theorem of ODEs. Owing to (A.3)-(A.4), this is a particular case of the more general semilinear PDE (21) in [START_REF] Fridman | Robust sampled-data control of a class of semilinear parabolic systems[END_REF]. It has been proved that a unique strong solution p[t] exists in the Hilbert space

H 1/2 = D((-Γ ) 1/2 ) =  w ∈ H 1 (0, D) : w x (0) = w(D) = 0  with Γ = ∂ 2 ∂x 2
, where (-Γ ) 1/2 is the square root of -Γ . Then, a similar result holds with u[t] due to (A.5).

To analyze the well posedness of the observer of Table 1, it is more judicious to start analyzing the (transformed) error system (14a)-(14f). First, the parabolic equation ( 14b)-(14d) has a well known solution in H 2 (0, D) that can be found in many places (see e.g. Smyshlyaev & Krstic, 2005, p. 315). Then, by the usual existence theorem of ODEs, (14a) has a unique solution Z ∈ C 1 ([0, ∞) : R n ).

For the same reason, the ODEs (14e)-(14f) have unique solutions θ ∈ C 1 ([0, ∞) : R m ) and R -1 ∈ C 1 ([0, ∞) : R m ) and, by ( 10) and (7b), one gets the unique existence of θ ∈ C 1 ([0, ∞) : R m ) and R ∈ C 1 ([0, ∞) : R m ). Well posedness of the parabolic PDEs (4i)-(4j) can also be established applying the analysis of [START_REF] Fridman | Robust sampled-data control of a class of semilinear parabolic systems[END_REF]. Accordingly, one gets that those equations have strong solutions λ 3 [t] and λ 4 [t] in the Hilbert space H 1/2 defined above. Then, by the usual existence theorem of ODEs, (4g)-(4h) have unique solutions λ 1 ∈ C 1 ([0, ∞) : R n×m 1 ) and λ 2 ∈ C 1 ([0, ∞) : R n×m 2 ). The results obtained so far, together with (12a)-(12b), imply the existence of a unique solution X ∈ C 1 ([0, ∞) : R n ) and unique strong solution ũ[t] in the space H 1/2 . Similar existence results hold with the states X and û[t], due to (7a).

Appendix B. Properties of the matrix function M (x)

The matrix function M(x) satisfies the following properties (see Lemma 1 in Ahmed-Ali, Giri, Krstic, & Lamnabhi-Lagarrigue, 2015):

(1) d 2 M dx 2 (x) = M(x)A, with M(0) = I, dM dx (0) = 0. (2) M(x) =  ∞ i=0 x 2i (2i)! A i .
(3) AM(x) = M(x)A and M -1 (x)A = AM -1 (x).

The first equality in (3) is obtained by pre-and postmultiplication of both sides of the development of Part 2 by A. The second equality is obtained by pre-and post-multiplication of both sides of the first equality by M -1 (x).

Appendix C. Proof of boundedness of λ i (t) (i = 1, 2) and λ i (x, t) (i = 3, 4)

As the vector signals φ 1 (t) are bounded and A -KCM -1 (D) is Hurwitz it follows from (4g)-(4f) that, λ 1 (t) and λ 2 (t) are bounded provided λ 3 (0, t) and λ 4 (0, t) are so. Then, one just needs to show that the latter are bounded. The proof will only be performed for λ 3 (0, t) (the proof for λ 4 (0, t) matches similar arguments). where the last term is obtained using an integration by parts and the boundary conditions (4k). Again using integration by parts and (4k), equality (C.2) further develops as follows:

Ẇ1 (λ 3 ) = -

Ẇ1 (λ 3 ) ≤ -  1 - 2Dξ π 2   D 0   λ 3,x (x, t)   2 dx -  1 - ς 2   D 0   λ 3,xx (x, t)   2 dx +  1 2ξ + 1 2ς   D 0   CM(x)M -1 (D)φ 1 (t)   2 dx. (C.4)
Let the still free scalars be selected as follows:

1 -2Dξ π 2 < 0; 1 -ς 2 < 0.

(C.5) Then, by applying Wirtinger's inequality (1a) to the two first terms on the right side of (C.4), it follows that:

Ẇ1 (λ 3 ) ≤ -  1 - 2Dξ π 2  π 2 4D  D 0 ∥λ 3 (x, t)∥ 2 dx -  1 - ς 2  π 2 4D  D 0   λ 3,x (x, t)   2 dx +  1 2ξ + 1 2ς   D 0   CM(x)M -1 (D)φ 1 (t)   2 dx ≤ - π 2 2D min  1 - 2Dξ π 2 , 1 - ς 2  W 1 (λ 3 ) +  1 2ξ + 1 2ς   D 0   CM(x)M -1 (D)φ 1 (t)   2 dx (C.6)
where the last inequality is obtained using (C.1). Note that φ 1 (t)

is also bounded by assumption and M(x) is bounded due to (4m).

Then, it follows from (C.6) that W 1 (λ 3 ) is bounded and, from (C.1), so is

 D 0   λ 3,x (x, t)
  2 dx. Applying Wirtinger's inequality (1b), it follows that ∥λ 3 (x, t)∥ is bounded. The proof is complete.

Fig. 1 .

 1 Fig. 1. System structure.

Fig. 2 .

 2 Fig. 2. System state variables X (t) = [x 1 (t) x 2 (t)] T (solid) and their estimates (dashed).

Fig. 3 .

 3 Fig. 3. System parameters (θ 1 , θ 2 ) (solid) and their estimates (dashed).and (4i)-(4j) (involving infinite-dimensional signals), on the other hand. This coupling is part of the current transformation novelty that makes the observer analysis harder.

  0) = 0. The resulting observer performances are illustrated by Figs. 2-5. Clearly, all state and parameter estimates converge to their true values confirming the theoretical result of Theorem 1.

Fig. 4 .

 4 Fig. 4. PDE states u(0.1, t) (top) and u(0.7, t) (bottom). Solid: true system signals. Dashed: signal estimates.

Fig. 5 .

 5 Fig. 5. Estimation error ũ(x, t) vs 0 ≤ x ≤ 1 and t ≥ 0.

  x, t)[CM(x)M -1 (D)φ 1 (t)] T dx - D 0 λ 3,x (x, t)λ T 3,tx (x, t)dx =  D 0 λ 3 (x, t)λ T 3,xx (x, t)dx x, t)[CM(x)M -1 (D)φ 1 (t)] T dx -

  x, t)[CM(x)M -1 (D)φ 1 (t)] T dx xx (x, t)(λ T 3,xx (x, t) -[CM(x)M -1 (D)φ 1 (t)] T)dx s inequality twice, where ξ > 0 and ς > 0 are arbitrary scalars. Applying Wirtinger's inequality (1a) to the second term on the right side of (C.3) yields:

Table 1

 1 Adaptive observer.

	State observer:

Appendix A. Existence analysis

The well-posedness analysis of the system (2)-( 3) can be performed in many ways. One simple way is to introduce the following backstepping transformation, for (x, t) ∈ [0, 1] × [0, +∞):

Then, it is readily checked using (A.1) and Appendix B (Parts 1 and 2) that, the system (2)-( 3) can be rewritten in the coordinates


Mickael Hilairet
email: mickael.hilairet@univ-fcomte.fr

Olivier Bethoux
email: olivier.bethoux@lgep.supelec.fr

Member, IEEE Malek Ghanes
email: malek.ghanes@ensea.fr

Member, IEEE Valentin Tanasa
email: valentin.tanasa@cti.pub.ro

Jean-Pierre Barbot
email: barbot@ensea.fr

Dorothée Normand-Cyrot
email: dorothee.normand-cyrot@lss.supelec.fr

Member, IEEE Mickaël Hilairet

Member, IEEE Olivier Béthoux

Senior Member, IEEE Jean-Pierre Barbot

Fellow, IEEE Marie-Dorothée Normand-Cyrot

M Hilairet

J.-P Barbot

M.-D Normand-Cyrot

Experimental validation of a sampled-data passivity-based controller for coordination of converters in a fuel cell system

Keywords: interconnection management, sampled-data controller, supercapacitors

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

standards, which are becoming more stringent. To develop environmentally friendly vehicles compatible with the current requirements concerning automotive sector, the combination of hydrogen tank and fuel cell (FC) is a promising path. It ensures no local emission, the traction power, and a high cruising range. To improve this solution energy efficiency, a bidirectional storage device has to be added: it enables to recover braking energy and assist the FC during rapid demand power changes, which significantly increases the FC lifetime [START_REF] Caux | Modelling and control of a fuel cell system and storage elements in transport applications[END_REF]- [START_REF] Motapon | A comparative study of energy management schemes for a fuel-cell hybrid emergency power system of more-electric aircraft[END_REF]. For this function, supercapacitors (SCs) are good candidates since they have a high power density and are able to produce a large number of charge/discharge cycles.

Many power structures can combine FC and SC power sources, as mentioned in [START_REF] Azib | An innovative control strategy of a single converter for hybrid fuel cell/supercapacitors power source[END_REF]. The one based on the doubleconverter structure [START_REF] Thounthong | Fuel cell highpower applications[END_REF] enables both energy management and bus voltage regulation, which is particularly important for some applications such as electric vehicles. This paper is based on this structure. The power demand can be supplied by both sources. The SCs can harvest the load regenerative power and the FC transient excess power delivery. Indeed, for life span reasons, FC has to avoid idle functioning or sharp power changes.

Power management of this system relies on frequency decoupling. The controller has to meet three main objectives: first, respect the FC dynamic by limiting FC power slope regardless of load demand; second, comply with the load demand; and third, control the SC charge. Among the many techniques that are available for this application (see, for instance, [START_REF] Hilairet | A passivity-based controller for coordination of converters in a fuel cell system[END_REF], for a short bibliography), passivity-based controllers are attractive because they guarantee stability properties [START_REF] Ortega | Interconnection and damping assignment passivity-based control: A survey[END_REF]. The practical implementation is actually based on a microcontroller, and hence depends on signals sampling and zero-order hold (ZOH) devices. It consequently involves discrete theory. In order to best approximate a pure continuous controller (which is called emulation process in a sampled-data context), the implementation may use either very small sampling period or a set of low-value gains [START_REF] Hilairet | A passivity-based controller for coordination of converters in a fuel cell system[END_REF], [START_REF] Monaco | Sampled-data stabilization; a PBC approach[END_REF]. Both cases do not offer optimal solutions. The first case requires a microcontroller with high performances (fast analog-to-digital conversion and a high clock frequency of the control unit), whereas the second leads to poor hybrid system performance with respect to selected criteria. The aim of this paper is to prove that a pure digital passivity-based controller can fulfill the hybrid source requirement, although the imposed sampling frequency of the control process is quite low.

Motivated by a recent theoretical result in [START_REF] Tiefensee | IDA-PBC under sampling for port-controlled Hamiltonian systems[END_REF], this study builds on a previous theoretical study, which demonstrates that a direct sampled-data controller based on the interconnection and damping assignment passivity [interconnection and damping assignment passivity-based control (IDA-PBC)] theory can successfully achieve stabilization without any overcurrent [START_REF] Tiefensee | Sampleddata energetic management of a fuel cell/supercapacitor system[END_REF] under large sampling time.

The main motivation of this work is the design of a controller that:

1) ensures the stability of the whole system; 2) generalizes regular linear controller such as PI controller (see, for instance, [19, eqs. (12) and ( 13)], which could be useful); 3) integrates the knowledge of the system to reproduce regular controllers; 4) allows large sampling time without degrading the performances. Items 1-3 are detailed in [START_REF] Hilairet | A passivity-based controller for coordination of converters in a fuel cell system[END_REF] for the same power architecture based on the double-converter structure, whereas item 4 is addressed in this paper. As a result, the four items have been followed and carried out leading to a complete procedure for designing a controller enabling the real-time coordination of many converters. The latter has been implemented on a test bench with two sources and a load and compared with the analog passive approach of [START_REF] Hilairet | A passivity-based controller for coordination of converters in a fuel cell system[END_REF].

This paper is organized as follows. In Section II, the modeling of hybrid source and the background of continuous IDA-PBC theory are recalled (see [START_REF] Hilairet | A passivity-based controller for coordination of converters in a fuel cell system[END_REF] for more details). Section III details the proposed sampled-data passivity-based controller. Section IV shows and discusses the actual experimental results achieved on a reduced-scale test bench based on a 1.2-kW Nexa Ballard FC and compared it with the analog passive approach of [START_REF] Hilairet | A passivity-based controller for coordination of converters in a fuel cell system[END_REF]. Section V provides concluding remarks.

II. SYSTEM ARCHITECTURE

A. Problem Settlement

The hybrid source consists of a low-transient unidirectional power flow source (FC) and a fast-transient bidirectional power flow source (SC) interconnected by a two-converter structure proposed in [START_REF] Thounthong | Fuel cell highpower applications[END_REF], [START_REF] Hilairet | A passivity-based controller for coordination of converters in a fuel cell system[END_REF], and [START_REF] Tiefensee | Sampleddata energetic management of a fuel cell/supercapacitor system[END_REF] and represented in Fig. 1. Such architecture permits the control scheme to properly manage the different powers. The control of the SCs converter has to regulate the dc bus voltage at a constant value equal to its reference v * b , whereas the control of the FC converter has to drive the SCs state of charge toward its constant target value. To work properly, the control scheme is divided in three subblocks.

• A digital nonlinear controller (outer loop) controls the dc bus voltage and the charge of SCs by computing current references i * fc and i * sc values. Saturations ensure that their values do not exceed [0; i fc max ] and [-i sc max ;+i sc max ], which are defined in order to protect FC, SCs, and converters.

• Two fast inner proportional-integral (PI) current loops featuring antiwindup compensation schemes ensure that whatever load requirements, i fc and i sc go rapidly to their reference values i * fc and i * sc , respectively, which are imposed by the outer loop, which is the digital IDA-PBC controller in this study. The outputs of both PI controllers define the duty cycles α fc and α sc of the two converters.

The complete "fuel cell-supercapacitors" system with the associated control scheme is represented in Fig. 1. The whole system is described by a fifth-order nonlinear state-space model called singular perturbed system [START_REF] Kokotovic | Singular Perturbation Methods in Control: Analysis and Design[END_REF], because of the timescale difference between the voltages and the currents. Consequently, assuming that currents i fc and i sc are instantaneous compared with voltages v b and v sc , the fifth-order differential system degenerates to a third-order differential system as follows [START_REF] Hilairet | A passivity-based controller for coordination of converters in a fuel cell system[END_REF]:

d dt v b (t)= 1 C v fc (t) v b (t) i * fc (t)+ v sc (t) v b (t) i * sc (t) -i l (t) d dt v sc (t)= - i * sc (t) C sc d dt i l (t)= -R l (t)i l (t)+v b (t) L (1) with x r (t)=[x 1 (t); x 2 (t); x 3 (t)] T =[v b (t); v sc (t); i l (t)] T , con- trol reference inputs u r (t)=[i * fc (t); i * sc (t)]
T , and measures y r (t) and z r (t), where y r

(t)=[v b (t); v sc (t); i l (t)] T , and z r (t)= [i fc (t); i sc (t); v fc (t)] T .
Remark 1: The charge has been considered as a R l L load in order to represent the current as a time-varying profile without abrupt change by a proper value of the inductance L.

B. IDA-PBC Outer-Loop Controller Design

The IDA-PBC needs to handle the state x r (t) to the reference equilibrium point

x * r (t)=[v * b (t); v * sc (t); (v * b (t)/R l (t))], with v * b (t) and v * sc (t)
being the dc bus and SCs reference voltages, respectively.

A storage energy function H d is defined as

H d (x r (t)) = (1/2)x r (t) T Qx r (t) with xr (t)=x r (t) -x * r (t) and Q = diag(C, C sc ,L),
so that this function is equal to zero at the equilibrium point x * r (t). In this context, the dynamical error equation is as follows:

ẋr (t)=[J-R ]∇H d (x r (t)) + g (x r (t)) u r (t)+d(t) (2) y r (t)=g T (x r (t)) ∇H d (x r (t)) (3)
with

J-R= ⎡ ⎣ 00 -1 LC 00 0 1 LC 0 -R l L 2 ⎤ ⎦ ∇H d (x r (t)) = [ C ṽb (t) C sc ṽsc (t) L ĩl (t)] t g (x r (t)) = ⎡ ⎣ v fc (t) Cv b (t) v sc (t) Cv b (t) 0 -1 C sc 00 ⎤ ⎦ d(t)= ⎡ ⎣ - i * l (t) C 0 0 ⎤ ⎦ .
An algebraic equation needs to be solved in order to obtain the controller, with the constraint of skew symmetry and positive semidefiniteness of J d and R d (x r (t)), respectively [START_REF] Ortega | Interconnection and damping assignment passivity-based control: A survey[END_REF], i.e.,

J d = ⎡ ⎣ 0 J 12 J 13 -J 12 0 J 23 -J 13 -J 23 0 ⎤ ⎦ (4) R d (x r (t)) = ⎡ ⎣ r 1 00 0 r 2 0 00r 3 ⎤ ⎦ (5)
leading to some solutions [START_REF] Hilairet | A passivity-based controller for coordination of converters in a fuel cell system[END_REF]. It follows that setting up

r 1 = x 2 α/x 1 C 2 , r 2 =0, r 3 = R l /L 2 l , J 12 = -(α/CC sc ), J 13 = -(1/
CL), and J 23 =0 with α>0 leads to the nonlinear control law (see [START_REF] Hilairet | A passivity-based controller for coordination of converters in a fuel cell system[END_REF] for more design details) that matches regular controllers [START_REF] Azib | Structure and control strategy for a parallel hybrid fuel cell/supercapacitors power source[END_REF] based on PI controller, i.e.,

i * fc = v b max{v fc ,v fc min } v * b R l -αṽ sc ,α > 0 (6) i * sc = -αṽ b . (7)
Finally, the closed-loop system error is as follows:

ẋr (t)=[J d -R d (x r (t))] ∇H d (x r (t)) (8)
with

J d = ⎡ ⎣ 0 -α CC sc -1 LC α CC sc 00 1 LC 00 ⎤ ⎦ R d (x r (t)) = ⎡ ⎣ αv sc C 2 v b 00 00 0 00 R l L 2 ⎤ ⎦ .
Remark 2: In [START_REF] Hilairet | A passivity-based controller for coordination of converters in a fuel cell system[END_REF], it has been shown that the outer closedloop system is globally asymptotically stable. Moreover, according to Tikhonov's theorem [START_REF] Tikhonov | Ordinary Differential Equations[END_REF], the whole system is locally asymptotically stable (see [START_REF] Hilairet | A passivity-based controller for coordination of converters in a fuel cell system[END_REF] for more details).

C. Practical Issues and Implementation

Equation [START_REF] Zhu | Design of the dynamic power compensation for PEM FC distributed power system[END_REF] shows that FC current reference u r1 = i * fc is highly dependent on the load conductance. Consequently, the IDA-PBC would guarantee the dc bus voltage control by an estimation of the charge R l . The proposed solution consists in the implementation of the following algorithm:

Ŷl = K Rl s + K Rl i l v b (9) i * fc (t)= v b (t) v * b (t) Ŷl (t) -αṽ sc (t) max {v fc (t),v fc min } (10) i * sc (t)= -αṽ b (t) (11)
where an estimate of the conductance Y l =1/R l is implemented in order to avoid a division that is more computationally expensive than a multiplication. Gain K Rl enables to adjust the sensitivity of control value u r1 (t)=i * fc (t) to R l . Choosing a small parameter value ensures slow i * fc (t) variations even in the case of large R l (t) changes. In such a situation, strong variations of i sc (t) ensure a good control of the bus voltage.

Remark 3: It could be noticed that the proposed controller needs an increase of sensors in order to measure the load current i l compared with regular controller proposed in the literature. Nevertheless, in industrial applications, such current measurement could be useful to protect the equipment and ensure its reliability.

Remark 4: Stability analysis with a load estimator. The outer-loop voltage control composed of the third-order system (1), the load estimator, and the IDA-PBC is globally asymptotically stable, invoking a theorem on stability of cascaded systems (see [START_REF] Hilairet | A passivity-based controller for coordination of converters in a fuel cell system[END_REF] for more details).

Remark 5: The two sources have current and voltage limits. Consequently, saturation is included in the algorithm, as in [START_REF] Azib | Structure and control strategy for a parallel hybrid fuel cell/supercapacitors power source[END_REF].

Remark 6: The tuning of nonlinear controllers such as IDA-PBC is not obvious and trivial. To analyze the influence of the tuning parameters on the closed-loop system, more specifically on the FC current dynamics, some simulations have been carried out. After trials and errors, a reasonable choice for the two adjusting parameters α and K r is (10,0.5). It ensures that the dynamic of the FC current is less than 4 A/s so that no significant impact on the ageing FC could be noticed [START_REF] Wahdame | Comparison between two PEM fuel cell durability tests performed at constant current under solicitations linked to transport mission profile[END_REF]. It follows that the closed-loop bandwidths of the dc bus and SCs have been roughly evaluated nearly 1.6 and 0.25 Hz, respectively, ensuring the frequency decoupling of the two sources.

Specifically, the controller [see ( 9)-( 11)] is implemented with a microcontroller, which involves sampled-data inputs v b [k] and v fc [k], and control values i * fc [k] and i * sc [k] remained 3 constant between two consecutive sampling times by means of a ZOH device. The controller is

Ŷl [k]=e -K Rl T s Ŷl [k -1] + (1 -e -K Rl T s ) i l [k] v b [k] (12) i * fc [k]= v b [k] v * b [k] Ŷl [k] -αṽ sc [k] max {v fc [k],v fc min } (13) i * sc [k]= -αṽ b [k] (14)
where T s is the sampling period of the outer loop.

III. SAMPLED-DATA IDA-PBC

A. Introduction

The passivity-based control technique developed in [START_REF] Ortega | Adaptive motion control of rigid robots: A tutorial[END_REF] has been further enhanced with the IDA-PBC methodology (see [START_REF] Ortega | Interconnection and damping assignment passivity based control of port controlled Hamiltonian systems[END_REF] and the references therein). Nevertheless, properties of the continuous controller are usually lost under sampling, leading to a decrement of the controller closed-loop performances. As a result, a direct sampled-data design, permitting the design of digital IDA-PBC in a quite systematic way, is more appropriate [START_REF] Tiefensee | IDA-PBC under sampling for port-controlled Hamiltonian systems[END_REF]. The sampled-data controller is designed in such a way that the energetic behavior of the sampled closed-loop system is equal to that of the continuous one at the sampling instant.

B. Sampled-Data Conception

The analysis of the interconnection and dissipation matrices shows that ṽb and ṽsc are connected by the two inputs (i * fc and i * sc ) and ensure the stabilization of the continuous closed-loop system at the desired equilibrium point x * r with a good transient response. Therefore, the digital controller must preserve this energetic behavior by an appropriate design method.

Let us denote xr (t) and u r (t) the closed-loop continuoustime state and the control input, respectively, as detailed in Section II. The sampled state under control u r [k] is denoted, at the time instant t = kT s ,b yx r [k]. As stated in [START_REF] Tiefensee | IDA-PBC under sampling for port-controlled Hamiltonian systems[END_REF], the main idea is to design a sampled-data controller that reproduces the energetic behavior of the Hamiltonian function at the sample instants in order to guarantee the stability of the equivalent sampled-time closed-loop system.

Given a Hamiltonian function H d : R n =⇒ R, H d (0) = 0, and a continuous input u r (x r (t)) that ensures Ḣd (x r (t)) ≤ 0,it is desired to design a constant piecewise control u r [k] to ensure, at the sampling instants

H d (x r [k +1])-H d (x r [k]) = (k+1)T s kT s Ḣd (x r (τ )) dτ (15) with Ḣd (x r (τ )) = L f (•) H d | xr (τ ),t=τ (16)
where L is the usual Lie derivative, and

f (•)=(J- R)Qx r (t)+g(x r (t)+x * r (t))u r (t)+d(t).
The evolution of H d in ( 16) reads

H d (x r ((k +1)T s )) = e T s £ f (•) H d (•)| xr =x r (kT s ),kT s . (17)
Similarly, the evolution xr (τ ) of the continuous closed-loop system is defined as

xr (τ )=e (τ -kT s )£ f (•) xr | xr =x r (kT s ),kT s (18)
where £ f (•) is the Lie-Backlund derivative, which is defined according to the following recall.

Recall: [START_REF] Fliess | A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems[END_REF] The first Lie-Backlund derivatives of any function Γ(x, t) with respect to any vector field ζ(x, t) is defined as

£ Γ(x,t) ζ(x, t)= ∂ζ(x, t) ∂x Γ+ ∂ζ(x, t) ∂t . ( 19
)
Iteratively, one has

£ j+1 Γ(x,t) ζ(x, t)=£ Γ(x,t) £ j Γ(x,t)) ζ(x, t) . ( 20
)
Now, considering the control u r (t)=u r [k], the equilibrium point x * r (t)=x * r [k], and the perturbation d(t)=d[k] constants on the sampling interval, one obtains in ( 17)

H d (x r [k +1])=e T s L f (•) H d (•)| xr [k],k (21)
where £ is replaced by the usual Lie derivative L (all the derivatives with respect to t are equal to zero), i.e., f (

•)= (J-R )Qx r (t)+g(x r (t)+x * r [k])u r [k]+d[k].
In order to design the enhanced sampled controller, we look for a controller defined as follows:

u r [k]=u r0 [k]+ i≥1 T s (i +1)! u ri [k] (22)
such that ( 16) holds when x r [k]=x r (t)| t=kT s . It follows that, by equating homogeneous terms in powers of T s in the right-and left-hand sides of ( 16), one iteratively computes each "corrective" term u ri of the sampled-data controller, thus obtaining the following first terms:

u r0 [k]=u c (t)| t=kT s (23)
u r1 [k]=u c (t)| t=kT s . (24)
The computation of higher order corrective term (i.e., i ≥ 2) is unnecessary in practice. Therefore, an interesting solution can be proposed at the first order of approximation, i.e.,

u r [k] T =[i * fc [k] i * sc [k]] T with Ŷl [k]=e -K Rl T s Ŷl [k -1] + (1 -e -K Rl T s ) i l [k] v b [k] (25)
i * fc [k]= v b max{v fc ,v fc min } v * b Ŷl -αṽ sc + T s 2! γ(•) t=kT s (26)
i * sc [k]= -αṽ b + T s 2 α C α v sc v b ṽb + αṽ sc + ĩl t=kT s (27)
with

γ(•)=u c = -α 2 v * b ṽb C sc v fc - 1 C sc v fc α v sc v b ṽb + αṽ sc + ĩl × v * b Ŷl + α v sc v b -2αṽ sc + αv * sc t=kT s . ( 28
)
As a result, the proposed sampled-data controller u r

[k] T = [i * fc [k],i * sc [k]]
T computed as in ( 25)-( 28) has better performances compared with the simple emulated controller because it improves to an error in O(T 3 s ) both the energetic performances ensuring convergency to the desired equilibrium (one step matching of the Hamiltonian function) and matching of the state variables evolutions at the sampling instants. These performances are discussed and well illustrated in Section IV through real implementation tests.

However, due to the interplay between the sampling period length and how far one is from the equilibrium, one cannot theoretically prove asymptotic convergency under approximate digital controller. It is reasonable to conjecture that, as usual, in such a context, some practical stability such as condition holds at the sampling instants.

C. Practical Implementation

In practice, the controller leads to a flat variation of the FC current. Hence, the corrective input γ(v b ,v sc ,i l ,Y l )= uc is rather low, thus possible to neglect without compromising on the performances. Conversely, the i * sc corrective term may produce high values. This latter becomes important in case of large sampling period.

Finally, the practical proposed sampled-data controller

u r [k] T =[i * fc [k],i * sc [k]]
T is defined as follows:

Ŷl [k]=e -K Rl T s Ŷl [k -1] + (1 -e -K Rl T s ) i l [k] v b [k] ( 29
) i * fc [k]= v b max{v fc ,v fc min } v * b Ŷl -αṽ sc t=kT s (30) i * sc [k]= -αṽ b + T s 2 α C α v sc v b ṽb + αṽ sc + ĩl t=kT s . (31)

IV. EXPERIMENTAL RESULTS

A. Test Bench Description

The experimental setup (see Table I) is based on a Nexa Ballard FC (46 A/1200 W) associated with two Maxwell SC modules in series (125 F). The FC and SC boost converters are based on metal-oxide-semiconductor field-effect transistor modules switching at 20 kHz. The load is emulated by a programmable electronic equipment from Höcherl & Hackl (model ZS1806, rated power of 1800 W, imax = 150 A, Vmax = 60 V). Finally, the controllers are implemented in a dSPACE DS1103 board where the voltage outer control loops sampling has been set to 20 kHz or 500 Hz. The current closed loops are based on PI controller updated at 20 kHz [START_REF] Azib | An innovative control strategy of a single converter for hybrid fuel cell/supercapacitors power source[END_REF], [START_REF] Hilairet | A passivity-based controller for coordination of converters in a fuel cell system[END_REF].

B. Result Analysis

All the experimental results detailed thereafter have been performed in order to validate the outer-loop control algorithm. Hence, in order to highlight the performances of the new digital controller, a comparison between the emulated controller based on the continuous-time design [implementation of the continuous IDA-PBC with a ZOH device, ( 12)-( 14)] and the sampleddata controller ( 29)-( 31) are detailed for different sampling periods.

Figs. 2 and3 show the responses of the emulated controller for sampling periods equal to 50 µs and 2 ms, respectively, where the dc bus voltage reference is set equal to 50 V, and the load current varies between 0 and 15 A (this is equivalent to a load admittance variation from 0 to 0.3 S).

First, all the abrupt variations of the power load at time 1, 16, 36, 76, or 116 s have been balanced by an abrupt variation of the SC current [see Figs. a slow decrease in FC current i fc and, thereafter, a recharge of the SCs with the excess of energy.

Second, we can note that the SC voltage is not perfectly equal to its reference at steady state. In fact, the controller design supposes that the converters are lossless (no perturbation) and all the parameters are well known. Therefore, in practice, this small steady-state error could be removed by adding a small integral term without deteriorating the stability [START_REF] Hilairet | A passivity-based controller for coordination of converters in a fuel cell system[END_REF], [START_REF] Donaire | On the addition of integral action to port-controlled Hamiltonian systems[END_REF].

Third, the remaining useful life of the FC is ensured by the smooth variation of the current i * fc (adjusted by gain K Rl ) despite rapid power load requests.

Fourth, it can be noticed that the increase in the sampling period amplifies the peak of the SCs' current and the signals' noise.

Figs. 4 and5 show the responses of the sampled-data controller for sampling periods equal to 50 µs and 2 ms, respectively. We can notice that the sampled-data controller prevents overshoots of the SC current and the dc bus voltage, although the sampling frequency is significantly reduced. In conclusion, the performances are maintained with the sampleddata controller" whereas the emulated algorithm greatly reduces them.

Indeed, for a sampling time equal to 2 ms, the emulated controller has an SC peak current i max sc at about 11.85 A at t =1s, as opposed to the sampled-data controller, which only induces a 9.67-A peak. An analog observation can be noticed at t =96s during a load power decrease. The emulated controller requests a i min sc = -25 A, whereas sampled-data controller requests i min sc = -20 A. These observations are illustrated in Fig. 6. This confirms that the proposed controller is actually effective in a sampled-data context. Using the sampled-data controller leads to very satisfactory performances of the equipment under control, and this is increasingly so when the sampling period is high.

V. C ONCLUSION

Motivated by recent theoretical results from sampled passivity-based control, a new controller for coordination of an FC and SCs has been detailed. Practical results show that the performances of the system under control with an emulated controller have been downgraded when the sampling frequency has been reduced. By contrast, in the same context, the sampled IDA-PBC has maintained the requested performance. It follows that such controller is suitable for high-power industrial applications where the switching and sampling frequencies are low.

Fig. 1 .

 1 Fig. 1. Studied system with its associated control scheme.

Fig. 2 .

 2 Fig. 2. Experimental result of the emulated controller-Ts =50 µs. (a) bus voltage v b . (b) SCs voltage vsc. (c) load current i l . (d) fuel cell voltage v fc . (e) fuel cell current i fc . (f) SCs current isc.

  2(f) and 3(f)], ensuring that the dc bus reference voltage is reached [see Figs. 2(a) and 3(a)]. On the contrary, any load power decrease (at t =56or 96 s) leads to

Fig. 3 .

 3 Fig. 3. Experimental result of the emulated controller-Ts =2 ms. (a) bus voltage v b . (b) SCs voltage vsc. (c) load current i l . (d) fuel cell voltage v fc . (e) fuel cell current i fc . (f) SCs current isc.

Fig. 4 .

 4 Fig. 4. Experimental result of the sampled-data controller-Ts = 50 µs. (a) bus voltage v b . (b) SCs voltage vsc. (c) load current i l . (d) fuel cell voltage v fc . (e) fuel cell current i fc . (f) SCs current isc.

Fig. 5 .

 5 Fig. 5. Experimental result of the sampled-data controller-Ts =2ms. (a) bus voltage v b . (b) SCs voltage vsc. (c) load current i l . (d) fuel cell voltage v fc . (e) fuel cell current i fc . (f) SCs current isc.

Fig. 6 .

 6 Fig. 6. Experimental result of the emulated and sampled-data controllers. (a) SCs current isc -Ts = 500 µs. (b) SCs current isc -Ts =2ms.

TABLE I ELECTRIC

 I CHARACTERISTICS OF THE HYBRID SYSTEM