Olivier Hugues
email: o.hugues@net.estia.fr

Philippe Fuchs
email: philippe.fuchs@ensmp.fr

Olivier Nannipieri

New Augmented Reality Taxonomy : Technologies and Features of Augmented Environment

This article has a dual aim: firstly to define augmented reality (AR) environments and secondly, based on our definition, a new taxonomy enabling these environments to be classified. After briefly reviewing existing classifications, we define AR by its purpose, ie. to enable someone to create sensorymotor and cognitive activities in a new space combining the real environment and a virtual environment. Below we present our functional taxonomy of AR environments. We divide these environments into two distinct groups. The first concerns the different functionalities enabling us to discover and understand our environment, an augmented perception of reality. The second corresponds to applications whose aim is to create an artificial environment. Finally, more than a functional difference, we demonstrate that it is possible to consider that both types of AR have a pragmatic purpose. The difference therefore seems to lie in the ability of both types of AR to free themselves or not of location in time and space.

Introduction

It is indisputable that augmented reality (AR) is the result of a series of technological innovations. However, does reducing AR to a set of technical characteristics in behavioural interfacing not lead us to underestimate the wealth of AR and its functionalities? We are therefore entitled to naively ask the following question: what is augmented in augmented reality? If reality is by definition everything that exists, then strictly speaking reality cannot be augmented since it is already everything. So what is augmented? Initially the answer seems obvious: it is not reality, but the perception of reality which is augmented. From this perspective, the question of what is real, existence and properties is not examined here. The issue concerns perception, ie. the phenomenon -reality as it is perceived, not the noumen -reality in itself, to use Kant's words [START_REF] Kant | Kritik der reinen vernunft[END_REF].

Augmented perception?

We can find in literature two main movements which define perception. On the one hand (a passive conception), the sensory system passively receives stimulations and processes this information so as to refer to internal representations. On the other (an active conception), it is the extraction of regularity between actions and stimulations which enable perception. It seems that the current movement is rather to consider perception using the sensorymotor approach (the second case), in opposition to the linear and sequential approach of the perception process, the first case [START_REF] Auvray | Perception, immersion et interaction sensorimotrices en environnement virtuel[END_REF]. Furthermore, this is confirmed by Gibson [3] who considers the senses as full perceptory systems ("perception is to extract, thanks to movements, this information by detecting its invariants"). It is important to emphasise that like Auvray and Fuchs [START_REF] Auvray | Perception, immersion et interaction sensorimotrices en environnement virtuel[END_REF] who used the theoretical framework proposed by Bergson in "Matrer and Memory" [START_REF] Bergson | Matière et mémoire. Essai sur la relation du corps à l'esprit[END_REF], that any perception and any knowledge have only one final aim -whether conscious or not: action. We do not perceive and do not attempt to familiarise ourselves in order to know but rather to act. Inherited from the pragmatic conception of William James [START_REF] James | Pragmatism: A new name for some old ways of thinking[END_REF], the Bergsonian theory insists on the nature itself of our perception and any search for information in the real world: perception is never disinterested, knowledge is only ever a means to acting better in the real world [START_REF] Bergson | Matière et mémoire. Essai sur la relation du corps à l'esprit[END_REF] -and surviving for wild animals or being happier for humans. Although any increase in the quantity of information -and consequently, any increase in our understanding of reality -admitted by AR aims for greater mastery of what is real, it is clear that, from a technological point of view, AR can offer interfaces which propose either, more explicitly, information, or, more explicitly, a better mastery of our actions with regard to real events. But how do technical devices modify our perception? According to [START_REF] Auvray | Perception, immersion et interaction sensorimotrices en environnement virtuel[END_REF], "using a new technical device modifies our sensory-motor relationship with the environment; and subsequently it modifies our perception". Technical tools modify our "perceptory space". The step is decisive. However, we cannot "only" endeavour to perceive better since perception is not a final aim in itself, but rather a means of achieving a target action [START_REF] Bergson | Matière et mémoire. Essai sur la relation du corps à l'esprit[END_REF]. In the manner of virtual reality (VR), augmenting reality may satisfy two objectives for the user: encourage understanding and mastery of the real world and therefore, an augmented perception of reality on the one hand and, on the other hand, propose a new environment whose aim does not appear to be obey either requirements in terms of knowledge or practical requirements.

Final aim and birth of a taxonomy

Globally, the aim of AR is to enable a person to carry out sensory-motor and cognitive activities in a new space by associating the real environment and a virtual environment. Yet, like VR [START_REF] Fuchs | Le Traité de la Réalité Virtuelle[END_REF], AR can propose either a modellisation of the real world based on an environment which imitates or symbolises the real world, or the creation of an artificial environment which does not correspond to anything which exists. It is on the basis of this distinction that it is possible to propose a functional taxonomy of AR and draw a certain number of relative implications, namely, with regard to variations in space and time references allowed by AR.

Taxonomy of Interactives Mixed Systems

There have been many attempts to classify augmented reality environments since this technology is in fact a sub-set of environments or mixed systems.

Whether technical, functional or conceptual, these taxonomies often aim to describe, compare and generate [START_REF] Beaudouin-Lafon | Instrumental interaction: an interaction model for designing post-wimp user interfaces[END_REF]. We present an overview of some of the taxonomies present in literature. Readers will find a more complete review of classifications in [START_REF] Nigay | Espaces conceptuels pour l'interaction multimédia et multimodale[END_REF].

Conceptual taxonomies

In [START_REF] Jacob | Reality-Based Interaction: A Framework for Post-WIMP Interfaces[END_REF], the authors proposed a framework for analysis with which they explore different mixed environments so as to identify the common use of human abilities linked with the physical world, the body and the social environment. However, since the advantage of mixed environments is their ability not to faithfully reproduce reality [START_REF] Dourish | Where the Action is : The Foundations of Embodied Interaction[END_REF], the authors proposed six factors like the power of expression, efficiency, ergonomics and accessibility enabling the use of the physical world to be weighted. This classification enables a large number of possibilities for interaction generated by mixed systems to be highlighted. In [START_REF] Mackay | Augmenting reality: A new paradigm for interacting with computers[END_REF], Mackay proposes a classification of mixed interfaces whose common denominator is the target of augmentations. Different targets are users, objects of interaction and the environment. Dubois [START_REF] Dubois | Classification space for augmented surgery, an augmented reality case study[END_REF] extends this classification by introducing a methodology rating known as ASUR, extended in 2003 [START_REF] Dubois | ASUR++: supporting the design of mobile mixed systems[END_REF] and whose discretisation includes the user, the adaptor4 , the system and real entities. Later, Renevier [START_REF] Renevier | Système Mixtes Collaboratifs sur Supports Mobiles : Conception et Réalisation[END_REF] defined methods of interaction as passive (determined by the system) or active (determined by the user). Creating, accessing, modifying and destroying links depends on the methods of interaction. A link is defined as being short-lived or persistent and can be located with regard to users. Different types of locating are possible. The authors also define three types of interaction: the users interact with objects via links and inversely, and links are used for communication between users.

Technical taxonomies

In [START_REF] Milgram | A taxonomy of mixed reality visual displays[END_REF][START_REF] Milgram | Augmented reality: A class of displays on the reality-virtuality continuum[END_REF], the authors propose a technological classification now well known under the name of "virtuality-reality continuum". They endeavour to discretise and classify the environment according to four categories. This continuum highlights the fact that there is a progressive transition from real to virtual and inversely. The authors of [START_REF] Benford | Expected, sensed, and desired: A framework for designing sensing-based interaction[END_REF] chose to analyse the user's movements in order to extract its framework based on expected movements, captured movements and desired movements. By using Norman's theory of action [START_REF] Norman | Book chapter of User Centered System Design[END_REF], the authors proposes dissociating the systems which augment performance from systems which augment evaluation.

Functional taxonomies

With the aim of clarifying concepts which underlie the combination of the real world and virtual world, Dubois et al. [START_REF] Dubois | Combinons le monde virtuel et le monde réel : Classification et principe de conception[END_REF] propose a classification broken down into two distinct parts: the first characteristic is the purpose of the task and the second is the type of augmentation. Whilst the purpose of the task enables the authors to augment and specify the Milgram continuum [START_REF] Milgram | A taxonomy of mixed reality visual displays[END_REF] by adding to it two continua so as to distinguish augmented reality from virtual reality, the second characteristic enables the existence of two different functional types of augmentations to be underlined. The first consists of an "augmented performance" whilst the second is an "augmented perception". According to the authors, the first type of augmentation enables users to carry out tasks in the real world in a new way like for example with the "Active Badge" [START_REF] Want | The active badge location system[END_REF] whilst the second, much more common concerning AR systems, enables relevant information to be provided for the task in hand.

Proposal for a functional taxonomy for AR

We propose distinguishing several AR functionalities whose organisation is presented in Figure 1.1.

Functionality 1: augmented perception of reality

This first functionality consists of highlighting the fact that AR constitutes a tool for assisting decision-making. It can provide information which will enable a better understanding of reality and which will ultimately optimise our action on reality. Fuchs [START_REF] Fuchs | Le Traité de la Réalité Virtuelle[END_REF] proposes a taxonomy which considers the means by which AR enables this objective to be achieved. Five types of environments can be distinguished. To these five type of environment, we can add another type to which Fuchs [START_REF] Fuchs | Le Traité de la Réalité Virtuelle[END_REF] attributes the functionality 0. This is the case where real images (RI) and virtual entities (VE) are displayed on the same screen, but without any relationship between the two. In this case, the user has a single screen with two independent displays. We then talk about functionality 0 because there is no relation between the content of VE and RI. With the exception of this case of a limited AR environment, which is possible but not relevant, other environments are possible and more relevant. Specifically, the augmented perception of reality functionality can be divided into five sub-functionalities.

Sub-functionality 1: Documented reality and documented virtuality

This is the minimum functionality of augmented reality: RI and VE are in two different display boxes, but their information is related. Augmentation thus consists of informing users, like for example, without the mediation of a technical device, an assembly manual for kit furniture. The aim of the second display box (mostly text) is to help users understand and guide their action. This is documented reality. Inversely, in some specific cases, for example, the synoptic of an industrial process (VE) with the "real time" incorporation of one or several windows displaying real parts of the process (RI) , the "document" no longer consists of a virtual environment, but of images of the real object. In this particular case, we can talk about documented virtuality. In both these cases it involves improving the understanding of the real or virtual scene by adding passive semantic information provided on another display support.

Sub-functionality 2: Reality with augmented perception or understanding

In this functionality, RI ad VE are in the same display box. Two levels can be distinguished according to the contribution of augmentation.

First level: Reality with augmented understanding.

This involves augmenting the understanding of images from the real scene by incrusting passive semantic information (Figure 1.2). VE (titles, keys, symbols, etc.), more or less visually close to real objects, providing complementary information on the latter (eg. functions, references). Inversing RI and VE in this functionality cannot be envisaged.

(a) Peak.AR [START_REF] Research | Peak.ar[END_REF]. (b) ARMAR [START_REF] Henderson | Augmented reality for maintenance and repair (armar)[END_REF][START_REF] Henderson | Evaluating the benefits of augmented reality for task localization in maintenance of an armored personnel carrier turret[END_REF]. Second level: Reality with augmented visibility. This is the augmented visibility of images from real scenes (if we limit ourselves to visual perception). VE (eg. "iron wire" model of real objects) geometrically match the contours of real objects (Figure 1.3). They enable objects to be highlighted so as to see them better. This involves, amongst other things, calibrating cameras. It is then possible either to improve the clarity of images by highlighting the apparent contours of objects so as to perceive them more easily or to improve the understanding of objects by virtually representing their visible and invisible contours.

Sub-functionality 3: Perceptual association of the real and virtual

In this functionality new virtual objects are added to the real scene. We can distinguish between both cases according to the level of integration of virtual objects in the real scene.

First level: Incrustation of virtual objects on real images.

Virtual objects are incrusted (overlay) on top of real objects. Therefore virtual objects are not concealed by real objects. We can refer to this as association by superposition (Figure 1.4a).

Second level: Integration of virtual objects on real images.

Virtual objects are integrated with real objects. The latter conceal virtual objects which are positioned behind them. This is a 3D association. However integration with real time occlusion remains a major technological hurdle (Figure 1.4b). In addition to the conditions of the previous functionality, we semantically modellise virtual objects by taking into account their physical properties according to the laws of gravity, contact, elasticity, fluidity, etc. so as to enrich the scene. Prior knowledge is used in the real scene and its objects. This functionality defines geometrical and physical interactions between real and virtual objects. For example, we can mention the behaviour and attraction functions of virtual objects with real objects.

1.5.1.5 Sub-functionality 5: substituting the real by the virtual or virtualised reality

If we know the geometrical modellisation of the real scene observed, we can replace the video image display for the real scene by the synthetic image of the model, determined from the same point of view. By going from the video image to the synthetic image, and vice versa, the understanding for the environment is augmented. In this case we can talk about virtualised reality. We can then also change the point of view for the synthetic image, without moving the camera filming the real scene, which enables a better understanding of the scene. For example, we can mention teleoperation work carried out at the DER by EDF (Figure 1.5). In their visualisation device, the video image display for the real scene is not replaced by the model's synthetic image, but they are displayed on two neighbouring screens. The robot, which the operator is handling, is simultaneously displayed in real 3D images and synthetic 3D images. Whatever the point of view of real cameras, the operator can always observe on a screen in virtualised reality the mechanical configuration of the robot with "real time" animated synthetic images.

Fig. 1.5: The teleoperated robot is displayed simultaneously in real 3D images and using 3D artificial images.

Functionality 2: create an artificial environment

In the first functionality, AR enables objects, beings or relations which exist in reality but which cannot be perceived by users to be visualised, eg. underground stations [START_REF] Presselite | Metro paris[END_REF] exist in reality, but are not always easily identifiable. An iPhone R application shows (or highlights) what exists in reality but cannot be seen. It is clear that with regard to the Bergsonian theory of perception, all our activities, even the most speculative (eg. thinking, modellising) concern actions [START_REF] Bergson | Matière et mémoire. Essai sur la relation du corps à l'esprit[END_REF]. However, humans have an ability which seems to partly escape this law: imagination. We can reasonably acknowledge that the production of mental images may not satisfy a practical objective, although what we imagine may or not have voluntary effects on our perception and actions in the real world. The production of unconscious mental images (ie. dreams) and conscious mental images (ie. fantasies, defined as the production of mental images during the awakening phase, eg. when we imagine that we are in other places) does not seem to serve any practical function. When we form mental images of an imaginary environment it is not in order to act in the real world, but perhaps, on the contrary, an attempt to escape reality. However, AR is able to produce such environments which are no longer subject to practical requirements. In such cases, AR offers environments which present not reality as they are perceived -by augmenting the amount of information which we are unable to perceive, but which, however, is present in reality -but reality, as it could be in the future or even an impossible reality.

1.5.2.1 Sub-functionality 1: imagine reality as it could be in the future by associating the real and the virtual Overall this is not about being informed, understanding or even acting with regard to what really exists, but digitally creating a possible world which combines either a real environment with virtual images (level 1) or a virtual environment with real images (level 2). Each level can be broken down into two sub-levels according to the type of association. This type of functionality can use two types associations: with occlusion versus without occlusion.

First level: imagine a possible future environment by incrusting virtual objects on/in real images.

First sub-level: no occlusion.

In this functionality, new virtual objects are added to the real scene by superposition and are not concealed, not to better understand and/or act in reality, but with a disinterested aim, for example, strictly aesthetic.

Second sub-level: with occlusion. This involves adding virtual objects which conceal part of the real environment. Obviously, this is more efficient at the sub-level than the real/virtual association because it enables users to visualise what the possible future environment looks like if these virtual objects incrusted in the real environment really existed. Thus, an interface enabling one of the rooms of a real apartment to be visualised (eg. the user's) by adding virtual furniture [START_REF] Wohnung | Click & Design[END_REF]would not enable them to know or master reality any better, but to imagine what the future reality would be like.

Fig. 1.6: Example of a possible future environment by incrusting virtual elements [START_REF] Wohnung | Click & Design[END_REF].

Second level: imagine a possible future environment by incrustrating real objects in/on a virtual environment.

First sub-level: without occlusion.

In this functionality, we added by superposition, without occlusion, new real objects to the virtual environment. Here again, for example, to satisfy an aesthetic objective, it would involve seeing in what type of room (virtual), the real object would be best suited, by revealing all its aesthetic qualities. We could, for example, virtually move the real image of a sculpture in virtual rooms of a future exhibition venue so as to estimate the optimal location (according to constraints such as light, access for visitors, etc.).

Second sub-level: with occlusion.

Here again, the this second sub-level takes on its full meaning with the integration of real images in a virtual environment: the designer of the future exhibition venue may visualise what visitors really see if they visited this venue with such a configuration.

1.5.2.2 Sub-functionality 2: Imagine reality as it was by associating the real and virtual.

Likewise, it is possible, either to associate virtual objects which no longer exist with a real environment (level 1) or associate objects which still exist but which were present in an environment which no longer exists and which is, consequently, digitally created (level 2). And in each of these cases, this association can be done without (sub-level 1) or with occlusion (sub-level 2).

First level: imagine a past environment by incrusting virtual objects on/in real images.

First sub-level: with occlusion.

With this objective in mind, the objective is to visualise what current scenes looked like in the past without incrusting virtual objects belonging to the past and concealing the current environment.

Second sub-level: with occlusion.

Partial occlusion of the real present environment by virtual objects enables the past to be imagined as faithfully as possible. AR therefore plays the role of a "virtual memory". For example, the mixed environment which proposes adding, with occlusion, to the current Cluny Abbey building architectural items which have since disappeared enables the abbey to exist as it did then [START_REF] Ensam -On-Situ | Dispositif de réalité augmentée pour l'abbaye de cluny : Ray-on[END_REF]. We can thus see Cluny Abbey as it was when it was built in the 10th century (Figure 1.7). Fig. 1.7: Present Cluny Abbey augmented with items belonging to its past [START_REF] Ensam -On-Situ | Dispositif de réalité augmentée pour l'abbaye de cluny : Ray-on[END_REF].

Second level: imagine a past environment by incrusting real objects in/on a virtual environment.

First sub-level: without occlusion.

This functionality is clearly very limited: it is certainly possible to put in its past context the real image, for example, of a column in the city of Ephesus in tact in a virtual environment by digitally creating the city of Ephesus (today in ruins) without occlusion. Yet, the advantage of such as association is almost non-existent.

Second sub-level: with occlusion.

On the other hand, incrusting with occlusion the real image of the column of Ephesus in a virtual environment representing the city of Ephesus before it was subject to the ravages of time enables us to much more easily imagine the place and function of this column at the time when it was built on the site of Ephesus.

Sub-functionality 3: imagine an impossible reality

This functionality makes better use of the potential of AR with regard to its possible distance from reality by taking advantage of the imaginary dimension of mixed environments. Here the objective is not to inform, or encourage understanding or optimise users' actions. This functionality emphasises the potentially disinterested nature of mixed environments: it is possible to create an environment which cannot really exist, eg. artistic creations or systems whose final aim is purely aesthetic. Designers therefore have greater relative freedom and are not limited to the types of environments which they can use: they can go beyond the possibilities that this environment really offers and the means used may be very varied. Concretely, it is possible to use all types of environments in the augmented reality perception functionality. Without systematically and exhaustively listing all the types of artificial mixed environments, it is possible to give a few examples. The principle for all possible sub-functionalities consists of diverting the primary function of AR, ie. augmented perception of reality. Creating an artificial AR environment can for example be envisaged thanks to a semantic gap produced by the incoherence of the overall meaning of the mixed environment. Diverting sub-functionality n • 1 (documented reality and documented virtuality) of the first functionality (augmented perception of reality) involves associating, for example, with a real environment a virtual "document" providing information out of step with this environment with the aim of provoking in users a modification of the meaning of reality as it is perceived by users due to the difference between the real image and virtual information. We could for example associate with a real environment a virtual "document" erroneously describing this environment. Synthesising these mutually incompatible meanings would provoke in users the representation of an impossible mixed artificial environment. We could create an environment where a real image is augmented with erroneous meanings (ie. "documents"), ie. incoherent with the real image, but this would be a minimum functionality. The potential of AR is best exploited in environments which for example propose diverting the primary function of integrating virtual objects in a real scene (second level of sub-functionality 3: perceptual association of the real and the virtual ). This is the case of the AR game developed by Dassault Systems for Nestlé [START_REF] Nestle | Minimoys[END_REF] (Figure 1.8a), where a virtual character moves, partially concealing the real image, on a cereal box. Although this environment obeys objectives defined by a marketing strategy (eg. increasing traffic to Chocapic R 's website and creating a strong emotional link between the brand and its target, in this case young children), these objectives are merely an indirect consequence of the experience of young users. Basically, this environment creates an impossible artificial universe where a Minimoys R character runs across a cereal box. The aim is neither about better understanding reality nor more effectively mastering it. Although the final objective is interested (ie. making young consumers loyal), the system in itself could well have not obeyed any practical objective. It suffices to see how adults (who are not the target of this PR operation) adapt this game.

In other words, what this type of environment attempts to create is a feeling of presence in an artificial world. And what at the end of the day this world contributes to strengthening the links between players and the brand of cereals is not fundamental, even though the game was created with this objective in mind. It is possible to distance oneself even further from the practical constraints when for example proposing an impossible mixed artificial environment such as the view of a US city "through the windows" of an apartment located in France [START_REF] Rationalcraft | Winscape[END_REF] (Figure 1.8b). An application which enables users to enjoy the view of a suspension bridge in the US from the window of their Parisian apartment has no practical, only an aesthetic use. With Wiimote R placed on the window ledge between two windows and an infrared sensor on the user's head, when moving around the room, the corner view of windows changes so as to give the illusion that the apartment is located in the US.

(a) Chocapic and Nestlé [START_REF] Nestle | Minimoys[END_REF].

(b) "Virtual windows" [START_REF] Rationalcraft | Winscape[END_REF].

Discussion

The distinction between AR environments whose function is to augment the perception of reality and those whose function is to immerse users in an artificial environment has a double implication. The first concerns the fact that the taxonomy cannot be designed as a classification with a merely descriptive vocation. Taxonomy has a generative dimension [START_REF] Beaudouin-Lafon | Instrumental interaction: an interaction model for designing post-wimp user interfaces[END_REF]. Concretely, more than just a simple classification tool of what already exists, it enables environments which it is possible to design to emerge. In this respect, taxonomy is a tool for assisting the creation of virtual and augmented reality environments.

This reflection leads to the second major implication of analysing what has been done here. The difference between the two types of AR goes further than a simple distinction in terms of functionality: if the first type of AR (ie. functionality 1: augmented perception of reality) is a prisoner of the present, the second type (ie. functionality 2: imagine) goes far beyond it. In short, AR enables us, admittedly, to see, understand and master the present better, and it also enables us to propose an environment with which the designer and, consequently, users can play with location in time. When I see, thanks to an AR interface today the ruins of Cluny Abbey augmented with parts of this abbey which have disappeared [START_REF] Ensam -On-Situ | Dispositif de réalité augmentée pour l'abbaye de cluny : Ray-on[END_REF], I see an environment which is neither pure present nor pure past, but a mixture of both the past and present. Likewise, when I see my currently empty living-room augmented with its future furniture [START_REF] Wohnung | Click & Design[END_REF], I am neither in the present nor the future, but in a reality where both present and future are mixed. And when "hrough" the windows of my Parisian apartment I see a US city [START_REF] Rationalcraft | Winscape[END_REF], I perceive a world which is simply not possible. This is why AR enables us, not only, to free ourselves from reality's time categories (ie. past vs present vs future), but also from the spatial unity which characterises reality, which, for example, means I cannot see a US city by looking through the window of my apartment. When, for example, I am looking for the closest underground station thanks to an application available on my mobile [START_REF] Presselite | Metro paris[END_REF], this enables me to go there here and now: the first functionality of AR (ie. AR with an informative and practical aim) does not modify the space perceived, it only aims to unveil parts of that space -in this case underground stations -which are not perceived, but which may rightly be perceived since they do exist. On the other hand, whether it is Cluny Abbey [START_REF] Ensam -On-Situ | Dispositif de réalité augmentée pour l'abbaye de cluny : Ray-on[END_REF] or a furnished room [START_REF] Wohnung | Click & Design[END_REF], the items added modify the structure of current real spatial relations. In short, from the spatial point of view, the first type of AR does not modify anything, it only reveals what already exists: it updates the spatial structure of reality, whether this structure can be directly perceived by the senses (eg. Paris underground stations), whether it is the expression of causal relations scientifically shown based on calculations (eg. an environment which shows the existing electrical voltage between two items on an electric meter) or whether it can only be apprehended via an artificial technique (eg. an AR environment which would make something visible thanks to a thermal camera for light waves whose frequency escapes human sight). On the contrary, the second type of AR modifies the spatial structure by adding objects, beings or relations which do not belong to it: it modifies the spatial configuration of reality.

Conclusion

To summarise, strictly speaking, there are two major differences between the two types of AR: it is functional in so far as both AR do not satisfy the same objectives -one is practical, the other imaginary. However, this distinction is not fundamental because it can be argued that, just like the interface enables me to locate underground stations on my mobile [START_REF] Presselite | Metro paris[END_REF], the aim of Cluny Abbey in mixed reality [START_REF] Ensam -On-Situ | Dispositif de réalité augmentée pour l'abbaye de cluny : Ray-on[END_REF] is to at least provide knowledge (ie. knowing what this abbey looked like when it was built) and even my virtually furnished has a practical function (ie. enabling me to buy furniture or not according to the layout of the room). The basic distinction between the two AR therefore seems to correspond to the ability of both to be go beyond location in space and time or not.

Fig. 1 . 1 :

 11 Fig. 1.1: Functional classification of AR. (RI : image with "real" content, VE : virtual entity)

Fig. 1 . 2 :

 12 Fig. 1.2: Example of applying the augmented understanding of reality.

  (a) Karma[START_REF] Feiner | Knowledge-based augmented reality[END_REF].(b) Visualisation of a mechanical part[START_REF] Fuchs | Le Traité de la Réalité Virtuelle[END_REF].

Fig. 1 . 3 :

 13 Fig. 1.3: Reality with augmented visibility with objects' visible and invisible contours represented.

  (a) Incrustation of objects. (b) Integration of objects.

Fig. 1 . 4 :

 14 Fig. 1.4: Examples of perceptual associations between the real and virtual (extract from[START_REF] Gibson | Photorealistic Augmented Reality[END_REF]).

( a )

 a Present environment. (b) Both past and present environment.

Fig. 1 . 8 :

 18 Fig. 1.8: Examples of artificial AR environments.

  Whilst the first type of AR presents us reality (what I can see, what I can do), the second type, on the contrary, presents us what is imaginary: what I cannot really see -since it is not actually real -but which I can, paradoxically, see thanks to an AR interface. In short, the second type of AR enables us to go from what is put into action (ie. current) to what is not: the imaginary. And this imaginary may be possible or not. Finally, AR makes the impossible possible.

Adaptors transfer data from one world to the other and may be the system's input or output.

New AR Taxonomy : Technologies and Features of Augmented Environment


A Guiro

A Iggidr
email: iggidr@loria.fr

D Ngom

H Touré

On the stock estimation for some fishery systems

Keywords: Fishery models, Stage-structured population models, Estimation, Harvested Fish Population, Observers

In this work we address the stock estimation problem for two fishery models. We show that a tool from nonlinear control theory called "observer" can be helpful to deal with the resource stock estimation in the field of renewable resource management. It is often difficult or expensive to measure all the state variables characterising the evolution of a given population system, therefore the question arises whether from the observation of certain indicators of the considered system, the whole state of the population system can be recovered or at least estimated. The goal of this paper is to show how some techniques of control theory can be applied for the approximate estimation of the unmeasurable state variables using only the observed data together with the dynamical model describing the evolution of the system. More precisely we shall consider two fishery models and we shall show how to built for each model an auxiliary dynamical system (the observer) that uses the available data (the total of caught fish) and which produces a dynamical estimation x(t) of the unmeasurable stock state x(t). Moreover the convergence speed of x(t) towards x(t) can be chosen.

1 Introduction and a short survey of observers design

The stock estimation is one of the most important problem in fishery science. One can quote J.A. Gulland [START_REF] Gulland | Fish Stock Assessment, a manual of basic methods[END_REF]: A major emphasis in fishery science has been on the problems of estimating current and past level using catch levels and fishing effort data.

To make a policy decision about the exploitation of renewable ressources, it is necessary to take into account the state of the resource stocks. This implies the need of a good estimate of the available resource. Mathematical models are more and more used to describe the evolution of biological systems. Here, we consider two mathematical models for fishery resources. The first one is a "stage structured" model [43,[START_REF] Touzeau | On the stock-recruitment relationships in fish population models[END_REF] that describes the dynamics of a population divided in stage-classes (according to age, length or weight) and submitted to the fishing action. The second model is a "global" model that describes the evolution of a fish population that can move between an area where it can be harvested and a reserve area where no fishing is allowed [START_REF] Dubey | A model for fishery resource with reserve area. Nonlinear Anal[END_REF]. Both models are given by systems of differential equations of the form

ẋ = f (x, E), (1)
where E is the fishing effort (it can be seen as a control or an input) and x(t) is the state of the system at time t. The state variable x(t) represents the density of the population or the number of individuals by stage. For both models, the state x(t) is not available for measurement. In practice, the only available information at time t is the value of the captures: this means that one can measure the total catch at each time t. The value of the captures can be seen as the measurable output of system [START_REF] Alcaraz-Gonzalez | A tunable multivariable nonlinear robust observer for biological systems[END_REF]. The output is in general a function of the state variable and the input, that is, y(t) = h(x(t), E).

Now assuming that (1) is a "good" model of the system under consideration, if it is possible to have the value of the state at some time t 0 then it is possible to compute x(t) for all t ≥ t 0 by integrating the differential equation with the initial condition

x(t 0 ). Unfortunately, it is often not possible to measure the whole state at a given time and therefore it is not possible to integrate the differential equation because one does not know an initial condition. One can only have a partial information of the state and this partial information is precisely given by y(t) the output of the system. Therefore we shall show how to use this partial information y(t) together with the given model in order to have a dynamical estimate x(t) of the real unknown state variable x(t). This estimate will be produced by an auxiliary dynamical system which uses the information y(t) provided by the system (1). This dynamical system is generally of the form ẋ = g(x, E, y).

(

) 2
It can be represented by Schema 1 The estimate error is given by e(t) = x(t) -x(t) and it satisfies the following "error equation" The function g has to be determined in such a way that the solutions of ( 1) and ( 2) satisfy x(t) -x(t) → 0 as t → +∞ regardless of the respective initial conditions of system (1) and system (2).

ė = g(x, E, y) -f (x, E) (3)
A dynamical system (2) satisfying this conditions is called an "observer" for system [START_REF] Alcaraz-Gonzalez | A tunable multivariable nonlinear robust observer for biological systems[END_REF]. When the convergence of x(t) towards x(t) is exponential, the system (2) is an "exponential observer". More precisely, system (2) is an exponential observer for system (1) if there exists λ > 0 such that, for all t ≥ 0 and for all initial conditions (x(0), x(0)), the corresponding solutions of ( 1) and (2) satisfy

x(t) -x(t) ≤ exp(-λt) x(0) -x(0) .
In this situation a good estimate of the real unmeasured state is rapidly obtained.

One must notice that we need not care about the choice of the initial condition of the observer since the convergence of x(t) towards the real state x(t) does not depend on this choice.

When the system under consideration is a linear system, i.e., it can be written as follows

             ẋ(t) = Ax(t) + Bu(t), y(t) = Cx(t), x(t) ∈ IR n , u(t) ∈ U ⊂ IR m , y(t) ∈ IR q ,
A, B, and C are respectively n × n, n × m and q × n matrices,

then an exponential observer (called Luenberger Observer) [START_REF] Luenberger | An introduction to observers[END_REF] for this system is given by

ẋ(t) = Ax(t) + Bu(t) + K y(t) -C x(t) (5)
where the n × q matrix K has to be computed.

O (C,A) =         C CA CA 2 . . . CA n-1        
is of rank n. In this case we say that the system (4), or the pair (C, A), satisfies the Kalman rank condition for observability (one can see for more details and examples [START_REF] Sontag | Mathematical control theory. Deterministic finite-dimensional systems[END_REF] or [START_REF] Iggidr | Controllability, observability and stability of mathematical models[END_REF]).

The construction of observers for highly nonlinear systems is still a very active research area in Control Theory. Several methods have been developed for some classes of systems (one can see for instance the references [START_REF] Luenberger | An introduction to observers[END_REF][START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF][START_REF] Krener | Nonlinear observer design in the Siegel domain[END_REF][START_REF] Kreisselmeier | Nonlinear observers for autonomous Lipschitz continuous systems[END_REF][START_REF] Zimmer | State observation by on-line minimization[END_REF][START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF] that represent different approaches). This is not an exhaustive list, because the literature on the subject is very extensive. This active research has resulted in the emergence of many nonlinear observer design techniques. The most classical one is based on the "feedback linearization" and the observer normal form (see for instance [START_REF] Bestle | Canonical form observer design for non-linear timevariable systems[END_REF], [START_REF] Isidori | Nonlinear control systems. 3rd ed. Communications and Control Engineering Series[END_REF], [START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF], [START_REF] Zeitz | The extended Luenberger observer for nonlinear systems[END_REF]) Roughly speaking, this method consists in finding change of coordinates

x = κ(z), u = ζ(E), y = η(w)
in the state space as well as in the input space and in the output space in such a way that equation ( 1) is transformed into

ẋ = Ax + χ(w, u), w = Cx. (6)
In this case a Luenberger type observer can be easily constructed. However the conditions under which the appropriate changes of coordinates exist are restrictive.

These changes of coordinates often exist only locally and hence the derived observer design works only locally.

The second famous method is the high gain construction ( [START_REF] Tornambe | Use of asymptotic observers having-high-gains in the state and parameter estimation[END_REF], [START_REF] Bornard | A high gain observer for a class of uniformly observable systems[END_REF], [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF], [START_REF] Gauthier | Observability and observers for nonlinear systems[END_REF], [START_REF] Gauthier | Deterministic observation Theory and Applications[END_REF], [START_REF] Iggidr | Exponential stabilization of nonlinear systems by an estimated state feedback[END_REF], [START_REF] Atassi | Separation results for the stabilization of nonlinear systems using different high-gain observer designs[END_REF]). A short survey is given in [START_REF] Atassi | Separation results for the stabilization of nonlinear systems using different high-gain observer designs[END_REF]. This method is developed hereafter and will be used in this paper.

Another design method uses an on-line optimization approach ([24] , [START_REF] Alamir | Further results on nonlinear recedinghorizon observers[END_REF], [START_REF] Michalska | Moving horizon observers and observer-based control. Automatic Control[END_REF], [START_REF] Moraal | Observer design for nonlinear systems with discretetime measurements[END_REF], [START_REF] Zimmer | State observation by on-line minimization[END_REF]) such as moving horizon observers that use the integral output prediction error in the estimation process, and the observer using Newtons method. In this case, the state is estimated by minimizing a certain norm of the difference between the ob-server output and the measured output. The advantage of the online optimization method is the capability of dealing with a variety of nonlinear systems including time-varying systems, chaotic systems, and systems with unknown parameters.

Moreover this method does not require the use of any canonical form. However, the corresponding observer computations are generally quite heavy and may prevent the use of these observers for systems with very fast dynamics.

Historically observability theory and observers design have been developed for artificial engineering systems but nowadays they are more and more applied to "natural systems". We outline here some applications of nonlinear observers to biological models. Once again the list is not exhaustive.

In [START_REF] Bernard | Nonlinear observers for a class of biological systems: application to validation of a phytoplanktonic growth model[END_REF] the well-known Droop model which describes the growth of a population of phytoplanktonic cells is considered. Observers for this model are built and are used to discuss the validity of this model by comparing the prediction of the state computed by the observer with direct measurements of this state.

In [START_REF] Farza | Simple nonlinear observers for online estimation of kinetic rates in bioreactors[END_REF], observers are used to estimate the kinetic rates in bioreactors. The efficiency of the observer design is illustrated with examples dealing with the microbial growth and biosynthesis reactions.

A robust nonlinear asymptotic observer with adjustable convergence rate has been proposed in [START_REF] Alcaraz-Gonzalez | A tunable multivariable nonlinear robust observer for biological systems[END_REF]. This observer has been applied to a model of an anaerobic digestion process used for wastewater treatment.

The authors of [START_REF] López | Monitoring in a lotka-volterra model[END_REF] consider a system of populations described by the classical Lotka-Volterra model with one predator and two preys. The only available information is the total quantity of population preys without distinction between them. An observer is constructed that allows to estimate all the state variables. It is also shown how the observer can be used for the estimation of the level of an abiotic effect on the population system. It must be, however, noticed that the proposed observer in [START_REF] López | Monitoring in a lotka-volterra model[END_REF] is a local observer, i.e., its convergence is guaranteed only if the initial estimate error is small.

A high gain observer is used in [START_REF] Torres | High-gain nonlinear observer for simple genetic regulation process[END_REF] to study a system describing a one-gene regulation circuit. The observer is used to to rebuild the non-measured concentrations of the mRNA and the protein.

The use of observer theory in fishery is scarce, we have done some works in this sense (see [START_REF] Ngom | Ouahbi An observer for a nonlinear agestructured model of a harvested fish population[END_REF], [START_REF] Guiro | A Non Linear Observer for a Fishery Model[END_REF]). In [START_REF] Ngom | Ouahbi An observer for a nonlinear agestructured model of a harvested fish population[END_REF], an observer has been constructed for a stage structured discrete-time fishery model that exhibits an unknown recruitment function. In [START_REF] Guiro | A Non Linear Observer for a Fishery Model[END_REF],

a stage structured continuous model is considered and it is assumed that only the last class (mature individuals) is harvested. The present work is a continuation and a generalization of [START_REF] Guiro | A Non Linear Observer for a Fishery Model[END_REF].

The goal of this paper is twofold. First we shall show that some tools from control theory are helpful to address the stock estimation problem for an exploited fish population. More precisely we shall built exponential observers for the two models under consideration. These observers will allow to give an estimate of the respective stocks. The second is to show that the application of mathematical tools to biological systems has to be done carefully. One of the most efficient way to build an observer for a nonlinear system has been given in [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF]. We briefly recall the method developed in [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF]. To simplify matters we consider systems without control. Roughly speaking, the result of [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF] concerns systems that can be written (possibly after a coordinates change):

                         ż(t) =       0 1 0 ... 0 0 0 1 ... 0 . . . . . 0 0 ... 0 1 0 0 ... 0 0       A z(t) +        0 0 . . . 0 ψ(z(t))        = X(z(t)) y(t) = z 1 (t) = (1, 0, . . . , O) C z(t). (7)
The state of the system at time t is z(t) = (z 1 (t), z 2 (t), . . . , z n (t)) ∈ IR n , and its measurable output is y(t). The fact that y(t) = z 1 (t) means that one can measure only the first component of the state and hence the other components are not available for measurement. Assume that the function ψ is globally Lipschitz on IR n , that is, there exists

K > 0 such that |ψ(z) -ψ(x)| ≤ K|z -x| for all (z, x) ∈ IR n × IR n .
It has then been proved in [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF] that for θ ≥ 1 large enough, an exponential observer (a Luenberger type observer) for the system ( 7) is given by the following dynamical system:

ż = X(ẑ) -S -1 θ C T (C ẑ -y), (8)
with S θ being the solution of

θS θ + A T S θ + S θ A = C T C.
System ( 8) is an exponential observer for system [START_REF] Bornard | A high gain observer for a class of uniformly observable systems[END_REF] means that the solutions of (8) converge to the solutions of system [START_REF] Bornard | A high gain observer for a class of uniformly observable systems[END_REF] with an exponential speed regardless the values of the respective initial conditions z(0) and ẑ(0). To prove this result the authors of [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF] use the fact that the function ψ is globally Lipschitz on the whole state space IR n . The global Lipschitz assumption is very restrictive. Biological systems always evolve in a bounded domain D of IR n and hence the global Lipschitz assumption is satisfied on D. However, it must be noticed that the fact that the domain D is positively invariant for system [START_REF] Bornard | A high gain observer for a class of uniformly observable systems[END_REF] and that the map ψ is globally Lipschitz on D does not guarantee the convergence of the observer (8) even if one take the initial values inside D. Indeed, the domain D is positively invariant for the system (7) but it is not a positively invariant set for the system (8) defining the equations of the observer. More precisely, for a given initial condition (z(0), ẑ(0)) ∈ D × D, the corresponding solution (z(t), ẑ(t)) of (7-8) can leave the set D × D in finite time: the component z(t) will actually belong to D for all positive time but there is no reason that the same property will be true for ẑ(t). In order to built an exponential observer for the considered system in this situation, one has first to extend the function ψ from D to the whole IR n by a function ψ which is globally Lipschitz on IR n and then to consider the systems (7-8) defined on IR n × IR n after replacing the function ψ by its prolongation ψ. The stage-structured fishery model we consider here will illustrate this fact. For this model, there is a domain D ⊂ IR 3 which is positively invariant, and the system dynamics are defined by a vector field X which is globally Lipschitz on D. We shall show that the observer works well when we extend the vector field X to the whole space IR 3 and it fails to work when the prolongation is not done. The same things are valid for the global model.

This shows that the Lipschitz extension of the vector field mentioned in [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF] is not only for mathematical sophistication purpose but it is also necessary for application purpose. Here we construct simply a continuous Lipschitz extension of the function ψ. For more details concerning the design of Lipschitz extensions one can see for instance [START_REF] Rapaport | Design of exponential observers for nonlinear systems by embedding[END_REF].

The paper is organized as follows. In Section 2, we present the stage-structured model and we built an observer for this system. The construction is made for a three stages model. It can be done for an arbitrary number of stages but the calculus are longer and more complicated. Section 3 is devoted to the stock estimation problem for a "global" model. Once again, for clarity reasons, we have preferred to deal with a model with two fishing areas but the observer construction can be done for a system describing the dynamics of a fish population that can move between different fishing zones (an example of such a system has been considered in [START_REF] Mchich | Optimal spatial distribution of the fishing effort in a multi fishing zone model[END_REF]).

A Stage-structured model

In this section, we consider a class of a structured model in fishery with three classes.

The first class x 0 is constitute of the pre-recruits i.e the eggs, larvae and the juveniles.

The second and the third classes are the post-recruits or the exploited phase of the population.

The dynamics of the system are modeled by the following three dimensional system (see [43,[START_REF] Touzeau | On the stock-recruitment relationships in fish population models[END_REF], [START_REF] Ouahbi | Stabilization of an exploited fish population[END_REF]) :

           ẋ0 (t) = -α 0 x 0 (t) + 2 i=1 f i l i x i (t) - 2 i=1 p i x i (t)x 0 (t) -p 0 x 2 0 (t) ẋ1 (t) = αx 0 (t) -(α 1 + q 1 E)x 1 (t) ẋ2 (t) = αx 1 (t) -(α 2 + q 2 E)x 2 (t) (9)
where :

x i : the number of fish in the stage i.

α : linear aging coefficient (in time -1 )

m i : natural mortality rate of class i (in time -1 )

α i = m i + α (in time -1 )
p 0 : juvenile competition parameter (in time -1 .number -1 ) f i : fecundity rate of class i (no dimension)

l i : reproduction efficiency of class i (in time -1 )
p i : predation rate of class i on class 0 (time -1 .num -1 )

q i : capturability coefficient of class i (in unit effort -1 )
E: instantaneous fishing effort.

(in unit effort×time -1 ).

We assume that the total catch is available for measurement. This total catch can be considered as a measurable output of the system(9) and it is given by

y(t) = q 1 Ex 1 (t) + q 2 Ex 2 (t) (10)
We then obtain the following coupled system:

                   ẋ0 (t) = -α 0 x 0 (t) + 2 i=1 f i l i x i (t) - 2 i=1 p i x i (t)x 0 (t) -p 0 x 2 0 (t) ẋ1 (t) = αx 0 (t) -(α 1 + q 1 E)x 1 (t) ẋ2 (t) = αx 1 (t) -(α 2 + q 2 E)x 2 (t) y(t) = q 1 Ex 1 (t) + q 2 Ex 2 (t) (11)
We consider system [START_REF] Gámez | Monitoring environmental change in an ecosystem[END_REF] which is a nonlinear system. Our aim is to construct an observer (estimator) i.e an auxiliary system which will give a dynamical estimate (x 0 (t), x1 (t), x2 (t)) of the state (x 0 (t), x 1 (t), x 2 (t)) of system [START_REF] Dubey | A model for fishery resource with reserve area. Nonlinear Anal[END_REF]. For the construction of such auxiliary system, we shall use a method called High Gain construction (see for instance [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF]). This construction provide an exponential observer; the estimation error will converges to zero with exponential speed, i.e.,

x(t) -x(t) ≤ exp(-λt) x(0) -x(0) .

High Gain observer design for (11)

The system [START_REF] Gámez | Monitoring environmental change in an ecosystem[END_REF] is the system (9) coupled with the output [START_REF] Farza | Simple nonlinear observers for online estimation of kinetic rates in bioreactors[END_REF]. For the observer design, we will use the High Gain observer techniques (Gauthier et al.([13])) to construct a High Gain observer for system [START_REF] Dubey | A model for fishery resource with reserve area. Nonlinear Anal[END_REF].

It has been proved in [43] that there is a positively invariant compact set for system [START_REF] Dubey | A model for fishery resource with reserve area. Nonlinear Anal[END_REF]. This set is of the form

D = [a 0 , b 0 ] × [a 1 , b 1 ] × [a 2 , b 2 ],
where the numbers a i can be chosen as small as we need and the numbers b i are function of the parameters f i , l i and p i . More precisely:

b i = π i µ with π i = α i i j=1 (α j + q j E) ,
and µ = min

i :p i =0 { f i l i p i }
Let us denote by F the vector field defining the dynamics of the system (9), and h the output function, that is y(t) = h(x(t)) = q 1 Ex 1 (t) + q 2 Ex 2 (t) and

F (x(t)) =        -α 0 x 0 (t) + 2 i=1 f i l i x i (t) - 2 i=1 p i x i (t)x 0 (t) -p 0 x 2 0 (t) αx 0 (t) -(α 1 + q 1 E)x 1 (t) αx 1 (t) -(α 2 + q 2 E)x 2 (t)       
Let Φ be the function Φ :

o D → IR 3 ( o D
is the interior of D), defined as follows:

Φ(x) =      h(x) L F h(x) L 2 F h(x)     
, where L denotes the Lie derivative operator with respect to the vector field F . Thus,

Φ(x) = E                   q 1 x 1 + q 2 x 2 αq 1 x 0 + αq 2 -q 1 (α 1 + q 1 E) x 1 -q 2 (α 2 + q 2 E)x 2 -α 0 αq 1 + α 2 q 2 -αq 1 (α 1 + q 1 E) x 0 + αq 1 f 1 l 1 -αq 2 (α 1 + q 1 E) + q 1 (α 1 + q 1 E) 2 -αq 2 (α 2 + q 2 E) x 1 + αq 1 f 2 l 2 + q 2 (α 2 + q 2 E) 2 x 2 -αq 1 p 0 x 2 0 -αq 1 p 1 x 1 x 0 -αq 1 p 2 x 2 x 0                  
The Jacobian of Φ can be written:

dΦ dx =     0 q 1 E q 2 E αq 1 E γ 1 γ 2 γ 3 γ 4 γ 5     , and
dΦ dx -1 = 1 Γ     β 1 β 2 β 3 β 4 β 5 β 6 β 7 β 8 β 9     ,
where:

Γ = Det dΦ dx = q 1 Eγ 2 γ 3 + αq 1 q 2 E 2 γ 4 -q 2 Eγ 1 γ 3 -αq 2 1 E 2 γ 5 γ 1 = αq 2 E -q 1 E(α 1 + q 1 E) γ 2 = -q 2 E(α 2 + q 2 E) γ 3 = α 2 q 2 E -α 0 αq 1 E -αq 1 E(α 1 + q 1 E) -2αq 1 Ep 0 x 0 -αq 1 Ep 1 x 1 -αq 1 Ep 2 x 2 γ 4 = q 1 E(α 1 + q 1 E) 2 -αq 2 E(α 1 + q 1 E) -αq 2 E(α 2 + q 2 E) + αq 1 f 1 l 1 E -αq 1 Ep 1 x 0 γ 5 = q 2 E(α 2 + q 2 E) 2 + αq 1 f 2 l 2 E -αq 1 Ep 2 x 0 β 1 = γ 1 γ 5 -γ 2 γ 4 β 2 = -q 1 Eγ 5 + q 2 Eγ 4 β 3 = q 1 Eγ 2 -q 2 Eγ 1 β 4 = -αq 1 Eγ 5 + γ 2 γ 3 β 5 = -q 2 Eγ 3 β 6 = αq 1 q 2 E 2 β 7 = αq 1 Eγ 4 -γ 1 γ 3 β 8 = q 1 Eγ 3 β 9 = -αq 2 1 E 2 .
The determinant of dΦ dx can be written

Γ(x 0 , x 1 , x 2 ) = Det dΦ dx = (c + a 0 x 0 + a 1 x 1 + a 2 x 2 ) E 3 ,
where c and a i are functions of the parameters. The map (x 0 , x 1 , x 2 ) → Γ(x 0 , x 1 , x 2 ) is affine on the polyhedron D, hence it reaches its extrema on the vertexes of D. For a given set of parameters, it is then sufficient to compute the values of Γ(x 0 , x 1 , x 2 )

on the vertexes of D in order to see if Γ(x 0 , x 1 , x 2 ) vanishes in D or not.

We assume that the parameters are such that the map Φ is a diffeomorphism from

o D to Φ( o D
). This implies that system (11) is observable.

In the new coordinates defined by (z 1 , z 2 , z 3

) T = z = Φ(x)=(h(x), L F h(x), L 2 F (x)) T ,
our system can be written in the canonical form as follow:

               ż(t) =   0 1 0 0 0 1 0 0 0   A z(t) +   0 0 ψ(z(t))   y(t) = z 1 (t) = (1, 0, O) C z(t). (12)
where :

ψ(z) = L 3 F h(Φ -1 (z)) = L 3 F h(x) = ϕ(x)
The function ϕ is smooth (it is a polynomial function of x = (x 0 , x 1 , x 2 )) on the compact set D. Hence, it is globally Lipschitz on D. Therefore it can be extended by φ, a Lipschitz function on IR 3 which satisfies φ(x) = ϕ(x), for all x ∈ D. In the same way we define ψ the Lipschitz prolongation of the function ψ.

So we have the following system (13) defined on the whole space IR 3 . The restriction of ( 13) to the domain D is the system (12):

           ż = Az +     0 0 ψ(z)     , y = Cz. (13)
Hence, we have shown that system [START_REF] Gámez | Monitoring environmental change in an ecosystem[END_REF] satisfies the conditions of the following result which provides the observer construction. Then an exponential observer for system ( 13) is given by the following system :

ż = Aẑ + ψ(ẑ) + S -1 (θ)C T (y -C ẑ). ( 14
)
where S(θ) is the solution of

0 = -θS(θ) -A T S(θ) -S(θ)A T + C T C,
and θ is large enough.

Here,

S(θ) =     θ -1 -θ -2 θ -3 -θ -2 2 θ -3 -3 θ -4 θ -3 -3 θ -4 6 θ -5     .
Precisely θ ≥ 2ncK √ S, where K is the lipschitz coefficient of the function ψ, n is the dimension of the space, and S = sup i,j |S(1) i,j |.

For the proof one can see [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF].

Going back to the our original system (9) via the transformation Φ -1 , we have :

ẋ = F (x) + dΦ dx -1 x=x × S(θ) -1 C T (y -h(x)) (15)
The restriction of this system to D is the following system :

                                   ẋ0 = -α 0 x0 + 2 i=1 f i l i xi - 2 i=1 p i xi x0 -p 0 x2 0 +(3θβ 1 + 3θ 2 β 2 + θ 3 β 3 )(y -q 1 E x1 -q 2 E x2 ) ẋ1 = αx 0 -(α 1 + q 1 E)x 1 +(3θβ 4 + 3θ 2 β 5 + θ 3 β 6 )(y -q 1 E x1 -q 2 E x2 ) ẋ2 = αx 1 -(α 2 + q 2 E)x 2 +(3θβ 7 + 3θ 2 β 8 + θ 3 β 9 )(y -q 1 E x1 -q 2 E x2 ) ( 16
)
which is the observer for the fishery model ( 9). This observer is particularly simple since it is only a copy of ( 9), together with a corrective term depending on θ.

Simulations and comments

We present here some simulation results that show the efficiency of the observer of system (9). The simulations have been done with the free software SCILAB. We use the following fishery parameters [START_REF] Ouahbi | Stabilization of an exploited fish population[END_REF], [43].

α 0 = 1.3; α 1 = 0.9; α 2 = 0.85; p 0 = 0.2;

p 1 = 0.1; p 2 = 0.1;
q 1 = 0.07; q 2 = 0.15;

f 1 = 0.5; f 2 = 0.5;

l 1 = 10; l 2 = 10; E = 0.5; α = 0.8.
For these parameter the Jacobian of the function Φ is expressed as:

dΦ dx =          0 0.035 0.075 0.028 0.027275 -0.069375 -0.01458 -0.0112x 0 0.0589979-0.0028x 0 0.191338-0.0028x 0 -0.0028x 1 -0.0028x 2         
The determinant of this matrix is:

Det( dΦ dx ) = 1.612 × 10 -6 + 0.00004697x 0 + 0.0000125265x 1 + 0.0000125265x 2 .
The states x 0 , x 1 and x 2 are time varying but remain in the positive orthant; so the Det( dΦ dx ) does not vanish. Therefore dΦ dx is invertible and then Φ(x) is a diffeomorphism.

With the parameters defined in the top of this section, we compute the coor-

dinates

A global model 3.1 The model and the observer

Here we consider the dynamics of a fish population moving between two zones (see [START_REF] Dubey | A model for fishery resource with reserve area. Nonlinear Anal[END_REF]). The first zone is a free fishing area, and the second zone is a reserve area where no fishing is allowed. Let x 1 (t) be the biomass density at time t of the fish population in the free fishing area and x 2 (t) be the biomass density at a time t of the fish population in the reserved areas. For (i, j) ∈ {1, 2} 2 , we denote by m ij the migration rate from the zone i to the zone j. In the free fishing area, the total fishing effort is denoted by E. The growth of the two sub-population in each zone follows logistic model. The dynamics of the fish subpopulations in unreserved and reserved ares are then assumed to be governed by the following autonomous system of differential equations [START_REF] Dubey | A model for fishery resource with reserve area. Nonlinear Anal[END_REF].

       ẋ1 = r 1 x 1 1 - x 1 K 1 -m 12 x 1 + m 21 x 2 -qEx 1 ẋ2 = r 2 x 2 1 - x 2 K 2 + m 12 x 1 -m 21 x 2 . (17)
r 1 and r 2 represent the intrinsic growth of each fish sub-population, respectively, K 1 and K 2 are the carrying capacities of fish species in the unreserved and reserved areas, respectively; q is the catchability coefficient of fish species in the unreserved area. The parameters r 1 , r 2 , q, m 12 , m 21 , K 1 and K 2 are positives constants.

To the system (17) we associate the capture (i.e. the output) y = qEx 1 (the total of caught fish in the unreserved area), with this output, we show the observability condition of system [START_REF] Gulland | Fish Stock Assessment, a manual of basic methods[END_REF] and construct an auxiliary system that will give a dynamical estimation of the state of system [START_REF] Gulland | Fish Stock Assessment, a manual of basic methods[END_REF].

It is possible to find a positive real number w 0 in such a way that for any w ≥ w 0 the following compact set D w is positively invariant for system [START_REF] Gulland | Fish Stock Assessment, a manual of basic methods[END_REF]. This compact set is given

D w = {(x 1 , x 2 ) ∈ IR 2 + : x 1 + x 2 ≤ w},
The proof of this fact as well as the computation of w 0 as a function of the parameters are given in Appendix A.. Let us denote by f the vector field that defines the system (17):

f (x) =      r 1 x 1 1 - x 1 K 1 -(m 12 + qE)x 1 + m 21 x 2 r 2 x 2 1 - x 2 K 2 + m 12 x 1 -m 21 x 2      . Let z = z 1 z 2 , y(t) = h(x) = qEx 1 (t) and Φ(x) = y ẏ =    qEx 1 r 1 qEx 1 1 - x 1 K 1 -(m 12 + qE)qEx 1 + m 21 qEx 2    . Therefore dΦ dx =    qE 0 r 1 qE - 2r 1 qEx 1 K 1 -(m 12 + qE)qE m 21 qE   
and Det

dΦ dx = q 2 E 2 m 21 .
As the parameters q, E and m 21 are positive ( = 0), we can conclude that Det dΦ dx = 0, and then, Φ is a diffeomorphism from IR 2 to Φ(IR 2 ), thus system ( 17) is observable.

Thanks to ( [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF]) the observer can be expressed as follows:

ẋ = f (x) + dΦ dx -1 × S(θ) -1 C T (y -h(x)), (18)
where f is a Lipschitz extension of the function f from the invariant domain D w to the whole IR 2 space, C = (1, 0) and

S(θ) -1 = 2θ θ 2 θ 2 θ 3 , with θ ≥ 1.
The restriction of the estimator [START_REF] Horwood | The optimal harvest from a multicohort stock[END_REF] to the invariant domain D w is given by the equations:

               ẋ1 = r 1 x1 1 - x1 K 1 -m 12 x1 + m 21 x2 -qE x1 + 2θ(x 1 -x1 ) ẋ2 = r 2 x2 1 - x2 K 2 + m 12 x1 -m 21 x2 +2θ( qE m 21 - 1 m 21 + m 12 qEm 21 + 2r 1 x 1 m 21 + θ m 21 )(x 1 -x1 ), (19)

Simulation

Simulations for the model [START_REF] Gulland | Fish Stock Assessment, a manual of basic methods[END_REF] together with its observer [START_REF] Horwood | The optimal harvest from a multicohort stock[END_REF] have been done with the following parameters : 17) with its observer (18): x 2 (solid line) and its estimate x2 (dashed line) when f is not extended

r 1 = 7

Conclusion

We have tried to combine modern Control Theory, Computer Science and Mathematics to address the state estimation problem for systems that model the dynamics of fish populations submitted to a fishing action. Indeed one of the important problems in fishery sciences is to estimate the state of the resource using the available data, in order to produce scientific opinions that can be helpful for developing management policies that need to have a good estimate of the available resource.

In this work, we have constructed High Gain observers for some fishery models.

With the use of judicious value of the gain parameter θ we obtain satisfactory es- In practice it is easier to measure the total catch 9without doing any distinction between individuals) then to measure the catch for each stage class. However the observers given in this paper assume that the model is good enough and that the parameters values are available.

Nonlinear control techniques are useful for studying and controlling complex systems.

Although they have been initially developed for mechanical and electrical systems their applications to biological and environmental problems are growing. Tools of optimal control theory have been extensively used in renewable resource management ( [START_REF] Clark | Mathematical bioeconomics. The optimal management of renewable resources[END_REF], [START_REF] Anit ¸a | Optimal harvesting for a nonlinear age-dependent population dynamics[END_REF] [23] [START_REF] Horwood | The optimal harvest from a multicohort stock[END_REF] [25], [START_REF] Marutani | On the optimal path in the dynamic pool model for a fishery[END_REF], [START_REF] White | A reassessment of equivalence in yield from marine reserves and traditional fisheries managament[END_REF], [START_REF] Gao | Optimal pulse fishing policy in stage-structured models with birth pulses[END_REF], [START_REF] Mchich | Optimal spatial distribution of the fishing effort in a multi fishing zone model[END_REF]). The present paper shows that the estimation problem in fisheries management can also be investigated from the point of view of control engineering. end.

Figure 1 :

 1 Figure 1: A schematic representation of an observer

Proposition 2 . 1 (

 21 [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF]) Under the assumptions thatH1: Φ is a diffeomorphism from o D to Φ( o D ). ( o D is the interior of D).H2: ϕ can be extended from D to IR 3 by a C ∞ function, globally Lipschitz on IR 3 .

Remarque 2 . 1

 21 For the simulations we extend the function ϕ by continuity in order to make it globally lipschitz on IR 3 in the following way: We denote φ the prolongation of ϕ to IR 3 and the function π the projection on the domain D and we construct φ = ϕ • π. The extended function φ has the same Lipschitz coefficient as ϕ. The projection π is defined as follows: for x ∈ IR 3 , π(x) = x, where x ∈ D is such that dist(x, D) = x-x , i.e., x satisfies x-x = min u∈D u-x . The extension algorithm is described in Appendix B..

  of the higher corner B of the parallelepiped D ([43]) and we get B = (25; 20.639; 17.868). The nontrivial equilibrium point is x * = (18.572; 15.89; 13.743). The construction of the high gain observer (15) is done with θ = 17. For the simulations we have taken x(0) = [21; 20; 15] and x(0) = [35; 40; 10].Comments: Using the same parameters values, when we do not use the Lipschitz prolongation of the function ϕ to the whole IR 3 , the state estimation x(t) computed by the observer tends to infinity in finite time. This actually happens in the beginning of the integration process as it can be seen in Figures2, 4 and 6. When the Lipschitz prolongation of the function ϕ to the whole IR 3 is done, the convergence of the estimates delivered by the observer is quite fast (Figures3, 5 and 7).

  [START_REF] Iggidr | Controllability, observability and stability of mathematical models[END_REF] we compute w 0 = 8.987 and we take w = 20.With these parameters, the invariant domain is the triangle defined by O(0, 0), A(w, 0) = A(20, 0) and B(0, w) = B(0, 20), and we take θ = 4.Using the SCILAB free software, the time evolution of the states as well as the respective estimates when the Lipschitz extension is done are drawn in Figures8 and 10. When the Lipschitz extension has not been done, the simulations are given in Figures9 and 11

  .

Figure 2 :Figure 3 :Figure 4 :Figure 5 :Figure 6 :Figure 7 :Figure 8 :Figure 9 :Figure 10 :Figure 11 :

 234567891011 Figure 2: Simulation of system (9) with its observer (15): x 0 (solid line) and its estimate x0 (dashed line) when ϕ is not extended

if x 2 ≤ a 2 x 2 ≤ b 2 x 1 ≤ a 1 then if x 2 ≤ a 2 x 1 ≤ b 1 then if x 2 ≤ a 2 x 2 ≤ a 2

 222212212222 then φ(x 0 , x 1 , x 2 ) = ϕ(x 0 , x 1 , a 2 ) else if x 2 ≤ b 2 then φ(x 0 , x 1 , x 2 ) = ϕ(x 0 , x 1 , x 2 ) else φ(x 0 , x 1 , x 2 ) = ϕ(x 0 , x 1 , b 2 ) then φ(x 0 , x 1 , x 2 ) = ϕ(x 0 , b 1 , x 2 ) else φ(x 0 , x 1 , x 2 ) = ϕ(x 0 , b 1 , b 2 ) then φ(x 0 , x 1 , x 2 ) = ϕ(b 0 , a 1 , a 2 ) else if x 2 ≤ b 2 then φ(x 0 , x 1 , x 2 ) = ϕ(b 0 , a 1 , x 2 ) else φ(x 0 , x 1 , x 2 ) = ϕ(b 0 , a 1 , b 2 ) then φ(x 0 , x 1 , x 2 ) = ϕ(b 0 , x 1 , a 2 ) else if x 2 ≤ b 2 then φ(x 0 , x 1 , x 2 ) = ϕ(b 0 , x 1 , x 2 ) else φ(x 0 , x 1 , x 2 ) = ϕ(b 0 , x 1 , b 2 ) then φ(x 0 , x 1 , x 2 ) = ϕ(b 0 , b 1 , a 2 ) else if x 2 ≤ b 2 then φ(x 0 , x 1 , x 2 ) = ϕ(b 0 , b 1 , x 2 ) else φ(x 0 , x 1 , x 2 ) = ϕ(b 0 , b 1 , b 2 )

  timation of the real state. The observer's convergence is quite fast and does not depend on the initial conditions choice. Therefore one can get a "good" estimate of the unmeasurable real state very quickly. It is interesting to notice that the state estimator built in this paper for the stage-structured model use only the total catch to give not only an estimate of the total stock but also an estimate of the number of individuals in each stage class. The classical techniques like the Cohort Analysis (CA) or the Virtual Population Analysis (VPA) use the total catch for each stageclass in order to give estimates of the number of individuals in each stage class.

Acknowledgments

We thank the anonymous referees for their valuable comments and suggestions that have helped us to improve the presentation of this article.

Appendix A. Positive invariance of D w

Let w be a positive real number, for N = w, we have

The function g is defined for 0 ≤ x 1 ≤ w.

The maximum value of the function g is then given by the expression

It is therefore clear that this maximum is non positive if w ≥ w 0 with

) This shows that for any real number w ≥ w 0 , the compact set

is positively invariant for system [START_REF] Gulland | Fish Stock Assessment, a manual of basic methods[END_REF].

Appendix B. Construction of the Lipschitz extension of ϕ

The function The problem of the extension is set for point x / ∈ D; in this situation we have 26 possibilities according to the situation of x. The different situations correspond to

The principle of this prolongation is to compose the function ϕ with the function π (the projection function of the point x on the domain D).

The extension of function ϕ is described by the following algorithm: