Vito Albino
email: v.albino@poliba.it

Claudio Achille

Vincenzo Alessio Garavelli
email: c.garavelli@poliba.it

Romano
email: va.romano@poliba.it

Romano Alessio

Dmmm

A Classification of Industrial Symbiosis Networks: A Focus on Materials and Energy Recovery

Keywords: sustainable production, industrial symbiosis, classification

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

Introduction

Nowadays, problems related to the natural resources consumption, energy usage, Greenhouse Gas (GHG) emissions, and fossil fuel utilization are becoming more and more crucial and relevant. Therefore, there is a need to give further impetus to efficient and eco-innovative production processes, to reduce dependency on raw materials, and to encourage optimal resource use and recycling (UNEP, 2009). Industrial symbiosis (IS) has been considered as one of the most effective solutions to reduce the impact of waste emissions and of primary input consumption moving towards sustainable production models. The Organization for Economic Co-operation and Development has recently identified IS as a tool for systemic innovation vital for future green growth (OECD, 2010). Furthermore, within the Resource Efficiency Flagship Initiative, IS is indicated as a tool that all member states should exploit to help companies work together to make the best use of the waste and by-products they produce (EU, 2011).

Industrial symbiosis has emerged as a sub-field of industrial ecology. "Industrial ecology is the science that provides the conceptual tools to analyze and optimize the flow of energy and materials in production systems" (The Interagency Workgroup on Industrial Ecology, Material and Energy Flows, 1998). In particular, IS operates at the inter-firm level [START_REF] Chertow | Industrial symbiosis: literature and taxonomy[END_REF]. However, despite over twenty years of industrial ecology, IS phenomenon is still not clearly defined.

IS has been inspired by the concept of "industrial ecosystem" in which "the consumption of energy and materials is optimized and the effluents of one process ... serve as the raw material for another process" [START_REF] Frosch | Strategies for manufacturing[END_REF]. More recently, IS was widely known as "engaging traditionally separate industries in a collective approach to competitive advantage involving physical exchange of materials, energy, water and/or by-products" [START_REF] Chertow | Industrial symbiosis: literature and taxonomy[END_REF], whose key success factors are "collaboration and the synergistic possibilities offered by geographic proximity". These keys for the success of IS differ according to the case-and location-specificity of IS, and they have been largely discussed within the recent literature, often achieving not fully agreed results.

Additionally, through the years, the IS network concept has been extended not only to all types of physical and unphysical resources that could be intentionally shared or exchanged (by-products, utilities, and ancillary services) [START_REF] Chertow | Uncovering" industrial symbiosis[END_REF], but also to all the other firm resources such as, for example, knowledge and social processes.

Therefore, IS has been approached from various perspectives (e.g. social, economic, environmental, spatial, organizational, technical). Nevertheless, with the increasing number of IS projects, many issues about IS boundaries, such as the entities involved (firms, organizations, industries, facilities) and the resources shared/exchanged (materials, energy, water, information, expertise), still continue to remain open.

With this regard, focusing on materials and energy physical flows only, and adopting an input-output (I-O) approach, we define an IS network as a set of one or more production chains utilizing their waste materials and residual energy in substitution of primary resources. In particular, we propose a technical classification model of IS networks based on the physical flow type involved, and the production chains sending and receiving it. The study aims at proposing a basic categorization of IS networks, a classification tool, which could be a common ground for successive structural network analyses.

In the next section a specific review of the literature is devoted to the materials and energy flows in IS networks. In the third section the classification model for IS networks is proposed. In the fourth section several case examples are provided. Finally, main conclusions are reported in the last section.

Materials and energy in IS networks

From a technical point of view, a production chain can be considered as an inputoutput system [START_REF] Storper | Flessibilità, gerarchie e sviluppo regionale: la ristrutturazione organizzativa dei sistemi produttivi e le nuove forme di governance[END_REF]) that describes the product flows existing among production processes. Input-output systems may involve many production processes, depending on a specific division and classification of production in proc-esses. In general, the chain may contain the extraction of raw materials, manufacturing, distribution and use of goods. Referring to the environmental sustainability, a sustainable production chain means favouring, for a given output, processes with less use of natural resources (energy and materials) and production of wastes. From the supply perspective, extended supply chains include the reduction and elimination of by-products through cleaner process technologies and lean production techniques, extending chain boundaries as far as to include the source and the destination of all the physical flows used and produced at each chain stage [START_REF] Albino | Input-output models for the analysis of a local/global supply chain[END_REF][START_REF] Albino | Analyzing Material and Energy Flows in an Industrial District using an Enterprise Input-Output Model[END_REF]. As the opportunity offered by the efficient use of materials (recycling, etc.) and energy stresses the relevance of network of actors, cooperation between actors becomes crucial.

Cooperation can be considered also in terms of industrial ecology. Industrial symbiosis is the main approach introduced by the industrial ecology to explain how industrial ecosystem [START_REF] Frosch | Strategies for manufacturing[END_REF]) may work. Two industrial actors operate in IS when they exchange materials and energy to reduce costs, create value, and improve the environment. Although the analysis of the role and level of cooperation between actors of production chains in permitting environmental and economic benefits needs to be explored further, ownership issues are not faced in our study. We aim at describing symbiotic relationships among industrial actors in terms of materials and energy flows only.

Assuming that symbiotic transactions are waste exchanges, the first technical classification of IS networks is proposed by [START_REF] Chertow | Industrial symbiosis: literature and taxonomy[END_REF], identifying ISs i) within a firm, facility or organization, ii) among firms collocated in a defined eco-industrial park, iii) among local firms that are not co-located, and iv) among firms organized "virtually" in a broader region. However, this categorization seems to be in contrast with the [START_REF] Chertow | Industrial symbiosis: literature and taxonomy[END_REF] IS definition, which indentifies the no well-quantified geographic proximity as the IS key driver. [START_REF] Lombardi | Redefining Industrial Symbiosis -Crossing Academic-Practitioner Boundaries[END_REF] redefine IS as "engaging diverse organizations in a network ... to yield mutually profitable transactions for novel sourcing of required inputs, and value added destinations for nonproduct outputs". As the decennary National Industrial Symbiosis Programme (NISP) experience in the United Kingdom demonstrates, the localization of the entities exchanging resources, although geographic proximity is often associated with IS, is not essential for the success of IS networks [START_REF] Lombardi | Redefining Industrial Symbiosis -Crossing Academic-Practitioner Boundaries[END_REF]. Distance between entities can be considered as a business-as-usual factor.

In this scenario, input-output matching is the most significant tool to analyze IS [START_REF] Chertow | Industrial symbiosis: literature and taxonomy[END_REF]. In input-output matching, firms systematically collect input and output data of companies and try to create links among partners. Recently, [START_REF] Bossilkov | Regional Synergies for Sustainable Resource Processing: A Status Report[END_REF] proposed a technical classification of existing IS networks in three dimensions: i) type of resource exchange (water, energy, process or non-process waste, utility sharing); ii) type of processing involved (direct use or reuse, energy recovery or alternative fuels, material recovery, etc.); and iii) type of synergy (bi or multilateral, etc.). However, this classification continues to leave ambiguity about IS boundaries. Exchanged/shared resources and involved actors, which should characterize IS networks and distinguish them from other types of resource synergy, are still not clearly defined. As [START_REF] Golev | Developing a classification system for regional resource synergies[END_REF] affirm, a common classification for IS networks does still not exist.

A classification model for IS networks

Adopting an input-output (I-O) approach, we define an IS network as a set of one or more production chains utilizing their waste materials and residual energy in substitution of primary resources. Thus, we propose a three-dimensional structural classification model for IS networks by tracking material and energy flows among symbiotic chains. This classification refers to an elementary IS network, i.e. a physical flow type (i.e. one waste-one primary input) shared by one or more actors (i.e. production chains). Starting from this basic categorization, we may represent and classify a more complex IS network as a composition of elementary IS networks.

Physical flow type

Let us consider an IS network constituted by one or more production chains.

Let us assume that the waste l output flow is the only waste material flow produced, and the primary input type k the only one required by the IS network. Assuming that the waste l can be used as primary input k, we may distinguish the i) materialmaterial, and ii) material-energy flow types. In the first case, the waste l can be directly used in production processes without further transformation as material input, or recycled obtaining the primary input required by the IS network. In the second case, the waste l can be used to produce energy (e.g. alternative fuels). Now let us suppose that the waste l output is an energy flow, and the primary energy input k the only required by the IS network. Assuming that the waste l can be used as primary input k, we may identify the iii) energy-energy flow type, i.e. an energy cascade, such as heat recovery.

Internal and external IS

Let us consider an IS network constituted by one production chain, α. Let us assume that the chain α only produces the waste type l, and only requires the primary input type k. Assuming that the waste type l can be used as primary input k, a form of internal IS may occur if the chain α uses the waste l as primary input k.

Let us consider now an IS network constituted by two production chains, α and β, where α ≠ β. Let us assume that the chain α only produces the waste type l, and no primary input is required, and the chain β only requires the primary input type k, and no waste is produced. Assuming that the waste type l can be used as primary input k, a form of external IS may occur if the chain β uses the waste l as primary input k.

IS network structure

Let us consider an IS network constituted by two or more production chains. Let us assume that the waste type l is only produced, and the primary input k only required by the IS network. Finally, let us assume that the waste l can be used as primary input k. For each physical flow type, depending on the number of production chains sending and receiving it, we may identify the following configurations: i) one-to-one, ii) oneto-n, iii) n-to-one, and iv) m-to-n, where m, n 2 ≥ . In the first case, let us suppose that the IS network is composed by two production chains, α and β. Moreover, let us assume that the chain α only produces the waste l, and no primary input is required, and the chain β only requires the primary input type k, and no waste is produced. The "one-to-one" configuration occurs if the chain β uses the waste l as primary input k.

In the second case, let us suppose that the IS network is composed by three production chains, α, β, and γ. Furthermore, let us assume that the chain α only produces the waste l, and no primary input is required, and the chains β and γ only require the primary input k, and no waste is produced. The "one-to-n" configuration occurs if the chains β and γ use the waste l as primary input k.

In the third case, let us continue to suppose that the IS network is composed by three production chains, α, β, and γ. Let us assume that chains β and γ only produce the waste l, and do not require any primary input, and the chain α only requires the primary input type k, and no waste is produced. The "n-to-one" configuration occurs if the chains α uses the waste l as primary input k.

In the fourth case, let us suppose that the IS network is composed by four production chains, α, β, γ, and δ. Let us assume that chains α and β only produce the waste l, and do not require any primary input, and chains γ and δ only require the primary input k, and do not produce any waste. The "m-to-n" configuration occurs if chains γ and δ use the waste type l as primary input k.

Case examples

As we have anticipated at the beginning of the previous section, a complex IS network may be a chain or a network of elementary IS networks. In this section, we describe several case examples of actual IS networks that better fit our classification model, based on a single one waste-one input physical flow type. [START_REF] Yang | A case study of industrial symbiosis: Nanning Sugar Co[END_REF] analyze the Nanning Sugar Co., Ltd. case, which is one of the most successfully examples of a new form of IS emerged in China, inspired by the "circular economy" philosophy. Black liquor generated during the alkaline pulping process passes through a recovery process. On the one hand, alkali recovered is reintroduced into the pulping process; on the other hand, lime sludge is used in substitution of limestone to produce cement. In this case, if we consider each physical flow type as a one waste-one input flow, we have two "one-to-one" configurations, one internal, and one external. However, if we categorize a physical flow type only based on the waste output flow, independently from the obtainable input types, this is an example of a "one-to-n" configuration, characterized by a material-material flow, and internal-external IS. [START_REF] Wolf | Industrial Symbiosis in the Swedish Forest Industry[END_REF] investigates the Swedish forest industry in the Östergötland region. Three actors are identified: the pulp mill, the saw mill, and the pellet production plant. Sawdust and wood chip by-products from the saw mill are delivered to the pulp mill to be used as biofuel. Thus, the pulp mill supplies steam and electricity to the saw mill, and to the pellet chain. As shown, this is a "one-to-n" IS network, both internal and external, with a material-energy flow, where the pulp mill operates as a conversion process.

Research Triangle Institute (1996) has developed a prototype eco-industrial park for the Brownsville (Texas, U.S.) / Matamoros (Mexico) industrial area, identifying opportunities for symbiotic exchanges among local existing companies. In particular, we focus on scrap plastic re-use. The proposed prototype suggests an IS network composed by a plastic recycler, and three other entities sending their plastic scrap: a textile company, and auto and discrete parts manufacturers. Therefore, plastic may be purchased in the form of plastic pellets by the discrete parts manufacturer, after being processed. This is a "n-to-one" configuration, with a material-material flow type, and internal-external symbiosis. [START_REF] Chertow | Assessing collective firm behavior: comparing Industrial Symbiosis with possible alternatives for individual companies in Oahu[END_REF] investigate some IS initiatives of a cluster of companies located in the Campbell Industrial Park in Honolulu, HI, on the island of Oahu. In particular, three organizations bring their used activated carbon to AES Hawaii cogeneration plant as alternative fuels: the Honolulu Board of Water Supply (BWS), Tesoro (i.e. an oil refining company), and the Kalaeloa Cogeneration Plant (i.e. an oilfired power plant). Both BWS and Tesoro pay AES Hawaii to take and burn the used carbon [START_REF] Mccann | AES Hawaii[END_REF]. We may categorize this case as a "n-to-one" configuration, with a material-energy flow type, and external IS only. [START_REF] Mirata | Industrial symbiosis networks and the contribution to environmental innovation: the case of the Landskrona industrial symbiosis programme[END_REF] study the case of the industrial town of Landskrona, located in south-western Sweden. Several sectors are included such as chemicals, waste management, metals recycling, and public infrastructures. However, if we consider the local community as a third actor, a heat cascading initiative involves other two entities: steel dust and lead battery recycling companies. Heat from lead acid battery and steel dust recyclers is used for district heating by the local community. A "n-to-one"/energy-energy/external IS occurs by tracking the heat flow.

Focusing on utility synergies, [START_REF] Van Beers | Industrial Symbiosis in the Australian minerals industry -The cases of Kwinana and Gladstone[END_REF] investigate the Kwinana Industrial Area, dominated by heavy process industries, and located in Western Australia. In particular, a "m-to-n"/material-material/internal-external IS occurs by tracking the wastewater flow. In fact, the cogeneration (i.e. Verve Energy) and the chlor-alkali (i.e. Nufarm Coogee) plants send their wastewater to the treatment plant of Tiwest, a company producing titanium dioxide. So, the potable and demineralized water returns to the plants that generate the wastewater flow. [START_REF] Korhonen | Regional industrial ecology: examples from regional economic systems of forest industry and energy supply in Finland[END_REF] analyzes the regional energy supply system of the Jyvaskyla region in Finland. The Rauhalahti CHP plant, which is a publicly owned body, receives waste wood from a plywood mill, located in the Saynatsalo suburb, and external fuel, such as peat, saw-mill and forestry waste from other actors. Thus, it produces electricity for all households, services and industry in the city, the energy produced by waste is used for district heating, and industrial steam is sent to the Kangas paper mill. This is an example of a "m-to-n" configuration, with a material-energy flow type, and external symbiosis.

Conclusions

Industrial symbiosis (IS) is emerging as one of the most effective tools to mitigate environmental impact of industrial activities, by reducing both primary resources consumption and waste emissions. However, no common classification model for IS networks exists [START_REF] Golev | Developing a classification system for regional resource synergies[END_REF]. In this paper, adopting an input-output approach, we have defined an elementary IS network based on the single physical flow type (one waste-one primary input) shared among production chains. Furthermore, we have distinguished between internal and external forms of IS. The former occurs when the waste output flow is sent and received by the same chain, the latter when the chain sending and the one receiving it are not the same. These two conditions can co-exist according to the structure of the network. Four possible configurations have been found depending on the number of production chains sending and receiving the physical flow type.

Several case examples of existing IS networks have been also provided. Focusing only on one specific physical flow type (one waste-one primary input), we have classified each case.

The proposed classification model for IS networks does not aim at being an exhaustive categorizing tool, but a basic categorization at the technical level useful for successive network analyses. It can support policy design as network identification provides guidelines for incentives in terms of transportation and environmental legacy issues, as well as of type and size of sectors to be involved in the IS network.

Acknowledgements

Financial support provided by the POR Puglia 2000-2006 (CIP_PE007 -"Metodologie innovative per lo sviluppo di mercati organizzati di servizi logistici", CIP_PS092 -"Produzione Distribuita come Sistema Innovativo -DIPIS and CIP_PE109 -"Development of an innovative model for self-evaluation of the degree of corporate social responsibility") is gratefully acknowledged.


Elias Matsas
email: imatsas@central.ntua.gr

Dimitrios Batras
email: dbatras@yahoo.gr

George-Christopher Vosniakos
email: vosniak@central.ntua.gr

Beware of the robot: a highly interactive and immersive Virtual Reality training application in robotic manufacturing systems

Keywords: Virtual Reality, Safe Human-Robot Cooperation, Manufacturing Training, Interaction, Serious Game

A highly interactive and immersive Virtual Reality Training System (VRTS) is developed, in terms of an educational serious game that simulates the cooperation between industrial robotic manipulators and humans, executing manufacturing tasks. "BeWare of the robot" application ultimately aims at studying the acceptability of human-robot collaboration, when both human and robot share the same workspace. The initial version of the application was evaluated by a group of users. Experimental results on usability and technical aspects are presented and several remarks about users' experience and behavior in the virtual world are discussed.

Introduction

In not-too-distant future manufacturing systems a need for cooperation and workspace sharing of industrial robots and humans to execute manufacturing tasks will arise. For a long time now, safety of the human interacting with industrial robots is addressed by segregation between humans and robots [START_REF] Helander | Ergonomics and Safety considerations in the design of robotics workplaces: A review and some priorities for research[END_REF]. Cooperation of human and robot may present an interesting solution that balances productivity, quality, initial capital cost, running cost and flexibility. However, all important physical and "mental" safety issues that arise must be successfully dealt with. Serious games and highly interactive and immersive Virtual or Mixed Reality training applications are preferentially deployed in such cases.

In terms of functionality, VR allows users to be extracted from physical reality in order to virtually change time, space, and (or) interaction type [START_REF] Fuchs | Le traité de la réalité virtuelle[END_REF]. Virtual Realitybased training systems (VRTSs) are advanced computer-assisted, interactive training systems using VR technology, e.g. allowing proper test and operation of new equip-ment before it is actually installed [START_REF] Lin | Developing virtual environments for industrial training[END_REF]. They need to have the necessary physical fidelity to mimic the resolution of the physical world in order to be effective for tasks that are characterized by a significant perceptual and/or motor component [START_REF] Nathanael | Cognitive task analysis for Virtual Reality Training: the case of CNC tool offsetting[END_REF]. Shukla [START_REF] Ch | Virtual Manufacturing: An overview[END_REF] points out the significance of VR in manufacturing training. All VRTSs can be decomposed into three distinct functional parts: (i) output devices, (ii) input devices (mainly for interaction), and, (iii) a VR engine including virtual scene, interaction models, and a graphical representation of the user (avatar) [START_REF] Burkhardt | Conception d'un système de RV pour la formation des agents de conduites aux opérations en milieu ferroviaire[END_REF].

In the literature, [START_REF] Pook | Deictic human/robot interaction[END_REF] first propose a deictic mode of sensory-motor control with gestures and a HMD for teleassistance, that can be useful for human/robot interaction and control. Morioka [START_REF] Morioka | A new cell production assembly system with human-robot cooperation[END_REF] developed a new cell production assembly system, in which physical and information supports are provided to the human operators to double productivity. Krüger [START_REF] Krüger | Cooperation of human and machines in assembly lines[END_REF] gives a survey about human-robot cooperation and all available technologies that support the cooperation. Oborski [START_REF] Oborski | Man-machine interactions in advanced manufacturing systems[END_REF] points outs the most important factors of man-machine interaction in advanced manufacturing systems and proposes a method of human attributes modeling for human-machine (robot) interaction. Corrales [START_REF] Corrales | Safe human-robot interaction based on dynamic sphere-swept line bounding volumes[END_REF] presents the implementation of real-time proximity queries between humans and robotic manipulators. From another point of view, [START_REF] Arai | Assessment of operator stress induced by robot collaboration in assembly[END_REF] deal with mental strains of human operators in human-robot cell production systems and propose metrics for a physiologically comfortable collaboration.

In this paper a serious game application named "BeWare of the robot" is presented, simulating tasks of human-robot collaborative tape laying for building aerospace fabric reinforced composite parts. Profiled fabric layers (patches/cloths) are stacked successively inside a die, one on top of the other, until the desired thickness is reached. Technical aspects of the application development are explained centering on safety issues, such as contacts and collisions that are tackled through "emergencies", i.e. visual stimuli and sound alarms. Preliminary evaluation results by a group of users are presented. Mental safety, i.e. human's awareness of the robot's motion, is the primary objective.

The Virtual Reality Training System Description and Analysis

Shop-floor environment and its components were developed using Rhinoceros™ and 3ds Max™ for 3D part design and mesh creation. Unity 3d™ game engine was used for assembly, rendering, lighting, physics, simulation, building compilation and programming. "BeWare of the robot" VRTS is a PC standalone build that can run in a typical PC with Windows XP™ and an nVidia™ graphics card supporting 3D stereoscopic vision. An immersive device (HMD) supporting head tracking and a Kinect™ sensor are needed to fully exploit navigation, tracking and immersion capabilities of the system. Communication between Unity 3d and the Kinect™ sensor is implemented with the OpenNI framework. The VRTS incorporates (i) the virtual model of a composites hand layout work-cell, (ii) the model of a Staübli RX90L robotic manipulator, (iii) the skinned model of an avatar created online in evolver.com, with a 3ds Max™ biped attached to it, (iv) interaction scripts in C#, (v) real-time shadows and lighting, and (vi) image, video and audio textures from the real working space. Interaction is mainly based on collision and ray-casting scripts, child/parenting functions, and skeletal tracking of 18 avatar's points. In order to describe, analyze and decompose the system, Object-Oriented Analysis (OOA) tools and UML 2.0 expressions are used. OOA methods divide the system to its components and make every relation explicit [START_REF] Deacon | Object-Oriented Analysis and Design[END_REF]. The Use Case Diagram presented in Fig. 2 defines the required tasks and interactions between the user and the system. The Activity Diagram is a graphical representation of system behavior: flow of actions, activities and transitions. The Structure Diagram provides a detailed analysis of system entities, their attributes, the methods used and the associations between the entities as shown in Fig. 3. OOA tools in VEs can be used as a path for the creation and/or the improvement of the virtual world. In addition, through OOA methods, system knowledge can be captured to build intelligence.

Experiment Set-Up and Results

To evaluate both the effectiveness and the usability of the application, as well as to investigate users' experience, a series of experiments were carried out. The participants of the experiment are a group of 30 senior mechanical engineering students (aged between 21 and 31 years), sufficiently familiar with theoretical Robotics and Manufacturing Systems design. None of the students had tested the application before the experiments.

The experimental task required subjects to individually test the "beware of the robot" application; that is to virtually collaborate in sequence with an industrial robot, while sharing the same physical environment. The experimental goal was to test/validate the following hypothesis: ─ Highly-interactive and immersive VRTSs may be a successful tool for motor skill training, such as human-robot collaboration testing. ─ Immersion and head tracking the user experiences with the use of the HMD boosts the feel of presence and realism. Manufacturing education is therefore enhanced when physical access to equipment is either not possible or potentially hazardous. ─ The use of additional audiovisual stimuli (e.g. HD video textures and 3d spatialized sounds from the real environment, real-time shadows and differed lighting) enhances user's experience of presence in the virtual world.

The testing platform consisted of: the "beWare of the robot" application, a PC running Windows XP™ equipped with an nVidia Quadro™ FX1700 graphics card, an eMagin Z800 HMD with stereo ear buds, a Microsoft Kinect™ sensor, keyboard and mouse. Furthermore, in order to be able to reproduce what the users sees through the HMD and to record the experiments with a video camera, we used a projector cloning the displays of the HMD in a wall behind the user.

Experiment's duration was 10 minutes and participants were given detailed instructions orally. After wearing the HMD, subjects were asked to stand in "Y" posture towards the Kinect sensor in order to be detected and calibrated. They, then, had to "discover" system's tracking, moving and rotating capabilities, as well as the objects of the virtual environment. Afterwards, participants had about 6-8 minutes to complete the tasks described in the Use Case Diagram (Fig. 2).

At the end of the experiment, each participant was asked to fill out and submit the online form (questionnaire). This was developed from scratch in order to examine user experience as well as system effectiveness. The questionnaire consisted of 42 questions, grouped into 3 thematic categories (i) participant's personal information, level and experience (14 questions), (ii) immersion, presence and realism (10 questions), (iii) usability, effectiveness, tracking and interaction quality, (18 questions).

Analysis was conducted on three levels: (i) online questionnaires, (ii) task execution video recording and a posteriori observation, and (iii) discussions with each participant.

Results and Discussions

Presence and Immersion. Table 1 shows the results of selected questions concerning users' experience in the VRTS. Some interesting findings rise from this table regarding the concepts of presence and immersion. Slater et al. [START_REF] Slater | A framework for immersive virtual environments (FIVE): Speculations on the role of presence in virtual environments[END_REF] define presence as a subjective phenomenon such as the sensation of being in a VE, while immersion as an objective description of aspects of the system such as field of view and display resolution. In the literature, several theories have been proposed on the nature of presence in immersive VR [START_REF] Schuemie | Research on Presence in Virtual Reality: A Survey[END_REF]. Our research focuses more on "presence by involvement"; that is, that both involvement and immersion are thought to be necessary for experiencing presence. Moreover, [START_REF] Prothero | Towards a robust, quantitative measure for presence[END_REF] claim that "presence" and "situation awareness" are overlapping constructs.

In our experiment, a vast majority of participants were feeling as they were really moving in the scene, "involved and present" in the virtual activity. In addition, 93% of the subjects answered that they did not lose their concentration at all during the test. The results also show that a large number of subjects felt like they were really moving an object with their hands, despite the fact that the object did not have physical mass. A posteriori video observation emphasizes the above finding with an intrinsic proprioceptive users' reaction: although subjects were told that they should use one hand, 20% of them used both hands in order to grasp and to carry the workpieces. We also noticed that some users (17%) spontaneously closed their hands and/or fingers in order to grasp the parts (as they would have reacted in the real world), although they were told that our system does not support fingers tracking. Usability and System effectiveness. Although 77% of the subjects had never used the Kinect™ sensor before, only 10% of them encountered some difficulties during the initial detection and calibration process. During the human-robot collaboration procedure, almost all subjects (97%) replied that they easily perceived the red transparent sphere that represented the robot's workspace, and 70% of subjects managed not to enter into the workspace (which was the potentially hazardous area). The survey results show that most of the subjects favorably accepted the use of visual and auditory stimuli (alarms) for the robot's workspace awareness. Note that the red sphere and visual alarms in general may be contradictory with fidelity and realism of VEs, but realism is not always the ultimate goal. In VRTSs learning procedure for example, exaggeration or deformation of real events is "authorized" for complex situations understanding [START_REF] Burkhardt | Conception d'un système de RV pour la formation des agents de conduites aux opérations en milieu ferroviaire[END_REF].

Concerning the Kinect's™ tracking quality, most of the participants considered that the avatar was following their (tracked) movements precisely, 23% of them, though, pointed out a slight "vibration" in the avatar's movements. On the other hand, observations revealed that 53% of the users sometimes "lost control" of their hands, even instantaneously. That is because experiment tasks required user body rotation towards the sensor, and at the extremes hands tracking is confusing: Kinect™ sensor cannot easily distinguish the left from the right hand. In addition when users turn their body at 180 degrees towards the Kinect sensor, skeletal tracking is lost and although they may continue moving, the avatar remains "frozen" for a while in the last detected posture. Furthermore, 43% of the subjects felt like their movements were altered compared with the real ones; with emphasis in hands "roll" rotation (50%) and body rotation (47%). Nevertheless, 70% of the subjects answered that they easily managed to pick-up the workpieces from the robot's end-effector, while 50% said that it was easy to navigate (walk, rotate, bend etc.) in the virtual world.

Concerning user motivation, 76% of the participants replied that during the experiment they were feeling more as if they were participating in an amusing game, and 90% of the subjects answered that training tasks requiring human-robot collaboration can be more attracting with the use of "serious games".

Eventually, the results suggest a positive prospect for the use of VR for training. In addition, moving around the scene and using real gestures (skeletal tracking) to manipulate objects and to complete the tasks makes learning more active and impressive for learners.

Conclusion and Future Work

Due to the notions of presence and realism VRTSs and highly-interactive serious games are probably an efficient and safe learning tool to study human-robot collaboration with overlapping workspaces. Through multi-level "invisible" interactions with the Kinect™ sensor skeletal tracking, user experience and behavior in the virtual world can be recorded, studied and analyzed, leading to new knowledge acquisition.

Experiment questionnaire analysis permitted to identify several remarks about user experience in the VE, and helped us pose the question of how virtual world affects user temporal identity in the VE, which will be tackled in an upcoming study.

Moreover, a Mixed Reality version of the application where the user would interact physically with real objects while being tracked and immersed in the virtual world is expected in the near future. Authors are also working on solving the skeletal tracking problems with the use of a second Kinect™ sensor.

Fig. 1 .

 1 Fig. 1. Avatar's hands trying to pick-up the part from the robot (left) and user trying to put the part in the metallic die (right).

Fig. 2 .

 2 Fig. 2. UML Use Case Diagram and Activity Diagram

Fig. 3 .

 3 Fig. 3. OOA Structure Diagram showing entities, attributes and associations.

Table 1 .

 1 Users' experience selected results.

Acknowledgements. This research has been co-financed by the European Union (European Social Fund-ESF) and Greek national funds through the Operational Program "Education and Lifelong Learning" of the NSRF -Research Funding Program: Heracleitus II, Investing in knowledge society through the European Social Fund.

We wish to thank the participants in the experiments for sharing their experience with us, as well as Dr. Dimitrios Nathanael for his helpful comments concerning the experiment set-up and the questionnaire. We would also like to thank Prof. Simon Richir and Matthieu Lépine for accepting and hosting our application in the exhibition section of Laval Virtual 2012 conference.


Wei Li

Mika Fukunishi

Benjamin J Morgan

Olaf J Borkiewicz

Karena W Chapman

Va- Lérie Pralong

Antoine Maignan

Oleg I Lebedev

Jiwei Ma

Henri Groult

Shinichi Komaba

Damien Dambournet

Anatase TiO2 is a potential negative electrode for sodium-ion batteries. The sodium storage mechanism is, however, still under debate, yet its comprehension is required to optimize the electrochemical properties. To clarify the sodium storage mechanism occurring in anatase, we have used both electrochemical and chemical routes from which we obtained similar trends. During the first discharge, an irreversible plateau region is observed which corresponds to the insertion of Na + within the interstitial sites of anatase and is accompanied by a drastic loss of the long-range order as revealed by x-ray diffraction, high resolution of high angle annular dark field scanning transmission electron microscope (HAADF-STEM) and pair distribution function (PDF) analysis. Further structural analysis of the total scattering data indicates that the sodiated phase displays a layered-like rhombohedral R-3m structure built from the stacking of Ti and Na slabs. Because of the initial 3D network of anatase, the reduced phase shows strong disorder due to cationic inter-mixing between the Ti and Na slabs and the refined chemical formula is (Na0.43Ti0.57)3a(0.22Na0.39Ti0.39)3bO2 where  refers to vacancy. The presence of high valence Ti ions in the Na layers induces a contraction of the c-parameter as compared to the ordered phase. Upon de-sodiation, the structure further amorphized and the local structure probed by PDF is shown to be similar to the anatase TiO2 suggesting that the 3D network is recovered. The reversible sodium insertion/de-insertion is thus attributed to the rhombohedral active phase formed during the first discharge, and an oxidized phase featuring the local structure of anatase. Due to the amorphous nature of the two phases, the potential-composition curves are characterized by a sloping curve. Finally, a comparison between the intercalation of lithium and sodium into anatase TiO2 performed by DFT calculations confirmed that for the sodiated phase, the rhombohedral structure is more stable than the tetragonal phase observed during the lithiation of nanoparticles.

In many areas of modern life, lithium-ion batteries are ubiquitous as energy-storage solutions. The growing demand for higher energy density and lower cost of electrochemical energy storage devices, however, has motivated a search for auxiliary technologies based on alternative chemistries. [START_REF] Dunn | Electrical Energy Storage for the Grid: A Battery of Choices[END_REF][START_REF] Larcher | Towards Greener and More Sustainable Batteries for Electrical Energy Storage[END_REF] One possible candidate is the sodium-ion battery, which is attractive because of the high earthabundance of sodium, and lower cost versus lithium-ion batteries, due to compatibility with aluminum as the anodic current collector. [START_REF] Yabuuchi | Research Development on Sodium-Ion Batteries[END_REF][START_REF] Pan | Room-Temperature Stationary Sodium-Ion Batteries for Large-Scale Electric Energy Storage[END_REF][START_REF] Kundu | The Emerging Chemistry of Sodium Ion Batteries for Electrochemical Energy Storage[END_REF] Development of sodium-ion batteries has been largely stimulated by knowledge of lithiumion analogues. The intercalation of Na + or Li + ions into a host lattice can, however, give qualitatively different voltage profiles, corresponding to different intercalation mechanisms. For example, lithium insertion in Li4Ti5O12 is accompanied by a spinel to rock-salt phase transition. [START_REF] Rousse | Rationalization of Intercalation Potential and Redox Mechanism for A2Ti3O7 (A = Li, Na)[END_REF][START_REF] Colbow | 4th International Meetings on Lithium BatteriesStructure and Electrochemistry of the Spinel Oxides LiTi2O4 and Li43Ti53O4[END_REF] The equivalent sodium insertion, however, proceeds via a complex three-phase-separation mechanism (spinel to two rock-salt phases of Li7Ti5O12 and Na6LiTi5O12). [START_REF] Scharner | Evidence of Two-Phase Formation upon Lithium Insertion into the Li1.33Ti1.67O4[END_REF] Such differences in intercalation behaviour can often be attributed to different properties of Li versus Na, such as ionic radius and polarizability. [START_REF] Sun | Direct Atomic-Scale Confirmation of Three-Phase Storage Mechanism in Li4Ti5O12 Anodes for Room-Temperature Sodium-Ion Batteries[END_REF][START_REF] Senguttuvan | Titanium(III) Sulfate as New Negative Electrode for Sodium-Ion Batteries[END_REF] In general, however, the performance of electrodes in sodium-ion batteries cannot be understood by simply extrapolating from their behaviour versus lithium, when it is necessary to carefully reexamine the sodium-intercalation behaviour.

In the search for new sodium-ion anodes, many titanium-based compounds, with diverse structures and chemistries, have been considered. These include titanium sulfate, [START_REF] Shirpour | New Materials Based on a Layered Sodium Titanate for Dual Electrochemical Na and Li Intercalation Systems[END_REF] phosphate, [START_REF] Wang | A Zero-Strain Layered Metal Oxide as the Negative Electrode for Long-Life Sodium-Ion Batteries[END_REF] sodium/lithium titanate, [START_REF] Rousse | Rationalization of Intercalation Potential and Redox Mechanism for A2Ti3O7 (A = Li, Na)[END_REF][START_REF] Shirpour | New Materials Based on a Layered Sodium Titanate for Dual Electrochemical Na and Li Intercalation Systems[END_REF][START_REF] Wang | A Zero-Strain Layered Metal Oxide as the Negative Electrode for Long-Life Sodium-Ion Batteries[END_REF][START_REF] Yu | An Ultrastable Anode for Long-Life Room-Temperature Sodium-Ion Batteries[END_REF][START_REF] Wu | Na-TiO2: A Layered Anode Material for Sodium-Ion Batteries[END_REF][START_REF] Kim | Anatase Titania Nanorods as an Intercalation Anode Material for Rechargeable Sodium Batteries[END_REF] and titanium oxides [START_REF] Kim | Anatase Titania Nanorods as an Intercalation Anode Material for Rechargeable Sodium Batteries[END_REF][START_REF] Xu | Nanocrystalline Anatase TiO2: A New Anode Material for Rechargeable Sodium Ion Batteries[END_REF][START_REF] Su | Anatase TiO2: Better Anode Material Than Amorphous and Rutile Phases of TiO2 for Na-Ion Batteries[END_REF][START_REF] Xiong | Amorphous TiO2 Nanotube Anode for Rechargeable Sodium Ion Batteries[END_REF][START_REF] Kavan | Lithium Insertion into TiO2[END_REF] . Of these, anatase TiO2 is attractive because of easy processing of nanoparticles and extensive knowledge regarding its intercalation properties versus lithium. [START_REF] Ren | Lithium Intercalation into Mesoporous Anatase with an Ordered 3D Pore Structure[END_REF][START_REF] Sudant | Electrochemical Lithium Reactivity with Nanotextured Anatase-Type TiO2[END_REF][START_REF] Wagemaker | Properties and Promises of Nanosized Insertion Materials for Li-Ion Batteries[END_REF][START_REF] Shen | Impact of Particle Size on the Non-Equilibrium Phase Transition of Lithium-Inserted Anatase TiO2[END_REF][START_REF] Wu | Anatase TiO2 Nanoparticles for High Power Sodium-Ion Anodes[END_REF] The electrochemical properties of anatase TiO2 versus Na offer an appealing working voltage of 0.8 V, and researchers have already demonstrated high-rate capability, suggesting potential use in commercial cells. [START_REF] Tahir | Extraordinary Performance of Carbon-Coated Anatase TiO2 as Sodium-Ion Anode[END_REF][START_REF] Ni | Self-Supported Nanotube Arrays of Sulfur-Doped TiO2 Enabling Ultrastable and Robust Sodium Storage[END_REF][START_REF] Hwang | Ultrafast Sodium Storage in Anatase TiO2 Nanoparticles Embedded on Carbon Nanotubes[END_REF][START_REF] Wu | Unfolding the Mechanism of Sodium Insertion in Anatase TiO2 Nanoparticles[END_REF] Optimising the electrochemical properties of anatase TiO2 for use in sodium-ion batteries requires a comprehensive understanding of the sodium intercalation mechanism, but this is still unclear despite previous studies. To the best of our knowledge, early studies on the sodium storage mechanism of TiO2 could not conclusively identify the phases involved during the sodiation/de-sodiation reactions. Kim et al. have proposed that the anatase framework is maintained during the sodium intercalation, i.e., via a solid-solution mechanism. [START_REF] Xu | Nanocrystalline Anatase TiO2: A New Anode Material for Rechargeable Sodium Ion Batteries[END_REF] Nevertheless, the ex-situ x-ray diffraction data published in support of this proposal did not show the main line of the tetragonal phase, which therefore cannot be conclusive. Subsequently, Wu et al., performed in-situ diffraction analysis showing that anatase amorphizes during the first discharge, yielding an amorphous electrochemically-active NaxTiO2 phase. [START_REF] Wu | Unfolding the Mechanism of Sodium Insertion in Anatase TiO2 Nanoparticles[END_REF] Louvain et al. have further confirmed the formation of an amorphous NaxTiO2 like phase using operando Raman spectroscopy. [START_REF] Louvain | On the Electrochemical Encounter between Sodium and Mesoporous Anatase TiO2 as a Na-Ion Electrode[END_REF] These studies point out the need to clarify the structure of the newly formed amorphous NaxTiO2 and to identify the sodium storage mechanism occurring in anatase .

To resolve the sodium storage mechanism occurring in anatase TiO2, we have used electrochemical and chemical routes to prepare reduced compounds, for which we obtained structural insights by combining high-energy synchrotron-based total scattering measurements, high resolution HAADF-STEM, magnetic measurements, and DFT calculations. Our analysis shows that intercalation of Na into anatase TiO2 proceeds by a different mechanism to Li intercalation. Where LiTiO2 maintains the pattern of edgesharing TiO6 octahedra of the anatase structure [START_REF] Borghols | The Electronic Structure and Ionic Diffusion of Nanoscale LiTiO2 Anatase[END_REF] , Na intercalation produces a rhombohedral R-3m structure, accompanied by a decrease of long-range order. We also find that further electrochemical cycling proceeds via a phase transition between an amorphous anatase-like TiO2 phase and disordered layered-like NaxTiO2 phase.

Synthesis and characterizations. Hydroxyfluorinated anatase was prepared by solvothermal synthesis according to reference [START_REF] Li | High Substitution Rate in TiO 2 Anatase Nanoparticles with Cationic Vacancies for Fast Lithium Storage[END_REF] . Pure anatase TiO2 nanoparticles was obtained by thermal treatment at 450 ºC under air for 4 hours. Powder X-ray diffraction (XRD) analysis was carried out using a Rigaku Ultima IV X-ray diffractometer equipped with a Cu Kα radiation source (λ = 1.54059 Å). Transmission Electron Microscopy (TEM) analysis was performed using a JEM ARM200F FEG double aberration corrected microscope operated at 200 kV, equipped with CENTURIO EDX detector and GIF Quantum. The sample was prepared in a glove box by suspending the powder in methanol and deposited it on holey carbon Cu grid. Afterwards the grid was immediately transport inside the microscope.

Electrochemical measurements. Electrodes were prepared by thoroughly mixing 80 wt. % active material, 10 wt. % black acetylene as the conductive agent and 10 wt. % sodium carboxymethyl cellulose [START_REF] Dahbi | Sodium Carboxymethyl Cellulose as a Potential Binder for Hard-Carbon Negative Electrodes in Sodium-Ion Batteries[END_REF] (CMC) as the binder. Then, the mixture was dispersed into a diluted H2O2 aqueous solution (H2O2 : H2O = 2.5 : 97.5 v/v %) and coated on an aluminum foil. The electrodes were dried at 80 °C under air for 6 hours and under primary vacuum for 12 hours. Sodium half cells were assembled inside a glove box filled with Ar by using sodium foil as the negative electrode. A solution of 1 M NaPF6 dissolved in ethylene carbonate and diethyl carbonate (EC:DEC) was used as the electrolyte. A sandwich-like film composed of glass filter in the two sides and polyolefin in the middle wasused as the separator. The Na cells were cycled between 0.0-2.0 V at a current density of 25 mA g -1 using galvanostatic discharge-charge testers (TOSCAT-3100, Toyo System Co. Ltd.). For ex-situ measurements, the cells were stopped at different state of discharge/charge, opened in a glove box. The electrodes were washed with EC:DEC and DEC, scrapped from the current collector and stored in a glove box filled with Ar for further characterizations.

Synchrotron diffraction. Samples were loaded inside Kapton capillaries. Data were collected at the 11-ID-B beamline at the Advanced Photon Source at Argonne National Laboratory, using high energy X-rays (λ = 0.2128 Å) allowing access to high values of momentum transfer. [START_REF] Chupas | Rapid-Acquisition Pair Distribution Function (RA-PDF) Analysis[END_REF][START_REF] Chupas | Applications of an Amorphous Silicon-Based Area Detector for High-Resolution, High-Sensitivity and Fast Time-Resolved Pair Distribution Function Measurements[END_REF] One-dimensional diffraction data were obtained by integrating the raw 2D total scattering data in Fit2D. [START_REF] Hammersley | Two-Dimensional Detector Software: From Real Detector to Idealised Image or Two-Theta Scan[END_REF] PDFs, G(r), were extracted from the background and Compton scattering corrected data following Fourier transformation using PDFgetX2. [START_REF] Qiu | PDFgetX2: A GUI-Driven Program to Obtain the Pair Distribution Function from X-Ray Powder Diffraction Data[END_REF] The Qmax was 22 Å. The PDFs were subsequently modelled using PDFgui. [START_REF] Farrow | PDFfit2 and PDFgui: Computer Programs for Studying Nanostructure in Crystals[END_REF] Chemical sodiation was performed using sodium naphtalenide as reducing agent. Anatase TiO2 nanoparticles (200 mg) were dispersed in 20 mL of THF. Naphtalene (0.5 g) and small pieces of Na metal (170 mg) were added to the mixture and stirred in a glove box during 6 days at room temperature. Finally, the reduced phase was washed with THF and dry under vacuum.

Magnetic properties. The magnetic susceptibility χ as a function of temperature was obtained from magnetic moment measurements performed by squid magnetometry with a MPMS-5T from Quantum Design. The data were collected upon warming from 5K to 290K in a 100 Oe applied magnetic field in zero-field-cooling (zfc) and fieldcooling (fc) processes. A diamagnetic correction was applied to χmol. Additional isothermal magnetic moment measurements were made as a function of the applied magnetic field H. In order to avoid air exposure, the preparation of the sample was made in a glove box. About 0.1 g of the sample powder was set within a gelatine capsule.

Computational. Density functional theory (DFT) calculations were performed using the plane-wave code VASP [START_REF] Kresse | Norm-Conserving and Ultrasoft Pseudopotentials for First-Row and Transition Elements[END_REF][START_REF] Kresse | Efficiency of Ab-Initio Total Energy Calculations for Metals and Semiconductors Using a Plane-Wave Basis Set[END_REF] with valence electrons described by a plane-wave basis with a cutoff of 500 eV. Interactions between core and valence electrons were described with the PAW method [START_REF] Kresse | From Ultrasoft Pseudopotentials to the Projector Augmented-Wave Method[END_REF] , with cores of [Mg] for Ti, [He] for O, and [Ne] for Na. The calculations used the revised Perdew-Burke-Ernzerhof generalized gradient approximation PBEsol [START_REF] Perdew | Restoring the Density-Gradient Expansion for Exchange in Solids and Surfaces[END_REF] , supplemented with a Dudarev +U correction applied to the Ti d states (GGA+U). We used a value of UTi,d = 4.2 eV, which has previously been used to model lithium intercalation in anatase TiO2 and in TiO2-B [START_REF] Morgan | GGA+U Description of Lithium Intercalation into Anatase TiO2[END_REF][START_REF] Morgan | Role of Lithium Ordering in the LixTiO2 Anatase → Titanate Phase Transition[END_REF][START_REF] Morgan | Lithium Intercalation into TiO2(B): A Comparison of LDA, GGA, and GGA+U Density Functional Calculations[END_REF] . All calculations were spin-polarized.

To model dilute Na-intercalated anatase TiO2 we considered a single interstitial Na ion in a 4×4×2 supercell (384 atoms). Optimised lattice parameters were obtained by performing a series of constant-volume calculations for a stoichiometric anatase Ti4O8 cell, with the resulting volume-energy data fitted to the Murnaghan equation of state. These calculations were considered geometry-optimized when all atomic forces were smaller than 0.01 eV Å - 1 . This procedure gives zero-pressure lattice parameters of a = 3.8495 Å and c = 9.5966 Å. These calculations used a 2×2×2 Monkhorst-Pack k-point mesh.

To model the fully sodiated phase we considered two structures for stoichiometric NaTiO2: the rhombohedral O3-type layered structure [START_REF] Clarke | Structure, and Magnetic Properties of NaTiO2[END_REF] (space group R-3m) identified in our experiments, as discussed below, and the tetragonal "LiTiO2" structure (space group I41/amd) observed by Wagemaker et al. for Li-intercalated anatase nanoparticles. [START_REF] Morgan | Lithium Intercalation into TiO2(B): A Comparison of LDA, GGA, and GGA+U Density Functional Calculations[END_REF] To compare the intercalation behaviour for sodium versus lithium, we performed calculations on both structures for compositions of NaTiO2 and LiTiO2. Calculations for the rhombohedral layered structure used a primitive cell (12 atoms) and a 4×4×1 Monkhorst-Pack k-point mesh. Calculations for the tetragonal "LiTiO2" structure used a 2×1×1 supercell and a 2×4×2 Monkhorst-Pack k-point mesh. In each case starting structures were taken from the literature, [START_REF] Borghols | The Electronic Structure and Ionic Diffusion of Nanoscale LiTiO2 Anatase[END_REF][START_REF] Clarke | Structure, and Magnetic Properties of NaTiO2[END_REF] and full volume and geometry optimisations were performed, with the same convergence criteria as for dilute NaxTiO2.

To calculate sodium and lithium intercalation energies, reference calculations for metallic Na and Li were performed for 2-atom cells, using the same convergence criteria as above, and with a 16×16×16 Monkhorst-Pack grid for k-space sampling.

Electrochemistry. The electrochemical reaction of anatase TiO2 nanoparticles (20 nm) with sodium was investigated using galvanostatic discharge-charge experiments. Details about material synthesis, characterizations and electrode preparation and testing can be found in the experimental section and Supporting Information. Figure 1 displays the galvanostatic discharge-charge curves of the anatase TiO2 (20 nm) electorde in Na half-cell tested between 0-2 V at a current density of 10 mA g -1 . During the first discharge, the potential of the cell decreases rapidly from the open circuit voltage to 0.3 V, delivering a capacity of ~100 mAh/g, which corresponds to 0.3 Na + per formula unit, which should include the formation of passivation layer. [START_REF] Komaba | Electrochemical Na Insertion and Solid Electrolyte Interphase for Hard-Carbon Electrodes and Application to Na-Ion Batteries[END_REF][START_REF] Dahbi | Effect of Hexafluorophosphate and Fluoroethylene Carbonate on Electrochemical Performance and the Surface Layer of Hard Carbon for Sodium-Ion Batteries[END_REF] Afterwards, an irreversible plateau region was observed at ~0.2 V and delivered a capacity of ~280 mAh/g, corresponding to 0.84 Na + per TiO2. The first discharge capacity was 380 mAh/g, exceeding the theoretical one, i.e. 335 mAh g -1 based on the Ti 4+ /Ti 3+ redox couple. The subsequent discharge-charge curves featuring a sloping profile and an average working voltage of ca. 0.8 V, differed from the first discharge indicating that an irreversible structural change occurring during the plateau region. Upon cycling, the Na-cell presents a reversible insertion/de-insertion of ca. 0.5 Na + per TiO2. A capacity of 165 mAh/g was obtained after 50 cycles demonstrating that the activation of the electrode enabled reversible sodium uptake and release (Figure S2 in Supporting Information). Thus, the nature of the structural changes occurring during the first discharge is key to understand the reversible sodium storage mechanism.

Sodium storage mechanism.

The structural changes associated with the electrochemical reaction with sodium upon the first discharge were investigated by high-energy x-ray diffraction. After full discharge, peaks from an anatase phase disappear and new broad Bragg peaks appear, indicating that sodium insertion induces a transition to a phase with decreased long-range order. The x-ray diffraction pattern of the discharged electrode was indexed with an O3-type NaxTiO2 rhombohedral structure (space group: R-3m) which captures the major features of the experimental data (Figure S3). [START_REF] Clarke | Structure, and Magnetic Properties of NaTiO2[END_REF] Note that the "O3" nomenclature referred to Delmas' notation indicating that the Na ions are octahedrally coordinated by oxygen and display a repetition period of three transition metal stacking along the c-axis. [START_REF] Delmas | Structural Classification and Properties of the Layered Oxides[END_REF] The structure of the electrochemically-sodiated NaxTiO2 phase was investigated through Pair Distribution Function (PDF) G(r), which was obtained from Fourier transformed high-energy x-ray total scattering data. By providing realspace structural information that is independent of the long-range order, PDF is a powerful tool to investigate the structure of amorphous and nanostructured materials. [START_REF] Billinge | Beyond Crystallography: The Study of Disorder, Nanocrystallinity and Crystallographically Challenged Materials with Pair Distribution Functions[END_REF] In keeping with x-ray diffraction analysis, the PDF was modeled using the O3-type NaxTiO2 rhombohedral type structure (space group: R-3m) which consists of a layered compound featuring sodium and titanium slabs with edgeshared (Ti/Na)O6 octahedra. In its ordered form (Inset Figure 2), Na and Ti occupy the 3a and 3b Wyckoff sites, respectively. Nevertheless, the observed transition from a tridimensional toward a layered like structure suggests a cation intermixing between the two slabs. Moreover, after the electrochemical sodiation of 0.3 Na + per TiO2, PDF refinement of the partially sodiated electrode revealed minor structural changes (Figure S4) suggesting that the region observed prior to the plateau region arises from the decomposition of the electrolyte in agreement with another report. [START_REF] Tahir | Extraordinary Performance of Carbon-Coated Anatase TiO2 as Sodium-Ion Anode[END_REF] Therefore, the first discharge corresponds to the reaction of 0.84 Na + per TiO2. Accordingly, we considered a structural model involving non-stoichiometry and cationic inter-mixing. The PDF data was fitted as follows: (i) the rate occupancy of the 3a and 3b cation sites was refined, (ii) we considered an intermixing of Na and Ti atoms in Na/Ti slabs according to (NaxTiy)3a(NazTiα)3bO2. The final refinement is shown in Figure 2. Table 1 gathered the structural parameters obtained after the refinement. A good fit for the r region beyond 4 Å was obtained with a reliability factor of Rw = 26.5% (4 -30 Å). Nevertheless, the PDF features in the local region, i.e. most notably from 1 to 4 Å but this extent up to 7 Å, cannot be well captured by the rhombohedral structure (Rw = 41.4% (1 -30 Å), indicating strong local disorder due to cation inter-mixing. The refinement of the 3a site (Na slabs) indicates full occupancy and strong disorder with 43% of Na and 57% of Ti. The 3b site is partially occupied with 39% of Na and 39% of Ti. The chemical formula of the resulting phase is (Na0.43Ti0.57)3a(0.22Na0.39Ti0.39)3bO2 where  refers to vacancy. Despite large standard deviation on the site occupancy, the number of Na deduced by the refinement agrees with the capacity obtained from the plateau region in Figure 1. As seen in Table 1, the unit cell parameters of (Na0.43Ti0.57)3a(0.22Na0.39Ti0.39)3bO2 are significantly lower than those of the ordered phase NaTiO2. In particular, the c-parameters are 15.07 and 16.24 Å for (Na0.43Ti0.57)3a(0.22Na0.39Ti0.39)3bO2 and NaTiO2, respectively. The strong contraction along the c-axis is consistent with the presence of high valence Ti IV ions in the Na layers. [START_REF] Maazaz | A Study of the NaxTiO2 System by Electrochemical Deintercalation[END_REF] To better understand the above structural analysis of the sodiated anatase, a scheme highlighting the structural relationship between the pristine tetragonal and sodiated rhombohedral phases is shown in Figure 2 (bottom). Along the (100) direction, the filling of the interstitial sites by sodium produces a structure that can be viewed as a layered compound, where slabs are equally occupied by Ti and Na atoms. This layered-type structure appears to be highly distorted which explained the large standard deviation obtained on the site occupancy and the loss of longrange order. Table 1. Structural parameters obtained by PDF refinement of the discharged electrode. For comparison purpose, the structural parameters of the ordered NaTiO2 phase [START_REF] Clarke | Structure, and Magnetic Properties of NaTiO2[END_REF] were added. Upon sodiation, the first peak at 1.941 Å (determined using Gaussian peak fitting), corresponding to Ti-O bonds, shifted to higher r-values with contributions at 1.937, 2.083 and 2.366 Å assigned to Ti 4+ -O, Ti 3+ -O and Na-O distances, respectively. This further supports that intercalation of sodium in anatase implies the Ti 4+ /Ti 3+ redox couple.

Upon de-sodiation, the peak associated with Na-O distance decreases in intensity indicating Na + de-insertion with a concomitant shift of the Ti-O bonds to lower r-value in agreement with the oxidation of Ti 3+ to Ti 4+ . Gaussian peak fitting of the Ti-O correlation indicates a remaining contribution of Ti 3+ -O bonds emphasizing an incomplete sodium extraction which agrees with the charge capacity corresponding to 0.6 Na + per formula extracted from the discharged electrode (Figure 1).

A comparison of the r-region >5 Å of the pristine and discharged/charged electrodes clearly demonstrates the dramatic loss of long-range order following the electrochemical reaction. While the pristine anatase appears to be effectively bulk, the ordered domains are reduced to 4 nm following sodium insertion. Strikingly, after sodium de-insertion, the length-scale further reduces to 2-3 nm (Figure 3). The anatase framework presents two types of Ti-Ti distances located at 3.04 Å (labeled A in Figure 3) and 3.79 Å (labeled B in Figure 3) corresponding to edge-and cornersharing TiO6 octahedra, respectively. Such features are partially recovered after de-sodiation indicating that the oxidized phase presents the short-range order of anatase. High-energy X-ray data of the charged sample showed a small peak which can be assigned to the (101) of the anatase type structure further supporting the recovery of the framework (Figure S5). Attempt to refine the PDF data of the charged electrode revealed that the recovered anatase is highly amorphous with a coherence length of less than 1 nm (Figure S6). Based on our results, the following mechanism accounting for the sodium storage in anatase can be proposed. Upon sodium insertion, the anatase type structure is converted into a layered like phase accompanied by a dramatic loss of long-range order due to strong cationic inter-mixing between the Ti and Na slabs. The first discharge corresponded to an activation step yielding an electroactive phase. According to Wu et al., the structural re-arrangement occurring during the first discharge suggests the release of O2, as detected by in-situ chromatography. [START_REF] Wu | Unfolding the Mechanism of Sodium Insertion in Anatase TiO2 Nanoparticles[END_REF] After sodium de-intercalation, the structure can no longer be considered as a layered compound but transforms to a heavily disordered three-dimensional network with a similar local structure to anatase. Reversible sodium insertion/de-insertion is insured by these two phases. This phase transition is, however, characterized by a solid solution-like potential curve. This phenomenon can be explained by the amorphous nature of the two phases, which leads to a spectrum of voltage profiles for the phase transition, that produces a sloping curve similar to a solid solution mechanism. [START_REF] Wagemaker | Properties and Promises of Nanosized Insertion Materials for Li-Ion Batteries[END_REF][START_REF] Van Der Ven | Effect of Surface Energies and Nano-Particle Size Distribution on Open Circuit Voltage of Li-Electrodes[END_REF] It should be noted that solid solutionlike voltage curve has been also observed in the case of a phase transition from a highly disordered amorphous Na3Fe3(SO4)2(OH)6 towards crystallized NaFe3(SO4)2(OH)6. [START_REF] Gnanavel | Reversible Transformation from Amorphouse Na3Fe3(SO4)2(OH)6 to Crystallized NaFe3(SO4)2(OH)6 Jarosite-Type Hydroxysulfate[END_REF] Chemical sodiation. Although attempts to reduce micron-sized anatase particles using a chemical route failed, [START_REF] Clarke | Structure, and Magnetic Properties of NaTiO2[END_REF] it can be anticipated that decreasing the size of the particles will increase the sodium solubility. Following Wagemaker's works on the chemical reactivity of TiO2 nanoparticles vs. Li, [START_REF] Wagemaker | Large Impact of Particle Size on Insertion Reactions. A Case for Anatase LixTiO2[END_REF] we employed a chemical route to reduce TiO2 nanoparticles using Na-Naphetlane as the reducing agent (see experimental method). The reaction was left for six days to reach equilibrium conditions. Figure 4 shows the x-ray diffraction powder pattern of the chemically reduced phase. The chemical sodiation induced a strong reduction of the crystallinity and similarly to the electrochemical route, a phase transition toward a NaxTiO2 like phase was observed. This result indicates that both electrochemical and chemical sodiation yielded NaxTiO2 type phase. Accordingly, the x-ray diffraction pattern was fitted using a rhombohedral structure and the following unit cell parameters were obtained a=3.05 and c= 14.60 Å. The c value indicates cationic inter-mixing in agreement with the threedimensional pristine structure. Moreover, the lower cvalue obtained by chemical sodiation indicates a larger amount of inserted sodium. Elemental analysis performed on reduced nanoparticles using energy-dispersive X-ray spectroscopy (EDX) yielded to a composition close to Na0.9TiO2. Elemental mapping on nanoparticles presented in Figure 5a shows a homogeneous distribution of all elements, which is further evidenced from the overlay color image (Figure 5b). Bright field HRTEM image and corresponding FT pattern (Figure 5c) of a fragment of nanoparticle oriented along the c-axis revealed lattice plane separation of 0.54 nm corresponding to the d002 spacing of the rhombohedral NaxTiO2 phase, in agreement with XRD analysis. Deeper atomic insights were obtained using high resolution HAADF-STEM performed on a single nanoparticle viewed along the [001] rhombohedral direction (Figure 6). A close inspection by high-resolution HAADF-STEM revealed non-homogeneity at the atomic level. Contrast in HAADF-STEM image is proportional to the thicknesses of the crystal and atomic number (~Z 2 ). There is clear evidence of Na(Ti) atomic columns contrast variation. Some columns appear very bright with respect to the surrounding ones, and can be attributed to Ti-rich (Z=22) columns (marked by white arrows in the Figure 8 inset). Moreover, the presence of black holes distributed over the nanoparticles (marked by red arrows head in Figure 8 inset) can be associated with vacancies thus concluding on similar structural arrangement adopted by the electrochemically and chemically reduced phases. Magnetic properties. The magnetic susceptibility measured as a function of the temperature of the sample chemically sodiated is shown in Figure 7. We noted that the small upturn in the χ(T) observed below ~50 K indicates a small Curie-Weiss paramagnetic component which could be attributed to a small amount of magnetic impurity, supported also by the slightly larger fc values. The M(H) curve collected at 5 K (figure 9 inset) confirms this feature as M maximum values in 5 T reach only 0.01 µB/f.u. Overall, the χ(T) curve shows a rather T-independent χ from 50K to 290K for both zfc and fc curves with low χ values of about 10 -3 emu.mol -1 . This result is consistent with Pauli paramagnetism resulting from the network of S=1/2 spins made by the majority Ti 3+ in the disordered structure on which charge carriers are delocalized with three possible type of MO6 with M=Ti, Na, and vacancy. The delocalized character ascribed to Ti-Ti distances is close to the Goodenough criterion. [START_REF] Goodenough | Magnetism and the Chemical Bond[END_REF] Figure 7. Susceptibility versus temperature for NaxTiO2 sample. Inset: M(H) curve at T=5K. Clarke et al, [START_REF] Clarke | Structure, and Magnetic Properties of NaTiO2[END_REF][START_REF] Clarke | Single-Phase Na1.0TiO2: Solid-State Synthesis and Characterisation by High-Resolution Powder Diffraction[END_REF] investigated the magnetic properties of ordered NaTiO2 and Na0.9TiO2 samples. For a NaTiO2 composition, they observed a magnetic transition at  260 K ascribed to a monoclinic transition of the RT structure. Furthermore, the authors observed a similar transition for their Na0.9TiO2 sample, i.e. a composition very close to that of our chemically prepared material. The absence of the magnetic transition for the sample under study suggests that the structural transition observed in references [START_REF] Clarke | Structure, and Magnetic Properties of NaTiO2[END_REF][START_REF] Gnanavel | Reversible Transformation from Amorphouse Na3Fe3(SO4)2(OH)6 to Crystallized NaFe3(SO4)2(OH)6 Jarosite-Type Hydroxysulfate[END_REF] is responsible for the  transition. Moreover, both the pseudo-layered character of as prepared sample offering a large degree of ionic disorder as well as the smallness of the crystallites can explain the absence of the magnetic transition.

DFT calculations. To better understand the intercalation of sodium into anatase TiO2, we performed DFT calculations for NaxTiO2 in the diluted limit and at the fully intercalated limit. To allow a direct comparison with the intercalation of lithium into anatase TiO2, we also performed equivalent calculations for LixTiO2.

To model intercalation in the diluted-limit, one Na or Li atom was placed at an octahedral interstitial site in a 4×4×2 Ti128O256 supercell, corresponding to x=0.008. For Na + intercalation, the Na + ion adopts a five-coordinate off-centre position (Figure 8), with average nearest Na-O separations of 2.13 Å, and a sixth Na-O distance of 3.28 Å. Constraining the Na ion to the centre of the interstitial octahedron increased the energy by 59 meV. In contrast, lithium does preferentially occupy the octahedron center, giving [4+2] coordination with average distances to the equatorial oxygen ions of 1.90 Å. [START_REF] Morgan | Role of Lithium Ordering in the LixTiO2 Anatase → Titanate Phase Transition[END_REF] The calculated intercalation energy for Na, relative to metallic sodium, is -0.08 eV. This is significantly smaller (less negative) than the corresponding intercalation energy for Li, relative to metallic lithium, of -1.529 eV, and corresponds to a much smaller thermodynamic driving force for the initial intercalation of Na into anatase TiO2 than for Li. To model the fully sodiated limit, we performed DFT calculations on fully stoichiometric NaTiO2 and LiTiO2 (Figure 9). For LixTiO2, the theoretical maximum of x=1 is only observed experimentally in sub-10 nm nanoparticles, [START_REF] Morgan | Lithium Intercalation into TiO2(B): A Comparison of LDA, GGA, and GGA+U Density Functional Calculations[END_REF][START_REF] Van Der Ven | Effect of Surface Energies and Nano-Particle Size Distribution on Open Circuit Voltage of Li-Electrodes[END_REF] The reported LiTiO2 structure is tetragonal (I41/amd), and preserves the edge-sharing between TiO6 octahedra exhibited by pristine anatase TiO2. In contrast, Na intercalation produces a rhombohedral R-3m phase. For our DFT calculations, we consider both MTiO2 structures for M=Li and M=Na. For NaTiO2, the rhombohedral structure gives an average intercalation energy of -1.247 eV Na -1 , and is more stable than the LiTiO2 I41/amd structure by 0.272 eV per formula unit. In contrast, for LiTiO2, we find the tetragonal I41/amd structure to be more stable by 0.110 eV per formula unit, with an average intercalation energy of -1.716 eV Li -1 . Left panel: rhombohedral R-3m layered structure. [START_REF] Clarke | Structure, and Magnetic Properties of NaTiO2[END_REF] Right panel: tetragonal I41/amd "LiTiO2" structure. [START_REF] Borghols | The Electronic Structure and Ionic Diffusion of Nanoscale LiTiO2 Anatase[END_REF] Here, we investigated the sodium intercalation mechanism in anatase TiO2 emphasizing a complex process. During the first discharge, an irreversible plateau region was observed at ~0.2 V and enabled the electrode activation which can be further cycled with discharge-charge curves featuring a sloping profile and an average working voltage of ca. 0.8 V. Using PDF analysis, we identified the structural changes associated with the plateau region of the electrochemical profile demonstrating the stabilization of a rhombohedral phase NaxTiO2 showing high degree of disorder. The stabilization of such a structure was supported by DFT calculations which points out the structural difference between sodiated. The chemical formula deduced by PDF refinement led to (Na0.43Ti0.57)3a(0.22Na0.39Ti0.39)3bO2 where  refers to vacancy, highlighting the strong structural disorder due to cationic inter-mixing. Moreover, the presence of Ti cations in the Na slabs induced a contraction of the c-parameters as compared to the ordered phase Na-TiO2. PDF data of the charged electrode showed that the local structure of the anatase is recovered with short-range order of about 1 nm. Thus, the electrochemical process enabling the reversible sodiation/de-sodiation reactions implied the rhombohedral phase and an amorphous anatase phase yielding to a sloping composition-potential curves. Moreover, we showed that similarly to lithium, using chemical sodiation, downsizing the particle size increases the sodium solubility. Magnetic measurements performed on the reduced phase did not show any phase transition which is in contrast with the ordered phase. Supporting Information. Detailed analysis such XRD pattern of the pristine, high-energy XRD of the discharged, charged electrode and PDF refinements. This material is available free of charge via the Internet at http://pubs.acs.org.

Figure 1 .

 1 Figure 1. Galvanostatic discharge-charge curves of Na//TiO2 cells. The cells were cycled at 25 mA.g -1 .

Figure 2 .

 2 Figure 2. (Top) PDF refinement of the fully discharged electrode using an O3-type structure NaxTiO2 (space group: R-3m) model. Inset: structural representation of an ordered layered like NaTiO2 phase. (Bottom) Structural relationship between the pristine tetragonal and sodiated rhombohedral phases, view is along the (100) direction.

  mechanism. To identify the mechanism enabling reversible electrochemical reactions with sodium, PDFs of the pristine, the fully discharged and fully charged electrodes were gathered in Figure3with an emphasize on the short-range, i.e. 1 ≤ r ≤ 5 Å, and long-range orders, i.e. r-values >5 Å.

Figure 3 .

 3 Figure 3. PDF profiles of the pristine (black), fully discharged (red) and fully charged (blue) electrodes. A and B refer to the interatomic distances of Ti-Na/Ti in edge-sharing and cornersharing (Ti/Na)O6 octahedra, respectively.

Figure 4 .

 4 Figure 4. Le Bail profile refinement of the x-ray diffraction powder pattern of the reduced NaxTiO2 obtained by chemical sodiation.

Figure 5 .

 5 Figure 5. (a) Low magnification bright field TEM image of chemically reduced TiO2 nanoparticles. Inset: HAADF-STEM image. (b) EDX elemental mapping for Ti K, Na K and O K. Overlay color image (right corner). (c) Bright field HRTEM image and corresponding FT pattern of fragment of nanoparticle oriented along the c-axis. Spots in the FT pattern depicted with white arrows correspond to the d002=0.54 nm of the rhombohedral Na0.9TiO2 structure.

Figure 6 .

 6 Figure 6. High-resolution HAADF-STEM image of NaxTiO2 nanoparticle viewing along 001 zone axis. Corresponding FT pattern is given as insert in right upper corner. Enlargement of HAADF-STEM image (right bottom corner) indicate the brightness variation of Na(Ti) columns: enhancement -by white and depletion -by red arrow heads.

Figure 8 .

 8 Figure 8. DFT optimized structure for Na in anatase TiO2, showing the preferred off-centre coordination.

Figure 9 .

 9 Figure 9. DFT optimized structures for stoichiometric NaTiO2.Left panel: rhombohedral R-3m layered structure.[START_REF] Clarke | Structure, and Magnetic Properties of NaTiO2[END_REF] Right panel: tetragonal I41/amd "LiTiO2" structure.[START_REF] Borghols | The Electronic Structure and Ionic Diffusion of Nanoscale LiTiO2 Anatase[END_REF]

* damien.dambournet@upmc.fr The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n°[321879] (FLUOSYNES). Support from the Royal Society (UF130329) and EPSRC (EP/L000202 & EP/N004302/1). This research used resources of the Advanced Photon Source, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under Contract No. DE-AC02-06CH11357. B. J. M. acknowledges support from the Royal Society (UF130329). This work made use of the ARCHER UK National Supercomputing Service (http://www.archer.ac.uk), via the membership of the UK's HPC Materials Chemistry Consortium, which is funded by EPSRC (EP/L000202).

Authors are required to submit a graphic entry for the Table of Contents (TOC) that, in conjunction with the manuscript title, should give the reader a representative idea of one of the following: A key structure, reaction, equation, concept, or theorem, etc., that is discussed in the manuscript. Consult the journal's Instructions for Authors for TOC graphic specifications.


Rabah Rabah
email: rabah@auto.emn.fr

Michel Malabre
email: malabre@lan.ec-nantes.fr

Structure at Infinity Revisited for Delay Systems

Structure at infinity in a strong sense is considered for delay systems. A sufficient condition for the existence of this canonical form is given. The loss of generality is balanced by the fact that some solutions to control problems by static state feedback without anticipation can be obtained.

Introduction

The structure at infinity or the Smith-McMillan form at infinity are well known tools for the characterization of the solvability of some control problems such as model matching, disturbance rejection, row-by-row decoupling,... For linear finite dimensional systems see [START_REF] Commault | Influence de la structure à l'infini des systèmes linéaires sur la résolution de problèmes de commande[END_REF][START_REF] Silverman | System structure at infinity[END_REF] for instance. The notion of zeros at infinity has been generalized to non-linear systems [START_REF] Moog | Inversion, découplage, poursuite de modèle des systèmes non linéaires[END_REF] and several concepts are available for singular systems [START_REF] Loiseau | A new canonical form for singular systems with outputs[END_REF]. For linear infinite dimensional systems and in the particular case of bounded operators, the structure at infinity was introduced in [START_REF] Hautus | The formal Laplace transform for smooth linear systems[END_REF], described in several equivalent ways and used to solve some control problems in [START_REF] Malabre | On infinite zeros for infinite dimensional systems[END_REF]. The particular case of delay systems was studied in [START_REF] Malabre | Structure at infinity, model matching and disturbance rejection for linear systems with delays[END_REF]. However the structure at infinity defined there is too weak to insure a good solution for control problems: indeed the potential compensators may be anticipative (see also [START_REF] Sename | Decoupling without prediction of linear systems with delays: a structural approach[END_REF]). In this paper we introduce the concept of strong structure at infinity which is more convenient to infinite dimensional systems (and to the delay systems as a particular case). This structure is only well defined for some classes of systems. The positive result is that if this structure at infinity is well available then all potential solutions of control problems are nonanticipative and may be realized by static state feedback.

Preliminaries

We consider delay systems described by

ẋ(t) = A0x(t) + A1x(t -1) + B0u(t) y(t) = C0x(t) (1)
where x(t) ∈ R n , y(t) ∈ R p and u(t) ∈ R m . Note that some results may be extended to systems with several * Supported by ESPRIT Basic Research project n 0 8924 (SESDIP) commensurate delays in state, input and output. The transfer function matrix of the system is

T (s) = C0(sI -A0 -A1e -s ) -1 B0
which may be decomposed as follows

T (s) = ∞ j=0 Tj(s)e -js , (2)
where Tj(s) = C0(sI -A0) -1 A1(sI -A0) -1 j B0.

Each matrix Tj(s) may be decomposed as a serie using the following constant matrices introduced by Kirillova and Churakova (see [START_REF] Tsoi | Recent advances in the algebraic system theory of delay differential equations[END_REF]):

Qi(j) = A0Qi-1(j) + A1Qi-1(j -1), Q0(0) = I, Qi(j) = 0, i < 0 or j < 0. (3)
We have

Tj(s) = ∞ i=0 C0Qi(j)B0s -(i+1)
Another expression which will be used in this paper is the following one

T (s) = ∞ i=0 i j=0 C0Qi(j)B0e -js s -(i+1) . (4)
Both expression may be obtained by a simple calculation using the relations (3), see for example [START_REF] Sename | Decoupling without prediction of linear systems with delays: a structural approach[END_REF][START_REF] Tsoi | Recent advances in the algebraic system theory of delay differential equations[END_REF]. Note that for the case of a system with simple delays we have Qi(j) = 0, for i < j. The situation is different for delay systems of neutral type.

Structure at infinity

In order to define the structure at infinity (weak and strong) we need the notion of proper functions.

Definition 3.1 A complex valued function f (s) is called weak proper if lim f (s) is finite when s ∈ R tends to ∞.
It is called strictly weak proper if this limit is 0. A matrix B(s) is weak biproper if it is a weak proper and its inverse is also weak proper. Weak proper is replaced by strong proper if the same occurs when e(s) → ∞.

It is obvious that strong properness implies weak properness. The strong properness was used in [START_REF] Hautus | The formal Laplace transform for smooth linear systems[END_REF] and [START_REF] Malabre | On infinite zeros for infinite dimensional systems[END_REF] in the description of the structure at infinity for infinite dimensional systems. In [START_REF] Malabre | Structure at infinity, model matching and disturbance rejection for linear systems with delays[END_REF] and [START_REF] Sename | Decoupling without prediction of linear systems with delays: a structural approach[END_REF] the weak notion was used in order to define the structure at infinity of delay system and to solve some control problem. However, this structure at infinity cannot be used, in general, to solve control problems with non predictive control laws in an efficient way. For example, if the transfer function of the system is T (s) = s -3 + s -2 e -s , the weak structure at infinity is s -3 since T (s) = s -3 (1 + se -s ), and since 1 + se -s is weak biproper. Suppose one has to solve the model matching problem for a given model Tm(s) = s -3 . As the structure at infinity of the plant and the model are the same, there exist a proper compensator C(s) such that T (s)C(s) = Tm(s), see [START_REF] Malabre | Structure at infinity, model matching and disturbance rejection for linear systems with delays[END_REF]. In fact the unique compensator solving this problem is C(s) = 1 + se -s which is not realizable by linear static state feedback even if one uses distributed delays.

If we consider the notion of strong properness to define the structure at infinity, some difficulties occur in the construction of the canonical form at infinity. For the given example the structure at infinity is not s -3 because 1 + se -s is not proper in the strong sense and it is not possible, in fact, to define strong structure at infinity for the given example. Is there a structure at infinity in the strong sense for every delay system? For the moment it is not possible to give a positive answer.

The importance of the strong properness is given by the following considerations. Consider the problem of rowby-row decoupling for a square system (m = p). If the problem is solvable by static state feedback (see Definition 4.1), then the corresponding compensator is given by C(s) = (W0 + W (s)) with a strictly strong proper matrix W (s). The same consideration may be made for other control problems: disturbance rejection, model matching.

As the matrices Ti(s) are rational functions with the degree of the denominator greater than the degree of the numerator, Ti(s) are strictly proper in the strong sense which is, in this case, the same as the weak sense and the structure at infinity is well defined in the classical (finite dimensional) sense. The orders of the zeros at infinity of the matrices Ti(s) may be compared in the following sense. We say that the orders of Ti(s) are increasing if the maximal order of the zeros at infinity of Ti(s) is less than or equal to the minimal order of those of Ti+1(s), for all i ≥ 0. This hypothesis excludes the example given above . The conterpart of this loss of generality allows us to solve control problem with good compensators. We have the following main result. Theorem 3.2 If the structure at infinity of each matrix Ti(s) are increasing, there then exist two strong biproper matrices B1(s) and B2(s) such that

B1(s)T (s)B2(s) =       ∆0(s) 0 • • • 0 0 0 ∆1(s)e -s • • • 0 0 . . . . . . . . . . . . . . . 0 0 • • • ∆ k (s)e -ks 0 0 0 • • • 0 0      
, where ∆i(s) = diag s -n i,1 , . . . , s -n i,j i and ni,j ≤ ni,j+1, i = 1, . . . , k and ni,j i ≤ ni+1,j i+1 .

Proof: Let Λ0(s) be the canonical form at infinity of the matrix T0(s), there then exist two rational strong biproper matrices B 0 1 (s) and B 0 2 (s) such that

B 0 1 (s)T (s)B 0 2 (s) = Λ0(s) + ∞ i=1 B 0 1 (s)Ti(s)B 0 2 (s)e -is , and
Λ0(s) = ∆0(s) 0 0 0 .
This may be written as:

T ( s) = B 0 1 (s)T (s)B 0 2 (s) = ∆0(s) + θ11(s) θ12(s) θ21(s) θ22(s) ,
moreover by our assumption, ∆ -1 0 (s)θ11(s), ∆ -1 0 (s)θ12(s) and θ21(s)∆ -1 0 (s) are strictly proper in the strong sense. This gives

T (s)B(s) = ∆0(s) θ12(s) θ21(s)β -1 (s) θ22(s)
, where β(s) = I + ∆ -1 0 (s)θ11(s) and B(s) being the strong biproper matrix:

B(s) = β -1 (s) 0 0 I .
Here and in what follows I denote the identity matrix with adequate dimension which is not precised in each case. Let us define B 1 1 and B 1 2 as follows:

B 1 1 (s) = I 0 θ21(s)β -1 (s)∆ -1 0 (s) I , B 1 2 (s)
I -∆ -1 0 (s)θ12(s) 0 I . Then B 1 1 (s)T (s)B 1 2 (s) = ∆0(s) 0 0 θ22(s) = T (s), i. e. T (s) = Λ0(s) + ∞ i=1 Ti(s)e -is , (5)
where

Ti(s) = 0 0 0 Ri(s) ,
the matrices Ri(s) being strictly proper. The same procedure may then be applied to the matrix

∞ i=1 Ri(s)e -is = e -s ∞ i=1
Ri(s)e -(i-1)s because the rational matrices Ri(s) have also increasing orders.

Control problems

We consider two control problems: row-by-row decoupling and disturbance rejection with static state feedback of the form:

u(t) = k i=0 Fix(t -i) + k i=0 Giv(t -i), t ∈ [0, k + 1]
with G0 invertible for the row-by-row decoupling problem and Gi = 0 when we consider only the disturbance rejection problem. which means that the global structure of TF,G is equal to the union of the rows' structures at infinity. On the other hand, as G0 is regular, one can see that

TF,G(s) = T (s)B(s),
with a strong biproper matrix B(s). This means that T and TF,G have the same structure at infinity. For the converse, assume now that Ti(s) have increasing orders and the condition of the theorem is satisfied. Let s -n i e -j i be the strong structure at infinity of the row i, i = 1, . . . , m. Our assumption gives

ciQ l (j)B0 = 0 forl < ni -1, j ≥ ji.
Then the first non zero moment of the decomposition ( 4) is

n i -1 j=0 ciQn i -1(j)B0e -js m i=1 , ( 6
)
where ci is a row of C0 and we have ciQn i -1(j)B0 = 0 for j < ji.

Let us denote

E l = [ciQn i -1(ji + l)B0] m i=1 , l = 0, . . . , ni -1 -ji.
The condition of the theorem yields to the invertibility of the matrix E0. The first moment of the decomposition (4) given in ( 6) is expressed by:

E = E0 + E1e -s + . . . + E k e -ks ,
where k = sup{ni -1, i = 1, . . . , m}.

Let us denote by F and G the needed feedback:

F = F0 + F1e -s + . . . , G = G0 + G1e -s + . . . ,
and by A the matrix A0 + A1e -s . A formal calculation gives, as in the classical paper [START_REF] Falb | Decoupling in the design and synthesis of multivariable control systems[END_REF]:

ci(A + B0F ) l = ciA l , l ≤ ni -1,
and then

ci(A + B0F ) l B0 = ciA l B0 = 0 l < ni -1,
and

ci(A + B0F ) n i -1 B0 m i=1 = ciA n i -1 B0 m i=1 = E diag {e -j 1 s , . . . , e -jms }. Let us remark that E is invertible. If G = E -1 , then ci(A + B0F ) n i -1 B0 m i=1 G = diag {e -j 1 s , . . . , e -jms }.
In the same way we can write:

[ci(A + B0F ) n i ] i=1 = ciA n i -1 (A + B0F ) m i=1 = [ciA n i ] m i=1 + ciA n i -1 B0 m i=1 F = [ciA n i ] m i=1 + EF.
Then F may be calculated from this relation:

F = -G [ciA n i ] m i=1 .
For the disturbance rejection problem the definition is Definition 4.3 The disturbance rejection problem for the system

ẋ(t) = A0x(t) + A1x(t -1) + B0u(t) + D0q(t) y(t) = C0x(t) (7)
is solvable if there exist a feedback

u(t) = k i=0 Fix(t -i), t ∈ [0, k + 1]
such that the output of the system is not affected by the disturbance q(t).

Our result is

Theorem 4.4 If the zeros at infinity of Ti(s) are increasing, then the disturbance rejection problem is solvable if and only if

Σ∞[ s -1 T (s) T D (s) ] = Σ∞[ s -1 T (s) 0 ]
where T (s) is the transfer function matrix of the control:

T (s) = C0(sI -A0 -A1e -s ) -1 B0
and T D (s) is the transfer function matrix of the disturbance:

T D (s) = C0(sI -A0 -A1e -s ) -1 D0.
Proof: Suppose that the problem is solvable. Then, for all k, the problem is solvable for the systems (see [START_REF] Malabre | Structure at infinity, model matching and disturbance rejection for linear systems with delays[END_REF])

żk (t) = A k z k (t) + B k u(t) + D k q k (t), w k (t) = C k z k (t), (8)
where A k , B k , D k and C k are given as in [START_REF] Malabre | Structure at infinity, model matching and disturbance rejection for linear systems with delays[END_REF]:

A k =     A0 0 • • • 0 A1 A0 • • • 0 . . . . . . . . . . . . 0 0 • • • A0     , B k =   B0 • • • 0 . . . . . . . . . 0 • • • B0   , C k =   C0 • • • 0 . . . . . . . . . 0 • • • C0   , D k =   D0 • • • 0 . . . . . . . . . 0 • • • D0   .
Let Φ k and Φ D k be the transfer function matrices of the control and disturbance of the systems [START_REF] Picard | Model matching for linear systems with delays[END_REF]. Then,

Σ∞[ s -1 Φ k (s) Φ D k (s) ] = Σ∞[ s -1 Φ k (s) 0 ]. As Φ k =   T0 • • • 0 . . . . . . . . . T k • • • T0   , Φ D k =   T D 0 • • • 0 . . . . . . . . . T D k • • • T D 0   ,
and according to the assumption on the zeros of Tj(s),

this gives Σ∞[ s -1 T (s) T D (s) ] = Σ∞[ s -1 T (s) 0 ].
The converse may be obtained in the same way and using the fact that, for the systems (8), we can choose the feedback with a lower triangular form which insures that for the system (7) the feedback is without anticipation [START_REF] Malabre | Structure at infinity, model matching and disturbance rejection for linear systems with delays[END_REF].

Note that the assumption on increasing zeros orders at infinity concerns the matrices Tj, but are verified for the matrices T D j in the context of the theorem. Other problems may be considered as invertibility, model matching. The result given in [START_REF] Malabre | Structure at infinity, model matching and disturbance rejection for linear systems with delays[END_REF] may be adapted to the strong structure at infinity. Remark 4.5 For systems without strong structure at infinity defined here, further investigations are needed to characterize the solution of control problems (see [START_REF] Picard | Model matching for linear systems with delays[END_REF] for another approach). Consider the above mentioned example: T (s) = s -3 + s -2 e -s . Let T m (s) = ∞ j=0 T m j e -js be a model. What are the conditions which insure the existence of a strong proper compensator C(s) = ∞ j=0 Cj(s)e -js such that T C = T m (it is the classical model matching problem). Suppose that T m (s) = s -5 -s -3 e -2s . A simple calculation yields to the following strong proper compensator: s -2 -s -1 e -s . We can see that [ T (s) Tm(s) ] B(s) = [ T (s) 0 ] , with strong biproper matrix B(s). Hence, [ T (s) Tm(s) ] and [ T (s) 0 ] are equivalent at infinity. The problem which needs some investigation is: how to define the canonical form corresponding to this equivalence relation?

Conclusion

For a class of delay systems we define the strong structure at infinity which allows to solve control problems by static state feedback without anticipation. However, it is not clear how to do when this structure at infinity is not defined. Further investigations on the structure at infinity in every time interval [k, k + 1] may give solution in a general case. The weak structure at infinity is always available in solving such problems but the solution must be taken with care.

Definition 4 . 1

 41 For m = p we say that the row-by-row decoupling problem is solvable iff there exist a feedback of the given type such that the transfer function matrix of the closed loop system is of the formTF,G(s) = diag [h1(s), . . . , hp(s)]where hi(s) are non zero strictly strong proper functions.The result isTheorem 4.2 If the zeros at infinity of Ti(s) are increasing, then the row-by-row decoupling problem is solvable if and only if Σ∞(C0, A0, A1, B0) = {Σ∞(c1, A0, A1, B0), . . . , Σ∞(cp, A0, A1, B0)}, where ci's are the rows of the matrix C0, and Σ∞ denotes the strong structure at infinity. Proof: Assume that the zeros at infinity of Tj are increasing and that the row-by-row decoupling problem is solvable. Then TF,G(s) = diag [h1(s), . . . , hp(s)] ,