Reidson Pereira Gouvinhas

Tatiana Reyes

Ricardo Manfredi Naveiro

Nicolas Perry

Eduardo Romeiro Filho

A Proposed Framework of Sustainable Self-evaluation Maturity within Companies: An Exploratory Study

Keywords: Ecodesign, Sustainable self-evaluation, Maturity framework

Companies are still struggling to implement ecodesign in their daily product development processes. It is believed that this occurs because companies should firstly improve their strategy and management skills towards sustainability before introducing any ecodesign tool. As a consequence, this paper aims to propose a self-evaluation framework to rank companies into different classes according to their level of maturity in terms of sustainability. It also discusses ¨minimum requirements¨ to move from one level to another towards sustainability. This will help companies to better understand which strategic and managerial actions they need to take before implementing any ecodesign.

INTRODUCTION

Corporate sustainability is nothing more than handling the business taking into account the sustainable development into its three fundamental principles: environmental protection, economic stability and social responsibility [START_REF] Grüninger | GRI -Caminhos e desafios para relatórios de sustentabilidade. Business meets social development[END_REF]. A literature review has indicated many reasons why companies should look to corporate sustainability to become more competitive [START_REF] Hart | Creating a sustainable value[END_REF][START_REF] Jackson | Motivating Sustainable Consumption: a Review of Evidence on Consumer Behaviour and Behavioural Change[END_REF][START_REF] Erscp | Meeting consumer demand for sustainable products[END_REF][START_REF] Barbieri | Desenvolvimento e meio ambiente: as estratégias de mudança da agenda 21[END_REF][START_REF] Medeiros | Sucessful Factors for Environmentally Sustainable Product Innovation: A Systematic Literature Review[END_REF]. As a consequence, the global sustainability awareness has evolved to a point where IJIDEM DOI (automatically inserted by the publisher) Review Article/Research Article Please choose one ------------Correspondence: reidson@ct.ufrn.br ; tatiana.reyes_carrillo@utt.fr ; ricardo.naveiro@poli.ufrj.br ; nicolas.perry@ensam.eu ; edu.romeiro@gmail.com businesses must consider the impacts of their processes and products throughout their lifecycle and supply chains. Unfortunately, in general, this process occurs in a heuristic manner, based on previous experience and general guidelines without the assurance that the designer will get a significant result [START_REF] Cross | Engineering Design methods: Strategies for Product Design[END_REF], and most of the time without a complete rethinking (and involvement) of all the supply chain. It follows that the product development activity is too risky for the product commercial success and it requires a suitable planning by the company.

RESEARCH GAP

A review on the literature has suggested many ways to deliver sustainability into companies [START_REF] Stubbs | Conceptualizing a `Sustenability Business Model[END_REF][START_REF] Wells | Business Model and Close-Loop Supply Chains : a typology[END_REF][START_REF] Hawken | Natural Capitalism: The Next Industrial Revolution Effect[END_REF][START_REF] Grassi | Business Models of Social Enterprise: a design approach[END_REF][START_REF] Tukker | Eight Steps of Product-Service System: Eight ways to Sustainability? Experiences from SusProNet[END_REF][START_REF] Mont | Product-Service Systems: Reviewing Achievements and Refining the Research Agenda[END_REF][START_REF] Pauli | The Blue Economy: 10 years, 100 innovations, 100 million jobs[END_REF] as well as has shown specific companies which have already tried to achieve that [START_REF] Baines | State-of-the-art in product-service systems[END_REF]. However, despite of these efforts, it seems that research on sustainability is presented on a widespread manner which indicates that the simple application of eco-innovation solutions, eco-efficiency and corporate responsibility practices alone are insufficient to deliver the holistic changes necessary to achieve a long-term social and environmental sustainability within companies [START_REF] Boken | A literature and practice review to develop sustainable business model archetypes[END_REF]. In other words, creating a sustainable corporate image it is not only a matter of developing some "sustainable products". It also requires that all management procedures within the company are based on a different philosophy and strategic vision towards sustainability. This new vision should permeate all company sectors and departments. This is particularly true for companies who are still struggling to implement ecodesign in their daily product development processes (PDP).

Therefore, our research argues that this gap occurs because companies should firstly improve their strategy and management skills towards sustainability before introducing any ecodesign tool. In other words, companies should improve the social environmental aspects of their products by considering a collaborative work among the entire organization. This involves not only the three levels of the company (strategic, tactical and operational) but also its stakeholders [START_REF] Zhang | Towards a Systematic Navigation Framework to Integrate Sustainable Development into the Company[END_REF]. In order to achieve this goal, a cultural change within the entire organization towards sustainability is required [START_REF] Cheng | The Driver of Green Innovation and Green Image -Green core Competence[END_REF]. As a consequence, to become more effectively, the PDP should be integrated into a broader framework within the organization in a holistic manner. This means that decisions taken during the development of the product should reflect other decisions taken at the strategic, tactical and operational levels and vice-versa. The decision process can be carried out ¨top-down¨ (from strategy to operational level), ¨bottom-up¨ (from operational to strategy level) as well as ¨middle-two sides¨ (from tactical to strategy and operational levels) in a complete synergy which involves the organization as whole [START_REF] Pigosso | Ecodesign maturity model: a management framework to support ecodesign implementation into manufacturing companies[END_REF][START_REF] Xu | Knowledge evaluation in Product Lifecycle Design and Support[END_REF]. This is a long process which demands patience and time to be completed and does not occur in a uniform manner within the entire organization. Some departments or sectors within the company may maturate these ideas quicker than others. As a consequence, the level of sustainable maturity may vary from one department to another and also at the different levels of the organization (strategic, tactical and operational) as well as among their personnel [START_REF] Drémont | A Meta-Model for Knowledge Representation Integrating Maturity for Decision Making in Engineering Design[END_REF][START_REF] Perry | Proposal of an architectural solution for economic and environment global eco-cost assessment: models combination analysis[END_REF].

Engert et al [START_REF] Engert | Exploring the integration of corporate sustainability into strategic management: a literature review[END_REF] argue that with respect to the strategy formulation and implementation, strategies can be either intended and deliberate or emergent.

In this paper, we are particularly interested in emergent strategy. This means that some organizations begin implementing strategies before they clearly articulate mission, goals, or objectives.

In this case strategy implementation actually precedes strategy formulation [24].

In addition, it seems that corporate sustainability is extremely complex once its parameters (e.g. technology, regime and visibility) may vary across industries, plants, countries and different point in time [START_REF] Salzmann | The Business Case for Corporate Sustainability: Literature Review and Research Options[END_REF].

Engert et al. [START_REF] Engert | Exploring the integration of corporate sustainability into strategic management: a literature review[END_REF] highlight the lack of effort regarding the development of potential solutions to integrate corporate sustainability dealing with complexity. "The integration of Sustainability assumes that a related strategic decision needs to be taken. This is essential for generating a sufficient level of company-wide commitment¨ [START_REF] Engert | Exploring the integration of corporate sustainability into strategic management: a literature review[END_REF]. For example, Rossi et al. [START_REF] Rossi | Review of ecodesign methods and tools. Barriers and strategies for an effective implementation in industrial companies Journal of Cleaner Production[END_REF] claim that despite the great number of existing ecodesign tools and methods, also in commercial tools available, companies still have difficulty in their practical and effective implementation and use. They also argue that new sustainable models should be based of the real needs of industrial companies, which at the moment are not included for example in the methods of development process.

Therefore, there must be a change on the way business is carried out. Companies should be able to review their business models to pursue a proactive attitude towards environmental regulations and incentives as well as market demands. In addition, consumers have to drastically change their approach to their consumption behavior. As a consequence, they may take part on the product development process themselves.

RESEARCH OBJECTIVES

This research proposes to develop an easy to use, not time consuming and suitable for all type of organisations self-evaluation framework to rank companies into different classes according to their level of maturity in terms of sustainability. This self-analysis will help companies to better understand which actions they need to take before implementing any ecodesign tool as well as to select which tool is more appropriate to their situation. The paper also discusses the ¨minimum requirements¨ that a company needs to attend before moving from one level to another showing a step-by-step evolution towards corporate sustainability.

RESEARCH METHODOLOGY

Blessing and Chakrabarti [START_REF] Blessing | DRM, a design Research Methodology[END_REF] argue that ¨Design research¨ (i.e. the study on how to research design) can be classified into three different main areas: Descriptive studies I and II as well as Prescriptive study (PS). Descriptive study I (DS-I) aims to increase the understanding of design and its successful factors by investigating the phenomenon of design through reviewing the literature about empirical research, undertaking empirical research, and, in addition, through reasoning. Prescriptive study (PS) indicates on how to develop a support to improve the design process, and Descriptive study II (DS-II) focuses on the evaluation of these different types of supports for the design process. In this paper, the term ¨support¨ can be used to describe simple guidelines, any design methodology, methods or tools to improve the design process. The authors argue that DS-I aims to understand an existing situation to provide suggestions for the support in order to improve the design process while DS-II aims to understand a situation in which the support is introduced, in order to assess its ability to improve the existing situation and to provide suggestions for improvement of the support.

Therefore, the proposed framework of this study can be classified as a DS-I once we propose a framework to understand how companies deal with aspects related to sustainability before introducing any ecodesign support.

In order to achieve the research aims, the following research methodology approach is divided into five phases as follows:

•

The initial stage of the research methodology is based on an extensive literature review. At this point, a number of different corporate sustainability models were analyzed (please, see section 5). This phase was useful to enable a better understanding regarding theories supporting corporate sustainable businesses;

Based on the literature review, a framework is developed aiming to evaluate companies in terms of their level of corporate sustainable maturity.

The framework is based on a questionnaire in which decision-makers from different areas of the organization (strategy, tactical and operational) will evaluate a series of statements against five aspects. According to the answers given, the framework generates 12 indicators which will be used to create a general index I. The value of this general index will then be used to rank companies into six levels of sustainable maturity.

•

The next step is to carry out an initial testing and validation of the proposed framework.

Based on that, ten manufactured companies were randomly selected and invited to answer the questionnaire. The proposed of this phase was to ¨pilot¨ the questionnaire to verify of it is well understandable and if it deliveries the results we expect from the framework.

•

The following step is to make some improvements on the questionnaire as well as the framework in general. Once completed, the framework will be applied to companies in France and Brazil for comparison. At the moment, the research is under the fourth step (i.e. initial testing and validation of the framework) and improvements within the questionnaire and framework will take place. More details on how the framework was built and how it works will be given in the section 6.

A MATURITY FRAMEWORK TOWARDS SUSTAINABILITY

Many frameworks aiming to introduce sustainable issues within different sectors of the economy have been proposed by the literature. For instance, some of them are mainly focused on a national, regional or community level [START_REF]Indicators of sustainable development: guidelines and methodologies[END_REF] while others are business level [START_REF]The sustainability metrics: sustainable development progress metrics recommended for use in the process industries[END_REF][START_REF] Sapangenberg | Sustainability indicators-a compass on the road towards sustainability[END_REF]31]. In addition, Global Reporting Initiative (GRI) uses a hierarchical framework consisting of categories, aspects and indicators [START_REF]GRI. Sustainability reporting guidelines[END_REF]. Nevertheless, some frameworks concentrate their efforts on one aspect of sustainability, such as Corporate Social Responsibility (CSR) [START_REF]Indicators of sustainable development: guidelines and methodologies[END_REF]. However, most of these frameworks are very complicated as well as time consuming and require some expertise to be performed [START_REF] Bocken | A literature and practice review to develop sustainable business model archetypes[END_REF].

Another group of frameworks is more related on improving sustainable products and services. For instance, Carlson and Rafinejad [START_REF] Carlson | Modelling Sustainability in Product Development and Comercialization[END_REF] developed a framework that enables companies to consider the impact of their product development process on future resource and environmental conditions (e.g. taking into account how a product can be designed for reuse and recycling). In addition, the Sustainability Maturity Model Research-on-Research (ROR) [START_REF] Hynds | A Maturity Model for Sustainability in New Product Development[END_REF] focuses primarily on the activities performed by Research and Development (R&D) department and New Product Development (NPD) functions or activities performed by other functions that support technology research and NPD. Also, the Ecodesign Maturity Model is a management framework based on a step-by-step approach aiming to support companies to carry out ecodesign implementation [START_REF] Pigosso | Ecodesign maturity model: a management framework to support ecodesign implementation into manufacturing companies[END_REF]. Its application comprises into six steps, including a set of face-to-face interviews with the selected employees.

However, all of these proposed frameworks concentrate their efforts only on improving products and services or on the product development process itself in terms of sustainability. It is believed that they should go beyond into their analysis by proposing a holistic description on ¨how a firm does business¨ [START_REF] Amit | Creating Value through Business Model Innovation[END_REF]. This vision requires a shifting from developing innovation and products towards creating new systems [START_REF] Johnson | How to Jump-Start the Clean Tech Economy[END_REF]. In addition, they should not have their focus only on the organization but also on a wider range of stakeholders in order to create a broader value-add network and transform the entire company [START_REF] Beattie | Value Creation and Business Models: Refocusing the Intellectual Capital Debate[END_REF][START_REF] Zott | The Business Model: Recent Developments and Future Research[END_REF].

Once sustainable practices and procedures are implemented, the company can also initiate some sustainable practice and procedures into their PDP. This will allow a changing culture towards sustainability within the entire organization. As a consequence, the initial focus is on the organization and its relation with its stakeholders towards sustainability rather than on its PDP. In this paper, the maturation process is understood as successive stages of acquiring knowledge and developing innovation in which sustainable aspects are incorporated, not necessarily simultaneously and uniformly, within the entire organization. It depends on the people who are taken part of the process, once it is a process of cultural changing. It is believed that companies have different levels of maturity in terms of implementation of environmental issues within their daily managerial procedures. Therefore, we proposed a theoretical classification according to these levels of maturity once they are hardly identified and formalized. This classification is described into details as follows:

5.1

Complete immature companies: type 1

It can be said that these companies are considered to be "completely immature" in terms of their engagement on sustainable issues. For this group of companies, ecodesign has not been implemented at all and they are still struggling to find out on how to cope with social environmental problems. At this stage the company is pressured by different stakeholders. These include customers' demands, shareholders, government, legislation and regulations (local, national or global), competitors, financial partners, unions, NOG's, suppliers and the society in general. Normally, companies' type 1 is not concerned on social environmental issues at all. Their normal practices are more related to profitability, cost reduction and production efficiency. As a consequence, they usually have a reactive posture towards social environmental issues and consider them as an "another inconvenient aspect to be managed and increase costs". This means that social environment issues are consider as being a barrier for their businesses and not an opportunity for market growth and increasing profitability.

In addition, these companies have a short vision of their future and all of their efforts are spent on dealing with daily operational drawbacks rather than having a more strategic approach for their businesses. Therefore, their efforts on ecodesign are very rare and probably none or few attempts have been made to include environmental concerns in their product development process.

To move from company type 1 to company type 2, the organization needs to consider some initial social environmental projects to improve its procedures (usually these projects are outsourced).

5.2

Immature companies: type 2

In the case of ¨immature¨ companies, some ecodesign projects may have already started, but they are very timid and no serious engagement from these companies has been observed so far. Generally, these projects are carried out by an external designer and they are not integrated into company's product development and production site.

Suffering the same pressure as companies' type 1, companies' type 2 have already started some timid experience on ecodesign. However, these companies still have the same reactive posture towards social environmental concerns as companies' type 1 and they see these experiences as a "rare market opportunity" to make profit on "green product" at that particular moment. This means that ecodesign has not been fully integrated into the product development process and environmental issues are seen as a unique opportunity to differentiate their products from their competitors. Therefore, these companies are still more concern on profitability, cost reduction and production efficiency issues rather than environmental aspects. However, they have already wakened up for new social environment market opportunities and try to experience few projects in this field. Probably, these experiences are motivated by initial market opportunities and/or pressure from competitors and new environmental legislation. In addition, it is observed that these experiences are carried out with the help of external consultants who do not have a fully understanding of companies' operational procedures. As a consequence, products are developed independently from decisions taken at company's production site. Therefore, some small operational problems may occur in areas such as manufacturing and assembling. Nevertheless, it is a start and if successfully implemented, these experiences can help these companies to fully integrate ecodesign in their product development management procedures in the near future. The step further, in terms of maturity, starts when the company begins to develop some ¨in-house¨ projects associated to social and/or environmental concerns.

5.3

Initial mature companies: type 3 These companies reached the point of "Initial maturation". This means that they have some "in house" sustainable projects. Ecodesign is integrated with company's production site and they may be working based on "simultaneous engineering" approach.

It can be noticed that in this case, the ecodesign process is now internally carried out within the company. In other words, ecodesign is now integrated into the product development process of the company. This means that the product development is integrated to the production process and management procedures such as simultaneous engineering may be used to reduce re-work. Also, it is perceived that sustainable issues are taken into account either on production level as well as on product development level. Finally, it can assume that environmental concerns are now taken into account at strategic level as well as part of the company's vision of the future.

This means that these types of companies have a more pro-active posture than companies' type 1 and 2. In addition, environmental issues are now considered as a market opportunity for these companies and they are keen to integrate these aspects as part of their daily normal management procedures.

As any other company, at any stage of sustainable maturation, these companies are still concern on profitability, cost reduction and production efficiency issues. However, they evolve to a point on which environmental issues play an important role on their decisions and should be taken into consideration regardless strategic, management or operational issues.

To move to the next step of the sustainable maturity process, the organization needs to stimulate its suppliers to also consider sustainable issues in their daily decisionmaking process.

5.4

Mature companies: type 4

For this type of company, they reach the point where environmental concerns have been taken into account at operational, management and strategic levels as well as during its product development process. It is the level of "Maturation". In other words, companies are seen as "mature companies" in terms of introduction of ecodesign and social environmental issues into their daily procedures. As a consequence, suppliers are stimulated to participate in this process by considering environmental aspects within their daily procedures. For companies' type 4 as for companies' type 3, ecodesign is already fully integrated into their product development process. Moreover, sustainable issues are part of their daily decision procedures (at operational, management and strategic level). However, in the case of companies' type 4, the suppliers have already been stimulated to also include social environmental concerns on their procedures. This means that social environmental performance will be used as one of strong the criteria for choosing their suppliers. A positive consequence is that this will affect management decisions downstream on the supply chain. In other words, more companies (the suppliers) will have to take into account social environmental issues when developing their own products and services. It is like a "domino effect" on the downstream of the entire supply chain.

In addition, companies' type 4 may include on their criteria a sort of "social and environmental performance" for their suppliers forcing these companies to prove that they are seriously considering environmental issues on their business procedures.

As a result, it can be said that companies' type 4 have a pro-active posture towards social environmental issues as well as these issues are seen as "unique market opportunity" for them to definitely take off in the marketplace.

To move from company type 4 to company type 5, the organization needs to ¨educate¨ their customers to consider social environmental aspects during their decisions.

5.5

Maturated and teaching companies: type 5

Companies from this group have not only taken on board their suppliers to consider environmental issues on their businesses decisions but also they have ¨educate¨ their customers to consider social environmental aspects during their consume decisions as well as integrated these demands by using marketing approaches and creating a new type of customer vision regarding their consumer behavior. It is considered that this process is "fully maturated" within the company and all of its procedures have taken into account social environmental concerns.

In this case, customers' demands and marketing procedures have been taken place. In other words, once ecodesign has already been fully integrated into their product development process and social environmental issues have been incorporated important issues of daily decision procedures (at operational, management and strategic level), the company starts to close its relationship with their customers as well as tries to "educate" them towards a more social environmental conscious consumer behavior. This means that the company starts to develop new marketing procedures, improve its approaches to have a closer relationship with its customers in terms of environmental concerns as well as develop new forms of commercial relationship with its customers. At this stage, new commercial approaches can be incorporate into company's commercial strategy. In some cases, companies have to convince their customers about the advantages of changing the way they relate with company's products and services. In this case, the information feedback collected from customers can passed on to the entire supply chain of the system involving a complete loop of information exchange among customers, companies and supply chain. This will give a futurologist vision on how social environmental concerns will influence business performance and what procedures companies will have to develop to achieve a complete "sustainable business approach". This means that new technologies, management procedures, marketing procedures, products and services will be develop to fulfill the requirements of this new "responsible consumer". These companies have a completely pro-active posture towards social environmental issues and their visionary posture towards future will change the way we consume goods and services. And finally, to move from company type 5 to company type 6, it requires an integration of many organizations which make business among themselves considering social environmental concerns as the most important aspect for them during theirs decision making process.

5.6

Integrated companies: type 6

Companies' type 6 are also considered to be "fully maturated" as in the case of companies' type 5. However, in this case, they have reached to a point on which social and environmental issues are the most important criteria for them to do business among themselves. They exchange sustainable experiences as well as stimuli their business partners to do the same, creating an integrated network of ¨sustainable companies¨.

At this point, companies trade with each order considering social and environmental aspects as well as searching solutions for a fully sustainable integration. They work for market which is fully conscious regarding their consume behavior and they develop new products and services to meet the demands of this new market.

MATURITY FRAMEWORK PROPOSAL

The proposed framework is based on a questionnaire on which decision-makers from different areas of the organization at strategy, tactical and operational levels will judge a series of statements according to their opinion. Each group of statements is classified according to 12 different categories of indicators, named as: company´s strategic vision; company´s values; company´s general policy; top management commitment; company relationship with stakeholders; company´s purchasing policy; company´s economic indicator performance; company´s environmental indicator performance; company´s social indicator performance; environmental communication; legislation; standards and company´s "green" marketing procedures.

To each judgment given by the decisionmakers, a number of points is associated to the statement as follows: a) When the respondent strongly agrees with the statement (I strongly agree: +2 points); b) When the respondent agrees with the statement (I agree: +1 point); c) When the respondent disagrees with the statement (I disagree: -1 points); d) When the respondent strongly disagrees with the statement (I strongly disagree: -2 points), and; e) When the responded does not have an opinion (I do not know: 0 points). For instance, if the indicator X is composed by four statements, the indicator will range from -8 points (i.e. -2 points versus 4 statements) to +8 points (i.e. 2 points versus 4 statements). However, each decision-maker of the company has different views and is interested in different indicator. Therefore, instead of using a range of points, one must normalize each indicator by dividing its value by the maximum number of points in which it can be achieved (in this particularly case, 8 points). Therefore, a normalized indicator will range in a scale from -1 to +1. Therefore, using the scale from -1 to +1, it can be said that if an indicator i reaches the value +1, then the company is considered "totally committed" (TC) to the specific issues of the indicator i. On the other hand, if an indicator i reach the value -1, then the company is considered "totally uncommitted" (TU) to this specific issues of the indicator i. Similarly, if an indicator i reach the value +0.5, it can be said that the company is considered "partially committed" (PC) to the specific issues of the indicator i and if an indicator i reach the value -0.5, it can be said that the company is considered "partially uncommitted" (PU) to the specific issues of the indicator i. In addition, companies can also be considered "nearly committed" (NC) if an indicator i reach any value between +0.5 and +1 or ¨nearly uncommitted( NU) if an indicator i reach any value between -0.5 and -1. Also, companies can be considered "nearly indifferent" (NI) if an indicator i reach any value between 0 and +0.5 or between 0 and -0.5. Finally, if the indicator i reach the value 0, the company is considered "indifferent" (I) to the specific issues of the indicator i. Once all weights are determined for each indicator, a weighted arithmetic mean can be used to calculate the generic index this index will range from -1 to +1. This means that the closer is the index I value -1, the more uncommitted is the company to sustainable issues. On the contrary, the closer is the index I to value +1, the more committed is the company to sustainable issues.

As a consequence, it is possible to interpret the final result of index I according to the classification presented in sections 4.1 to 4.6. Therefore, the analysis will help decisionmakers to consider each one of the 12 indicators separately as well as a generic index. As a consequence, companies will have the opportunity to evaluate their performance against important issues for sustainability individually as well as a global evaluation regarding sustainability in a holistic manner.

Initial test and validation

In order to test and validate our framework, ten medium to large size manufactured companies from various sectors (e.g. automotive and heavy goods vehicle, high and mediumvoltage products, camping gear and outdoor equipment, kitchen furniture, light sport airplanes, pharmaceuticals and so on) were randomly selected and contacted. As a consequence, members form these companies were invited to answer the questionnaire as well as complete the framework.

These members were chosen due to their managerial position within the company as well as their general overview of its procedures (e.g. product design managers, R&D managers and manufacture managers).

Initial results have indicated that the implementation of environmental aspects within companies does not necessarily follow a logical pathway. For some companies, environmental operational procedures are well established (i.e. mature) and some managerial environmental procedures are still in process of implementation (i.e. immature) while to others this process is inverse.

In addition, there is no apparent connection to the different categories of indicators. In other words, there is no connection between one indicator to another.

For instance, ¨top management commitment¨ is not necessarily related to ¨company´s values¨ or ¨company´s general policy¨.

Companies have difficulties to deliver holistic changes necessary to achieve a long-term sustainability in order to remain competitive. They should firstly improve their strategy and management skills towards sustainability before introducing any ecodesign tool.

Conclusions

Companies have difficulties to deliver holistic changes necessary to achieve a long-term sustainability in order to remain competitive. They should firstly improve their strategy and management skills towards sustainability before introducing any ecodesign tool. It requires a collaborative work among the entire organization involving not only the three levels of the company (strategic, tactical and operational) but also all its stakeholders. In general, it is a cultural change within the entire organization towards sustainability. The self-evaluation framework proposed in this paper aims to rank companies into different classes according to their level of maturity in terms of sustainability. It is a complementary work to other models already existent in the literature. For instance, it can be used as a first step stage to before implementing the Ecodesign Maturity Model proposed by Pigosso [START_REF] Pigosso | Ecodesign maturity model: a management framework to support ecodesign implementation into manufacturing companies[END_REF]. At a more advanced stage, this simple and practical framework will facilitate a changing on the ¨culture environment¨ within the company, making it more suitable towards sustainability. Therefore, more complex models of sustainable integration within companies such as the ¨Convergence¨ Research Project [START_REF] Zhang | ¨Towards a Systematic Navigation Framework to Integrate Sustainable Development into the Company[END_REF] can be implemented.

This ¨Convergence¨ model aims to integrate sustainability into the three levels. At the strategic level, it proposes a diagnosis of the current situation within the company to define sustainable strategies which can aid with the maturities grids of integration and governance. At the tactical level, a pilot assistance and optimization tool by offering roadmaps (selection of methods, resources and initial steps) adapted to the context of the company, embodied in an action plan. And finally, at the operational level to identify ecodesign actions to be taken and tools to be used at each stage of the PDP and promote an exchange of information between business experts within the PDP.

  Similarly, companies have different strategies of sustainable development and they focus on different indicators attributing different weights to individual indicator. Therefore, if necessary, the next procedural part it would be calculating a generic index I which involves determining weights with each indicator. As a consequence, it is suggested the use of the Analytic Hierarchy Process (AHP) [40]. The weight of each indicator can be delivered by the prioritization of their impact to the overall sustainability assessment of the company. Therefore, let us assume that N indicators of sustainable development are being considered with the goal of providing and quantifying judgments on the relative weight of each indicator with respect to all the other indicators. The first step is to set up the overall sustainability assessment of the company hierarchically, where the topmost aspect X is the main objective of the company, while subsequent aspects at lower levels consist of the criteria used in arriving at this overall sustainability assessment. The second step requires pair-wise comparisons to be made between each pair of indicators (of the given level of the hierarchy). The comparisons are made by posing the question which of the two indicators i or j is more important with respect to the sustainable development of the company, respectively. For instance, the intense of preference can be expressed on a factor scale from 1 to 9 as suggested by Hafeez et al. [41]. This scale was chosen, because in this way comparisons are being made within a limited range where perception is sensitive enough to make a distinction.

Table 1 .

 1 Relationship between Index and Company Type

	Index I	Company type
	-1 to 0.0	1
	>0.0 to 0.2	2
	>0.2 to 0.4	3
	>0.4 to 0.6	4
	>0.6 to 0.8	5
	>0.8 to 1.0	6

Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org)

ACKNOWLEDGEMENTS

The authors would like to thank the Direction de L´Enseignement Superior de la Recherhe et de L´Innovation, Region Champagne-Ardene and the École National Supérieure d´Arts et Métiers for the financial support of this research.


Youssef Benama

Thecle Alix

Nicolas Perry
email: nicolas.perry@ensam.eu

Framework definition for the design of a mobile manufacturing system

Keywords: production system, mobile manufacturing system, design of manufacturing plant

The concept of mobile manufacturing systems is presented in the literature as an enabler for improving company competitiveness by cost reduction, delay respect and quality control. In comparison with classical sedentary systems, added characteristics should be taken into consideration, such as the system life phases, the dependency to the production location, human qualification as well as means supply constraints. Such considerations might be addressed as soon as possible in the design process. This paper aims at presenting a contribution for the design of mobile manufacturing systems based on three analysis: (1) an analysis of the mobile manufacturing system features (2) an identification of the attributes enabling the system mobility assessment, and (3) the proposal of a framework for mobile production system design considering new contextspecific decision criteria.

Introduction

Ensuring shipment of bulky and fragile product can be economically and technically challenging. The solution that can be adopted is to conduct production activities close to the end client. In case of a onetime demand, implanting a permanent production plant may seem unrealistic and then the concept of Mobile Manufacturing System (MMS) that consists in using the same production system to satisfy successively several geographically dispersed customer orders, directly on the end client location can be a good alternative.

The use of mobility of production systems has been encountered in many industries: construction industry [START_REF] Erwin | Mobile onsite factoriesscalable and distributed manufacturing systems for the construction industry[END_REF], shipyard industry, etc. As interesting as it seems, the concept has been rarely discussed in the literature. The few existing definitions of mobility depend on authors and contexts [START_REF] Stillström | The concept of mobile manufacturing[END_REF]. Mobility is also defined at different levels for manufacturing system. There is an internal mobility concerning manufacturing system modules (machinery, material handling modules, etc.) and a global or external mobility concerning the movement of the whole manufacturing system. This last level is analyzed across geographic areas and underpins strategic considerations with medium to longterm implications.

In order to facilitate the movement of the manufacturing system to a new geographical location, Rösiö [START_REF] Rösiö | Supporting the design of reconfigurable production systems[END_REF] evokes three required characteristics that are: the mobility of module, the modularity [START_REF] Flores | Contribution aux methods de conception modulaire de produits et processus industriels[END_REF] and the integrability of modules.

In this paper, a holistic view of the manufacturing system is adopted. The mobility of the manufacturing system is defined as the ability of a manufacturing system, defined by its technical, human and information components, to move and produce on a number of successive geographical locations. The definition includes two aspects:  Transportability: the manufacturing system must be transportable and must be able to adapt to the requirements of the different transportation modes (road, sea, etc.)  Operationality: the system must be able to be quickly operational on different locations for which it is designed. The following section proposes to discuss how a Mobile Manufacturing System (MMS) may differ from a sedentary manufacturing system.

Requirements for manufacturing system mobility

The concept of mobility implies to consider some additional system lifephases regarding traditional manufacturing systems: mobility of modules, onsite maintenance management, organizational aspects and training needs, energy supply.

Manufacturing system design

The production system design process is based on four macro phases [START_REF] Pahl | Engineering Design: A Systematic Approach[END_REF]: (1) initialization, (2) preliminary design, (3) embodiment design and, (4) detailed design. Each of these phases consists in selection, evaluation and decision activities. Taking into account the characteristics of mobility is built through each phase of the MMS design process. Obviously, in a context of mobility, the production system environment changes from one implementation location to another and the analysis of the system' environment is of huge importance.

A production system is currently seen as a system composed of several subsystems generally analyzed through an external and internal views coupled with other physical system, decisions and informational views. Then the production system design depends on the design (or selection of items when solutions may exist in the market) of each system component, but also depends on the connection between these subsystems for their integration into the overall system.

A production system can be defined as a system of systems to the extent that on the one hand, it is composed of a set of subsystems. Each one has its own life cycle and each one may be defined independently from the others. On the other hand, interactions between these subsystems define constraints for the system of systems, affecting the performance of the overall system [START_REF] Alfieri | Usage of SoS Methodologies in Production System Design[END_REF].

The Systems engineering adopts two complementary points of view for systems analysis [START_REF] Fiorèse | Découvrir et comprendre l'ingénierie système[END_REF]:  An external view or "black box" approach defining the system boundaries used to identify the external environment elements that force the system and that the system must respond to by providing the expected services. The environment is defined by all factors that might influence or be influenced by the system [START_REF] Fiorèse | Découvrir et comprendre l'ingénierie système[END_REF].  An internal view or "white box" approaches that considers the internal system interacting elements, which define its organization (architecture) and its operation.

Additional system life-phases

During its operation, the MMS is first put into service on its implantation site before being used for production. Throughout this phase, maintenance and configuration operations carried out in order to adapt its behavior to meet at best the expected performance. However, unlike to sedentary manufacturing systems, Mobility requires additional operational phases:  Transportation phase (a): the MMS is packaged and transported to its implantation location.  Onsite installation phase (b): the MMS arriving on site is composed of independent modules and components that are integrated and lead to the plant installation. Upstream, operations to prepare the site are performed. Downstream, the factory is installed and operations of verification and commissioning are carried out.  Onsite production phase (c): the plant is used to produce locally. In parallel, maintenance operations are necessary to maintain high system performance.  Diagnosis and control phase (d): at the end of the production phase, a diagnosis of all modules is carried out to ensure that the mobile plant will be operational for the next production run. The modules requiring heavy maintenance or replacement are identified. Replacement and procurement orders are launched during this phase.  Dismantling phase (e): the plant is dismantled. Various modules and components are conditioned and prepared for the transportation phase.  Transportation phase (f): the modules are placed in the transportation configuration, two scenarios are then possible depending on the business strategy of the company: o A new order arrives and a new site is identified. The MMS is routed to the new location and the operational cycle resumes at the phase (b);

o
No new order is identified and thus no new implantation location is identified. The MMS is then routed to its storage location that corresponds to the phase (g). Depending on negotiations with the manager (client, institution, etc.) of the site where the system has been used, the MMS storage phase could take place in the former location in the expectation of a new order.  Storage phase (g): During the MMS inoperability period, the modules have to be stored until a new order. The storage can take place on the stationary basis, or on the latest operating location in order to stay closer to a potential market. During this phase, heavy maintenance operations can be conducted such as: maintenance or replacement of machines, modules reconfiguration, etc.

The identification of the lifephases is important as evaluating the overall performance (cost, delay, etc.) of the system depends on it.

Organizational aspects and training needs

Geographic mobility of the manufacturing system requires adapting the automation level required to the qualification level of the personnel available on site. To ensure the production system independence regarding the onsite operator qualification, the level of the manufacturing system automation must be adapted. An independent production system to operator's qualification can be imagined as a highly automated system. However, too many automation leads to a complexity requiring some expertise to ensure MMS maintenance operations. A tradeoff must be achieved between the required automation level and the onsite available qualification. An onsite operator' training offer facilitate this tradeoff.

System mobility means that each time a new team is involved in the system for a new implantation location [START_REF] Olsson | Experience reuse between mobile production modulesan enabler for the factoryinabox concept[END_REF]. Hence, the need to provide operator training for running the manufacturing system is crucial. Moreover, Fox recalls the need for a qualified local middle management which makes the link between foreign personnel and the local population, and who could be also responsible for applying best practices [START_REF] Fox | Moveable factories: How to enable sustainable widespread manufacturing by local people in regions without manufacturing skills and infrastructure[END_REF].

Mobility of modules

Mobility of the manufacturing system modules implies that each module is being transportable and operational on site. Modularity is an enabler of component mobility. The weight and volume of each module must be compliant to transportation modes. In addition, the modules must withstand different transportation constraints (mechanical shock, tightness constraints, etc.). Finally, the equipment onsite operationality (equipment) must adapt to onsite available energy sources. The equipment must be easily integrated and commissioned.

On-site maintenance management

On the one hand, maintaining system performance during the operation phase implies to adopt a comprehensive strategy that takes into account the duration of the manufacturing system presence on a specific implantation location in order to minimize the need for shutdown. On the other hand, in order to carry out onsite interventions, spare parts supply chain management must be adapted according to the manufacturing system mobility.

Energy supply

Depending of each implantation location characteristics, the energy supply issue arises each time. The MMS autonomy depends on its ability to be independent in supplying the necessary energy required for the operations of its resources [START_REF] Erwin | Mobile onsite factoriesscalable and distributed manufacturing systems for the construction industry[END_REF]. The energy supply system can be based on diesel generators or by using solar panels to provide the necessary power [START_REF] Fox | Moveable factories: How to enable sustainable widespread manufacturing by local people in regions without manufacturing skills and infrastructure[END_REF]. The issue of energy consumption (nature and quantity) can be a determining factor for choosing the MMS constituent resources.

After reviewing the requirements to be taken into account into a mobile manufacturing system analysis, we propose to discuss in the next section the system design issue

A design framework adapted for a single implantation location

The sequence of the key steps in the MMS design process [START_REF] Benama | Formalisation de la demarche de conception de système de production mobile : integration des concepts de mobilité et de reconfigurabilité[END_REF] (figure1) starts with 1) a refinement of the requirement specification 2) the determination of what is to be carried out inhouse or is to be outsourced and 3) some technical solutions proposal (MMS configuration design). These three steps are discussed hereafter.

Requirements specification refinement

The design activity starts from the requirements specification that contains a description of the product to be manufactured (BOM) and details the client's request (production volume, delays, requirements, etc.). The initial requirements specification will be supplemented with information and details obtained after MMS and implantation location environment analysis. This first enhanced specifications version (noted CdC_1 in figure 1) allows imagining a first configuration of MMS. This MMS configuration, not economically efficient, represents a generic definition able to satisfy the demand on the proposed location.

Manufacturing strategy analysis

The MMS generic configuration will be then refined through an analysis of what is relevant to produce on site or what needs to be outsourced. This analysis involves several criteria and requires the establishment of evaluation process and decision support [START_REF] Benama | Supporting make or buy decision for reconfigurable manufacturing system[END_REF]. The analysis of the make or buy strategy enables to decide the MMS functionalities, i.e. operations that the MMS should be able to carry on the implantation location. The description of the necessary MMS functionalities supplements the previous requirements specification (noted CdC in figure 1). The MMS design activity can be now conducted.

Design of MMS configuration

This activity considers as input the latest requirements specification version and technical data about all the resources that will be integrated into the MMS configuration as well as the production management information. The choice of MMS configuration is based on classical criteria such as cost, quality and delay; in addition to other criteria specific to the mobility concept, namely: the mobility indicator

Design of MMS configuration

This activity considers as input the latest version of the specification and the technical data about all resources that will be integrated into the MMS configuration as well as production management information and assumptions. The choice of the MMS configuration is based on several decision criteria.

In addition to the typical cost, quality and delay requirements, the proposed approach incorporates new criteria that are specific to the context of mobility [START_REF] Benama | Formalisation de la demarche de conception de système de production mobile : integration des concepts de mobilité et de reconfigurabilité[END_REF]: the mobility index, the integrability index and the criterion of onsite resources availability

Mobility index

Analyzing mobility during the embodiment design phase concerns the whole production system defined by all its components. These components can be classified into two categories: technical equipment and human modules. The assessment of technical equipment and human modules mobility is based on different approaches involving several criteria. It is therefore necessary to evaluate each category and then aggregate the results to give a unique appreciation of the whole manufacturing system mobility [START_REF] Benama | Formalisation de la demarche de conception de système de production mobile : integration des concepts de mobilité et de reconfigurabilité[END_REF]. This appreciation can be expressed by a quantitative value between 0 and 1 that indicates a satisfaction index. The index construction approach is based on a multicriteria analysis. Two important concepts are used: the expression of preference and the criteria aggregation.

On the one hand, the mobility of MMS technical module has to be satisfied through all its life phases. To be mobile, a technical module must be: transportable, mountable on site, operating on site and dismantled. On the other hand, the human system operates by providing flexible working ability to carry out simple or complex operations contributing to the functioning of MMS. This requires skills acquired or developed onsite during the onsite production phase. The human system mobility can be understood as the mobility of one or more skills necessary for the manufacturing system operation.

Integrability index

Generating a MMS configuration consists in the integration of various independent modules (machines, operators, conveyors, etc.). In order to have feasible configurations, it is necessary to ensure that the selected modules can be integrated with each other. Each module has one or more interfaces to bind to other modules. The Integrability evaluation process of a MMS configuration combines two approaches [START_REF] Benama | Formalisation de la demarche de conception de système de production mobile : integration des concepts de mobilité et de reconfigurabilité[END_REF]:  A decomposition analysis approach (topdown): The MMS configuration is broken down into individual modules. Each module integrates common interfaces with one or more other MMS modules. The analysis of Integrability is carried out at the level of each MMS configuration' elementary module.  An assessment approach based on integration (bottomup): it is based on the definition and evaluation of all nodes in the system configuration. Individual measurements will be aggregated to give a single measure of the MMS configuration' Integrability.

Criterion of on-site resource availability

For a given MMS configuration, the evaluation phase of the availability of the competences starts with the assessment of required skills in this configuration. Thus, for each configuration' entity, required skills are identified based on the attribute "needed sills" contained in the description of each resource. This attribute is faced to available competences on the implantation location. An evaluation method is proposed to ensure that the required resources by the MMS configuration that had been suggested are available on the implantation location [START_REF] Benama | Formalisation de la demarche de conception de système de production mobile : integration des concepts de mobilité et de reconfigurabilité[END_REF]. The assessment of skills availability is split into three stages: identification of the required skills, identification of relevant actor profiles and assessment of the profiles availability on the implantation location.

Conclusions

In this communication, we discussed the concept of mobile manufacturing system. The mobility requirements were addressed and a mobile manufacturing system design framework is presented. The design process is based on some decision criteria. In addition to the typical cost, quality and delay criteria, three other decision criteria are proposed: the mobility index, the integrability index and a criterion of onsite resources availability. The proposed design approach is limited to the consideration of a single implantation location. However, the concept of successive mobility requires that the same production system is operated successively on several implantation locations. The design approach must be adapted to the multisites context by integrating the concept of reconfigurability. A first analysis of this issue is presented in [START_REF] Benama | Formalisation de la demarche de conception de système de production mobile : integration des concepts de mobilité et de reconfigurabilité[END_REF]. This issue of successive multi sites mobility will be addressed in future communications.

Fig. 1

 1 Fig. 1 Mobile Manufacturing system design framework adapted for single implantation location


Guillaume Mercère
email: guillaume.mercere@univ-poitiers.fr

Régis Ouvrard
email: regis.ouvrard@univ-poitiers.fr

Marion Gilson
email: marion.gilson@cran.uhp-nancy.fr

Hugues Garnier
email: hugues.garnier@cran.uhp-nancy.fr

Subspace based methods for continuous-time model identification of MIMO systems from filtered sampled data

teaching and research institutions in France or abroad, or from public or private research centers.

I. INTRODUCTION

Subspace based identification methods (SIM) have received an increasing attention in the last fifteen years [START_REF] Van Overschee | Subspace identification for linear systems[END_REF], [START_REF] Katayama | Subspace methods for system identification[END_REF]. These techniques have indeed the property to supply a linear state space model of the system from input output (I/O) data without non linear optimization generally required by the prediction methods [START_REF] Ljung | System identification. Theory for the user[END_REF] with the help of computational tools such as the RQ factorization and the singular value decomposition (SVD). Most of these methods have been developed for discrete-time (DT) models [START_REF] Viberg | Subspace based methods for the identification of linear time invariant systems[END_REF], [START_REF] Bauer | Asymptotic properties of subspace estimators[END_REF]. However, in many applications (modelling, diagnosis, control), it is interesting to deal with a continuous-time (CT) model in order to have access to the physical parameters of the system. To reach this goal, a first indirect approach consists in firstly determining a DT model via conventional identification algorithms [START_REF] Ljung | System identification. Theory for the user[END_REF], then converting this representation in a CT model. The statistical properties in this framework are indeed well known and explain the affinity with this point of view. Unfortunately, this indirect approach suffers from many limitations, as [START_REF] Unbehauen | Continuous time approaches to system identification: a survey[END_REF] All these restrictions can be alleviated with the direct approach which consists in directly estimating a CT model from sampled I/O data [START_REF] Unbehauen | Continuous time approaches to system identification: a survey[END_REF] [START_REF] Garnier | Identification of continuous-time models from sampled data[END_REF]. The main problem of these techniques is the computation of the successive I/O time derivatives. The solution often rests on the use of linear filters, integral methods or modulating functions [START_REF] Garnier | Continuous-time model identification from sampled data: implementation issues and performance evaluation[END_REF]. In this paper, the reinitialised partial moments [START_REF] Djamai | Continuous time model identification using reinitialized partial moments. Application to power amplifier modeling[END_REF] and the Poisson moment functionals [START_REF] Saha | Identification of continuous dynamical systems -The Poisson Moment Functionals (PMF) approach[END_REF] are more particularly considered. Despite a development in a different framework, these techniques can be rewritten as an I/O filtering approach and supply a solution to the time derivative calculation which is relatively straightforward.

The direct CT estimation methods have essentially been developed for SISO systems. The MIMO framework is comparatively not so developed [START_REF] Haverkamp | Identification of continuous time MIMO state space models from sampled data in the presence of process and measurement noise[END_REF], [START_REF] Johansson | Stochastic theory of continuous time state space identification[END_REF], [START_REF] Haverkamp | State identification: theory and practice[END_REF], [START_REF] Bastogne | A PMF-based subspace method for continuous-time model identification. Application to a multivariable winding process[END_REF], [START_REF] Ohsumi | Subspace identification for continuous time stochastic systems via distribution based approach[END_REF], [START_REF] Li | Subspace identification of continuous time models for process fault detection and isolation[END_REF]. In this article, this problem is considered by associating respectively the reinitialised partial moments and the Poisson moment functionals to a particular class of SIM: the MOESP algorithms [START_REF] Verhaegen | Subspace model identification part 1: output error state space model identification class of algorithms[END_REF], [START_REF]Subspace model identification part 2: analysis of the elementary output error state space model identification algorithm[END_REF], [START_REF] Verhaegen | Subspace model identification part 3: analysis of the ordinary output error state space model identification algorithm[END_REF], [START_REF]Identification of the deterministic part of MIMO state space models given in innovations form from input output data[END_REF]. These subspace techniques, mainly based on the observability subspace estimation, use orthogonal projections which increases the global numerical robustness. The disturbances problem is treated by introducing a particular instrumental variable for the estimation of the system observability matrix basis.

The paper is organized as follows. Section II introduces the identification problem. Section III briefly presents both techniques used to circumvent the time derivative problem and gives the filtered data equation used in CT subspace identification. Then, the development of two new estimation algorithms is considered in Section IV. The problem of the instrumental variable construction for consistently modelling the system from noisy I/O data is more particularly analyzed. Some practical conditions for the choice of the user parameters are also given. In Section V, the performances of these new techniques are illustrated from numerical simulations.

II. PROBLEM FORMULATION AND NOTATIONS

Consider a linear time invariant CT system described by the following state space equations

ẋ(t) = Ax(t) + Bu(t) + w(t) (1a) y(t) = Cx(t) + Du(t) + v(t) (1b)
where u ∈ R nu and y ∈ R ny are respectively the input and output signal, x ∈ R nx the state vector and w and v the zero mean process and output measurement stochastic disturbances. Following the same reasoning as in [START_REF] Johansson | Identification of continuous time models[END_REF], it is supposed that the process and the measurement noise are constant between sampling instances: w(t) = w(kT s ) and v(t) = v(kT s ) for kT s ≤ t < (k + 1)T s , where T s denotes the sampling period. These disturbances are furthermore assumed to be DT white noise sequences with the following correlation

E v(kT s ) w(kT s ) v T (lT s ) w T (lT s ) = R v R vw R T vw R w δ(k, l) (2)
where δ(k, l) is the Kronecker delta function and E {•} the mathematical expectation. The justification for assuming such noise models for v(t) and w(t) are given in [START_REF] Johansson | Identification of continuous time models[END_REF], [START_REF] Haverkamp | Identification of continuous time MIMO state space models from sampled data in the presence of process and measurement noise[END_REF]. The goal of CT subspace identification is to estimate the state space matrices {A, B, C, D}, up to a similarity transformation, from DT I/O data {u(t k ), y(t k )} N k=1 with t k = kT s .

In order to introduce the CT subspace identification problem and the link with the DT approach, assume firstly that

• the disturbances v and w are equal to zero,

• the signals u and y are differentiable at least to the (i -1)th time derivative with i > n x . Then, the following yields [START_REF] Bastogne | A PMF-based subspace method for continuous-time model identification. Application to a multivariable winding process[END_REF]

y i (t k ) = Γ i x(t k ) + H i u i (t k ) (3)
where

y i (t k ) (resp. u i (t k ), w i (t k ) and v i (t k )) is a stacked vector composed by i -1 time derivatives of y (resp. u, w and v) at t = t k y i (t k ) = y T (t k ) y (1)T (t k ) • • • y (i-1) T (t k ) T , (4)
Γ i is the extended observability matrix

Γ i = C T (CA) T • • • CA i-1 T T (5)
and H i is the block Toeplitz matrices of the impulse response from u to y. For N -k + 1 samples, (3) becomes

Y i = Γ i X + H i U i (6)
with1

Y i = y i (t k ) y i (t k+1 ) • • • y i (t N ) (7)
and

X = x(t k ) x(t k+1 ) • • • x(t N ) . (8)
Equation ( 6) has the particularity to be structurally equivalent to the so called data equation used in DT subspace identification [START_REF] Viberg | Subspace based methods for the identification of linear time invariant systems[END_REF]. Thus, it seems possible to apply the classical following steps to estimate the system state space matrices [START_REF]Identification of the deterministic part of MIMO state space models given in innovations form from input output data[END_REF] • a compression of the I/O data via an RQ factorization,

• the estimation of a basis of the extended observability matrix from specific submatrices of the resulting lower triangular factor in a consistent way via singular value decomposition (SVD),

• the extraction of the A and C matrices via the Ainvariance of Γ i and the estimation of B and D from a linear regression optimization. However, this algorithmic procedure is based on an unavoidable assumption: the successive time derivatives of the data have to be measured to obtain equation [START_REF] Unbehauen | Continuous time approaches to system identification: a survey[END_REF]. Unfortunately, in practice, the direct computation of the time derivatives of the I/O is seldom feasible, particularly in a noisy framework. It is therefore essential to resort to particular techniques which are able to evaluate these time derivatives from measured sampled data. Many algorithms have been developed to reach this goal during the last three decades (see [START_REF] Garnier | Continuous-time model identification from sampled data: implementation issues and performance evaluation[END_REF] for an overview). A comparative study based on Monte Carlo simulations has shown that six CT methods present the best performances. Two2 of them are considered in the following: the Poisson moment functional (pmf ) approach [START_REF] Young | Parameter estimation for continuous time models: a survey[END_REF] and the reinitialised partial moments (rpm) method [START_REF] Jemni | A comparative study of the deterministic accuracy of two identification methods for continuous time systems[END_REF].

III. APPROXIMATION OF THE I/O TIME DERIVATIVE DATA

A. The reinitialised partial moments (rpm)

The reinitialised partial moments method [START_REF] Djamai | Continuous time model identification using reinitialized partial moments. Application to power amplifier modeling[END_REF] can be firstly considered as an integration method, i.e. a technique which avoids the differentiation of the data by performing a sufficient order integration. Let s(t) be a function defined on [0, ∞[. Then, the nth reinitialised partial moment is defined as3

M n {s(t)} = T 0 τ n n! s(t -T + τ )dτ. (9)
This integration function is an extension of the classical moment ∞ 0 τ n s(τ )dτ . The main characteristics of the rpm are to perform a time weighing of the I/O signals and a time shifting before integrating them. Concerning the later, the prediction error estimation variance is minimal for an optimal value of T = T . In order to take advantage of this property, the integration is reinitialised at each sampling time. Although the determination of T is essential, experiences show that the rpm method is not too sensitive to this user parameter. In practice, it can be chosen about twice as the main time constant for an aperiodic system or twice as the rising time for an underdamped system.

Secondly, the rpm method can be introduced from a filtering point of view. Indeed, the integration can be expressed as a time convolution product

Mn {s(t)} = T 0 ( T -τ ) n τ n-1 T n (n -1)! s(t -τ )dτ = [h n ⋆ s](t) (10)
where

h n (t) = ( T -t) n t n-1 T n (n -1)! for t ∈ [0, T ] (11a)
h n (t) = 0 elsewhere. (11b)

Thus, since the integration window is a time shifting window, the use of the rpm in the considered identification framework amounts to apply a finite impulse response filter to the I/O data. This property results in making the initial conditions vanish. Furhermore, for the same reason, the first T samples must be removed.

B. The Poisson moment functionals (pmf)

An alternative solution to handle the time derivative problem is to use the Poisson moment functionals approach which can also be written in terms of filtering. The nth order pmf transform of the signal s(t) over [0, t] is indeed given by the following convolution product [START_REF] Saha | Identification of continuous dynamical systems -The Poisson Moment Functionals (PMF) approach[END_REF]

M n {s(t)} = t 0 g n (τ )s(t -τ )dτ = [g n ⋆ s](t), (12)
where

• g n (t) = β n+1 t n n! e -
L(g n (t)) = β p + λ n+1 ( 13
)
where p is the Laplace variable. The nth order Poisson moment functional of s(t) can be considered as the output of a cascaded low-pass filter chain of n + 1 identical stages, each with a transfer function given by β/(p + λ). The user parameter is the cut-off frequency of the filter. It should be chosen in order to emphasise the frequency band of interest and it is advised in general to choose it a little bit larger than the bandwidth of the system to be identified. Moreover, it has been shown that the initial conditions can be neglected after a short period of time [START_REF] Garnier | Continuous-time model identification from sampled data: implementation issues and performance evaluation[END_REF].

C. Filtered data equation

The description of these two transformations via a time convolution product allows us to formulate their use for our identification problem in a global way. More precisely, if the impulse response m n (t) respectively stands for

m n (t) = h n (t) for the rpm g n (t) for the pmf , (14)
for all vector z i (t) ∈ R nzi defined as

z i (t) = z T (t) z (1) T (t) • • • z (i-1) T (t) T , (15)
it is straightforward to check that

[z i ⋆ m n ](t) =      [z ⋆ m n ](t) [z (1) ⋆ m n ](t) . . . [z (i-1) ⋆ m n ](t)      =      [z ⋆ m n ](t) [z ⋆ m (1)
n ](t) . . .

[z ⋆ m (i-1) n ](t)      :=      M n 0 {z(t)} M n 1 {z(t)} . . . M n i-1 {z(t)}      (16)
with n ≥ i + 1 by assuming that4

m (k-1) n holds for k ∈ [1, i] (17)
m (k-1) n (t = 0) = 0 for k ∈ [1, i] (18)
m (k-1) n (t = θ) = 0 for k ∈ [1, i] (19)
where θ = T for the rpm and θ = t for the pmf. This transformation, which is linked to the properties of the convolution product, allows to transfer the derivative operation on the function m n which is differentiable at order n by construction. The I/O measurement time derivative problem is also circumvented. Thus, equation ( 6) becomes (in a noisy framework)

M {Y i } = Γ i M {X} + H i M {U i } + G i M {W i } + M {V i } (20)
where M {•} denotes the transform applied to sampled data to approximate the I/O data time derivative

M {Y i } =      M n 0 {y(t k )} • • • M n 0 {y(t N )} M n 1 {y(t k )} • • • M n 1 {y(t N )} . . . . . . M n i-1 {y(t k )} • • • M n i-1 {y(t N )}      , (21) where
M {X} = M n 0 {x(t k )} • • • M n 0 {x(t N )} (22)
and where G i is the block Toeplitz matrices of the impulse response from w to y.

IV. SUBSPACE IDENTIFICATION ALGORITHM

Consider from now the equation of the filtered data [START_REF] Verhaegen | Subspace model identification part 3: analysis of the ordinary output error state space model identification algorithm[END_REF]. Then, assuming that the input is sufficiently exciting (see [START_REF] Jansson | On consistency of subspace methods for system identification[END_REF] for conditions), that the initial condition are zero and that the transformation does not remove any state variable, (i.e. rank {M {X}} = n x ), it is possible to apply the DT identification algorithms to the CT problem. The MOESP schemes [START_REF] Verhaegen | Subspace model identification part 1: output error state space model identification class of algorithms[END_REF], [START_REF]Subspace model identification part 2: analysis of the elementary output error state space model identification algorithm[END_REF], [START_REF] Verhaegen | Subspace model identification part 3: analysis of the ordinary output error state space model identification algorithm[END_REF], [START_REF]Identification of the deterministic part of MIMO state space models given in innovations form from input output data[END_REF] are more precisely considered in the following. The algorithms gathered under this acronym have indeed some interesting numerical advantages due to the use of orthogonal projections (see [START_REF] Goethals | Subspace identification for linear, Hammerstein and Hammerstein Wiener systems[END_REF]Chapter 5] for an interesting study).

A. Identification in a noisy framework

In a noisy general case 5 , i.e. when both v and w are non zero, the DT MOESP algorithms (PI MOESP and PO MOESP [START_REF] Verhaegen | Subspace model identification part 3: analysis of the ordinary output error state space model identification algorithm[END_REF], [START_REF]Identification of the deterministic part of MIMO state space models given in innovations form from input output data[END_REF]) are based on a particular RQ factorization where an instrumental variable is introduced to remove the effects of the noise term. In the CT framework considered in this paper, we suggest to use the following RQ factorization

  M {U i } Ξ M {Y i }   =   R 11 0 0 R 21 R 22 0 R 31 R 32 R 33     Q 1 Q 2 Q 3   ( 23
)
where 6 Ξ is made of instruments chosen such that the disturbances effects are asymptotically cancelled without modifying the informative part of the signal terms, i.e.

lim N →∞ 1 N M {V i } Ξ T = lim N →∞ 1 N M {W i } Ξ T = 0 (24a) rank lim N →∞ 1 N M {X} Ξ T = n x . (24b)
Indeed, since this RQ factorization gives access to the following relation [START_REF] Viberg | Subspace based methods for the identification of linear time invariant systems[END_REF]

R 32 Q 2 = M {Y i } Π M{Ui} ⊥ Ξ T P d ( 25
)
where P d is a particular weighting matrix depending on M {U i } and Ξ. Then, it is relatively straightforward to show that (see Appendix)

lim N →∞ 1 N R 32 Q 2 = lim N →∞ 1 N Γ i M {X} Π M{Ui} ⊥ Ξ T P d (26)
under the assumptions that the noise terms are uncorrelated with the input signal and the instruments. Hence, equation [START_REF] Jansson | On consistency of subspace methods for system identification[END_REF] proves that it is possible to estimate a basis of the observability matrix from R 32 when

rank M {X} Π M{Ui} ⊥ Ξ T P d = n x . ( 27
)
This basis is obtained by applying a SVD to this block matrix. At the same time, the order estimation can be realised by analyzing the singular values which follow from this decomposition. Some estimation techniques, based on particular information criteria, are available in the literature [START_REF] Bauer | Order estimation for subspace methods[END_REF], [START_REF] Bauer | Asymptotic properties of subspace estimators[END_REF].

B. Choice of the instrumental variable

The algorithm developed in the previous subsection is based on an essential assumption: the instrumental variable has to be chosen such as equation ( 24) is fulfilled. In general, in a CT framework, the instruments are built up by increasing the time derivative order i and, by extension, the number of rows of the matrices composing the filtered data equation [START_REF] Verhaegen | Subspace model identification part 3: analysis of the ordinary output error state space model identification algorithm[END_REF] [START_REF] Johansson | Stochastic theory of continuous time state space identification[END_REF], [START_REF] Haverkamp | State identification: theory and practice[END_REF], [START_REF] Ohsumi | Subspace identification for continuous time stochastic systems via distribution based approach[END_REF]. For example, R. Johansson et al recommend to choose i ≥ n u + n y + n x . The main drawback of this approach is that time derivative of high order have to be computed. Moreover, it is well known that using many approximations of time derivative practically leads to numerical performances corruption. Two solutions are therefore presented is the following to overcome this drawback.

1) Use of past inputs as IV:

It is firstly suggested to use in a first time shifted version of the sampled input data and build up a particular Hankel matrix

Ξ = U p =    u((k -p)T s ) u((k + 1 -p)T s ) • • • u((N -p)T s ) . . . . . . . . . u((k -1)T s ) u(kT s ) • • • u((N -1)T s )    (28)
as considered in the DT case [START_REF] Verhaegen | Subspace model identification part 3: analysis of the ordinary output error state space model identification algorithm[END_REF]. A straightforward way to study the efficiency and properties of this instrumental variable consists in analyzing the following limits

lim N →∞ 1 N M {V i } U T p (29a) lim N →∞ 1 N M {W i } U T p . (29b)
Since v and w have the same characteristics, only equation (29a) is used. Let us consider the following qth block row and rth block column entry of

1 N M {V i } U T p 1 N M {V i } U T p qr = 1 N N j=1
M n q {v(t j )} u T ((j + r -p)T s ). [START_REF] Gilson | Instumental variable methods for closed loop system identification[END_REF] This leads to

1 N M {V i } U T p qr = θ 0 m (q) n (τ )   1 N N j=k v(jT s -τ )u T ((j + r -p)T s )   dτ (31)
where m

(q)
n is defined as in ( 14) according to the chosen filtering method and θ = T for the rpm (32a) θ = t for the pmf .

(32b)

By assuming that

• the input and noise signals are ergodic processes,

• the system operates in open loop, it holds that

lim N →∞ 1 N N j=k v(jT s -τ )u T ((j + r -p)T s ) = 0. (33) Hence, since each component of lim N →∞ 1 N M {V i } U T p tends to 0 when N → ∞, equation (24a) is satisfied.
Concerning the value of the user parameter p, it has been shown in the DT framework [START_REF] Viberg | Subspace based methods for the identification of linear time invariant systems[END_REF], [START_REF] Jansson | On consistency of subspace methods for system identification[END_REF]

that choosing p ≥ nx nu guarantees that lim N →∞ 1 N X f Π U ⊥ f U T
p exists and has full rank n x , (34)

with U f the Hankel matrix of "future" sampled inputs and X f composed by "future" state vectors. Although the CT problem considers filtered versions of the input and the state signals, the condition p ≥ nx nu is a necessary requirement to have the number of rows of U p at least equal to n x . Thus, since the instrumental variable is made up with the same reasoning as in the DT case, this condition on p can be used in our CT identification problem. An easy way to choose p consists, for example, in using p = i since i > n x .

2) Use of past outputs as IV:

In the DT framework, it is often suggested to introduce past output data among the instruments in order to get more accurate estimates [START_REF]Identification of the deterministic part of MIMO state space models given in innovations form from input output data[END_REF]. Thus, it is interesting to analyse the effect of the use of Y p as instrumental variable. First of all, consider the following relation

Y p = Γ p X p + H p U p + G p V p + W p ( 35
)
where Γ p , H p and G p are defined as previously with i = p, where Y p , V p and V p are built up as U p and where

X p = x((k -p)T s ) • • • x((N -p)T s ) . (36)
Then, we have 7

1 N M {V i } Y T p = 1 N M {V i } X T p Γ T p + M {V i } U T p H T p +M {V i } V T p G T p + M {V i } W T p . ( 37
)
By assuming that the input and noise signals are uncorrelated, the second term of the right-and-side (rhs) of (37) vanishes as N tends to infinity. The third and fourth elements of the rhs of (37) can be studied in the same way. Consider e.g.

1 N M {V i } V T p .
Then, following the same approach as used previously, its qth block row and rth block column entry is

1 N M {V i } V T p qr = θ 0 m (q) n (τ )   1 N N j=k v(jT s -τ )v T ((j + r -p)T s )   dτ (38)
Under the assumption of ergodicity, it is straightforward to show that (see (2)) 7 The same approach is applied for M {W i }.

lim N →∞ 1 N N j=k v(jT s -τ )v T ((j + r -p)T s ) = R v δ(τ, (p -r)T s ). (39)
Thus

lim N →∞ 1 N M {V i } V T p qr = R v θ 0 m (q) n (τ )δ(τ, (p -r)T s )dτ. (40)
The problem of relation ( 40) is that the components of M {V i } V T p can be bounded asymptotically. However, since τ belongs to [0, θ], requiring (p -r)T s > θ guarantees that, for all p, δ(τ, (p -r)T s ) equals zero and, by extension,

lim N →∞ 1 N M {V i } V T p = 0.
The condition (p-r)T s > θ is unfortunately very conservative in practice. Thus, in general, reasonable values of p in the past output scheme will lead to estimates of the system matrices with a bounded bias. A good way to circumvent this difficulty is to consider a classical bootstrapping approach, i.e. using noise free output data as instruments constructed from the simulation of a (biased) model estimated beforehand. This type of iterative scheme can be used to introduce optimal instrumental variable [START_REF] Stoica | Optimal instrumental variable methods for identification of multivariable linear systems[END_REF], [START_REF] Gilson | Instumental variable methods for closed loop system identification[END_REF], [START_REF] Gilson | Subspace based optimal IV method for closed loop system identification[END_REF].

V. SIMULATION EXAMPLE

Consider the following 3rd order state space system

ẋ(t) =   0 1 0 -3 -2 -1 -1 -2 -1   x(t) +   1 1 2 1 1 2   u(t) (41a)
y(t) = 1 0 0 0 0 1 x(t) + v(t). (41b)
coming from one of the demonstration program available in the CONTSID Toolbox. The input signal is made up with two independent pseudo random binary sequences of size N = 1000. The sampling period is chosen equal to 0.1s. A white Gaussian noise is added on both outputs such that 8 the SNR=20dB. The initial state vector is equal to zero. Monte Carlo simulation (MCS) analysis is used to evaluate the performance of the proposed estimation schemes (named RPM+MOESP and PMF+MOESP) which will be firstly compared to two other CT identification methods

• the PMF+GSVD algorithm [START_REF] Bastogne | A PMF-based subspace method for continuous-time model identification. Application to a multivariable winding process[END_REF] which combines the Poisson moment functionals and a subspace based method using a generalized singular value decomposition, • the DT PI MOESP algorithm [START_REF]Subspace model identification part 2: analysis of the elementary output error state space model identification algorithm[END_REF] leading to a discrete state space model converted into the CT domain with the d2c function of Matlab. For this first MCS of 100 runs, the instrumental variable is built up from past inputs (see [START_REF] Bauer | Order estimation for subspace methods[END_REF]) with p = 4. The user parameters of RPM+MOESP and PMF+MOESP are respectively fixed as T = 25 and λ = β = 3. The same value of λ is used for PMF+GSVD. The filter order is chosen as i = n x = 3. good estimates of the complex poles but meet more difficulties to find accurately the real pole. It is however interesting to notice that the estimated real values are unbiased (centred around the expected one). On the contrary, the PMF+GSVD algorithm leads to biased values. Concerning the combination PI MOESP+d2c, the results are poor since the estimated poles are mistaken. Concerning PI MOESP+d2c, it is well known that the DT subspace algorithms need I/O Hankel matrices of relatively large size to guarantee good numerical properties. The requirement i ≥ n x is also not sufficient. Simulations show that increasing i improves the accuracy of the estimates. However, even in this case, the variance of the PI MOESP+d2c estimated poles is quite significant. This first analysis shows that using CT subspace identification algorithms (with good user parameters values) can be considered as a good approach for the CT MIMO system identification.

The aim of the second MCS analysis is the study of the instruments choice influence on the poles estimation consistency. In addition to the instrumental variable proposed in this paper (see [START_REF] Bauer | Order estimation for subspace methods[END_REF]), two other instrumental variable methods are considered • a time derivative approximation input matrix M {U i } with i = n u +n y +n x +1 used in the subspace algorithm proposed by R. Johansson et al as a correlation matrix (see [START_REF] Johansson | Stochastic theory of continuous time state space identification[END_REF] for details).

Figure 2 (in complement with figure 1) shows that, as expected, the use of relatively large number of time derivative inputs as instruments leads to poor numerical efficiency and produces less accurate estimates than considering time shifted inputs (see case 2 in fig. 2). Notice finally that the use of a time shifted version of M {U i } as instruments yields to estimated real poles with a bit larger variance than with Ξ = U p .

VI. CONCLUSION

In this paper, the identification of CT MIMO state space models is considered. The proposed approach consists more precisely in introducing the reinitialised partial moments or the Poisson moment functionals to build a particular sampled I/O algebraic relationship from which a MOESP like algorithm can be applied. A specific attention has been concerned with the construction of the instrumental variable. This study, illustrated with a simulation example, has shown that choosing past sampled inputs as instruments leads to good performances, particularly in comparison with classical approaches using high order time derivatives.

APPENDIX

Consider the rhs of [START_REF] Jemni | A comparative study of the deterministic accuracy of two identification methods for continuous time systems[END_REF]. Then, from (20), we asymptotically get

lim N →∞ 1 N M {Y i } Π M{Ui} ⊥ Ξ T P d = lim N →∞ 1 N Γ i M {X} Π M{Ui} ⊥ Ξ T + H i M {U i } Π M{Ui} ⊥ Ξ T + [G i M {W i } + M {V i }] Π M{Ui} ⊥ Ξ T P d . ( 42
)
By definition of the orthogonal projection

Π M ⊥ = I -M T MM T -1 M, (43)
the second term of the left-hand-side of the (42) vanishes as N tends to infinity. Likewise, we have 9

M {V i } Π ⊥ M{Ui} Ξ T = M {V i } Ξ T -M {V i } M T {U i } M {U i } M T {U i } -1 M {U i } Ξ T . ( 44
)
Since the instruments are chosen such as lim N →∞ 1 N M {V i } Ξ T = 0, the first term of the rhs of the previous equation is asymptotically cancelled. The last problem concerns the second term of the rhs of (44) and more precisely 1 N M {V i } M T {U i }. In order to prove that this matrix product vanishes as N tends to infinity, consider its qth block row an rth block column entry n and m (r)

n are defined as in equation ( 14) according to the chosen filtering method and θ = T for the rpm (47a) θ = t j for the pmf .

(47b)

Now, by ergodicity and since the disturbances are asymptotically independent of the input, lim N →∞ 1 N N j=1 v(t j -τ )u T (t j -µ) = 0 and, by extension lim N →∞ 1 N M {V i } M T {U i } = 0. All these steps prove the relation [START_REF] Jansson | On consistency of subspace methods for system identification[END_REF]. 9 The same approach is applied for M {W i }.

  Figure 1 plots the estimated poles obtained via the considered algorithms. It clearly appears that the RPM+MOESP and PMF+MOESP techniques give relatively 8 SNR=10 log " Py d Pv " where y d is the deterministic part of y.

•

  a time shifted version of M {U i } with a time delay ℓT s = 10T s and i = 3 developed [12],

1 N

 1 M {V i } M T {U i } qr = 1 N N j=1 M n q {v(t j )} M n r T {u(t j )} . (45)By introducing the convolution products linked with the filters M n q {•} and M n r {•}, we get1 N M {V i } M T {U i } qr = j -τ )u T (t j -µ)

Fig. 1 .

 1 Fig. 1. Estimated poles obtained with RPM+MOESP, PMF+MOESP, PMF+GSVD and PI MOESP+d2c for i = 3. The system poles are symbolized with grey crosses.

Fig. 2 .

 2 Fig.2. Estimated poles obtained with RPM+MOESP and PMF+MOESP when the instrumental variable is built up from a time shifted version (i = 3 and ℓ = 10) of M {U i }[START_REF] Haverkamp | Identification of continuous time MIMO state space models from sampled data in the presence of process and measurement noise[END_REF] (case 1) and from a filtered version of the inputs with i = nu + ny + nx + 1[START_REF] Johansson | Stochastic theory of continuous time state space identification[END_REF] (case 2).

U i is built up in a similar way as Y i .

Both transformations are available in the Matlab toolbox CONTSID[START_REF] Garnier | Latest developments for the Matlab CONTSID toolbox[END_REF] (see http://www.cran.uhp-nancy.fr/contsid/).

Mn {s(t)} is defined for t ≥ T .

By definition, these properties are verified by the rpm. Furthermore, gn(0) = 0 by construction. gn(θ) can be considered as equal to zero if θ = t is sufficiently big.

The noise free case is not introduced in this article due to the lack of space. This situation can however be considered as a special case of the algorithm developed in the noisy framework.

The problem of the instrumental variable construction is studied in subsection IV-B.