Tomas Grubessich
email: tomas.grubessich@usm.cl

Raul Stegmaier

Pablo Viveros

Fredy Kristjanpoller

Vicente González-Prida

François Pérès

Tomás Grubessich

Raúl Stegmaier

Expertos De

Sistemas De Información En

L A Gestión

D E Activos

Y Mantenimiento

Methodological proposal in order to increase the organizational learning based on experts' knowledge and information systems in

Keywords: Conceptual Model, Transformation of Data into Information, Understanding of Complex Systems Modelo Conceptual, Transformación de Datos en Información, Comprensión de Sistemas Complejos

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

INTRODUCTION AND PROBLEM STATEMENT

Now days there is a broad consensus on the importance of the information related to the operation of the organization in its own management, which has led to the creation of various tools that support in this line [START_REF] Park | An empirical investigation of the effects of data warehousing on decision performance[END_REF][START_REF] Sh | Internal control framework for a compliant ERP system[END_REF]. Tools with databases that handle large amounts of information from different areas have emerged, such as ERP and other software that are more specialized on the different workspaces. As an example, the following table 1 shows some of the trends that the information systems have taken in different areas of the organizations:

Types of Information Systems

Areas in which it operates Business intelligence (BI) Process, strategic and knowledge management

Table. 1. Information system trends in different areas of organizations

Another factor that every day has taken on greater significance is the recognition of the importance that the organizational learning has as a mean to maximize performance. Due to the large numbers of definitions for this concept, the organizational learning will be understood for the purpose of this study as "an organization that is able to create, acquire and transfer knowledge as well as to modify its behavior in order to reflect new knowledge and insights" [START_REF] Garvin | Building a learning organization[END_REF]. In this context, efforts have been made to promote the transfer of tacit knowledge of people [START_REF] Polanyi | The Tacit Dimension[END_REF] to the rest of the organization to achieve greater effectiveness and efficiency in usual activities and in the decision-making processes [START_REF] Li | Study on Knowledge Sharing Behavior Engineering[END_REF][START_REF] Zhao | A Game between Enterprise and Employees about the Tacit Knowledge Transfer and Sharing[END_REF][START_REF] Al-Qdah | A Conceptual Framework for Managing Tacit Knowledge through ICT Perspective[END_REF][START_REF] Zhong | Co-Insightsframeworkforcollaborativedecisionsupportandtacitknowledgetransfer[END_REF]. On the other hand it has been noted that, given the complexity of productive systems and the increasing amount of data, it is difficult to reach a level of knowledge that facilitates the process of decision making and to achieve a greater efficiency and effectiveness in the management of the organization. This is clearly reflected in the lack of clarity of the effects that certain decisions would have in the system, such as:

• the changes in operational policies,

• investment in new equipment,

• new inventory policies,

• among many others.

When there is no clear effect of some of the decisions mentioned, it is because a sufficient level of understanding has not been reached.

It is in this line that different methodologies aimed at managing and analyzing large amounts of information to better understand the systems have been created, among which stand:

• Knowledge Discovery in Data Base [START_REF] Nemati | Knowledge warehouse: an architectural integration of knowledge management, decision support, artificial intelligence and data warehousing[END_REF],

• Decision Support System [START_REF] March | Integrated decision support systems: A data warehousing perspective[END_REF][START_REF] Anaby-Tavor | Insights into enterprise conceptual modeling[END_REF][START_REF] Guillemette | Defining decision making process performance: Conceptualization and validation of an index[END_REF][START_REF] Hong | The critical success factors for ERP implementation: an organizational fit perspective[END_REF],

• Data Mining [START_REF] Marbán | A Data Mining & Knowledge Discovery Process Model, Data Mining and Knowledge Discovery in Real Life Applications[END_REF],

• Among others.

These methodologies are able to reach a high impact on the organizations, but they have a strong focus on information systems.

The above factors have been observed by the authors in different organizations in the field of asset management and maintenance very strongly and with peculiarities. When different methodologies are applied [START_REF] Viveros | [END_REF][START_REF] Khare | Automotive Field Failure Analysis based on Mileage -Feasibility & Benefits[END_REF][START_REF] Lawless | Methods for the estimation of failure distributions and rates from automobile warranty data[END_REF][START_REF] Neely | Performance measurement system design: a literature review and research[END_REF], problems arise from the different views between external experts and members of the organization. Many of these problems are due to the knowledge of the organization are scattered in inaccessible sources; also the knowledge doesn't have a clear structure; and the knowledge isn't in a friendly format to be transmitted and understood easily. This causes that projects related to asset management and maintenance need to go through a process where definitions are set and the information is obtained from data processing, for then starting applying the methodology itself. Given this context and understanding the difficulties faced by organizations in management and administration of data, and the creation and transfer of knowledge from expert contributors, which as a result leads to poor performances and often unassertive decisions, a methodological proposal for the analysis and the increase of the level of understanding a system is presented under a comprehensive and flexible approach. The increased level of understanding system has a focus on:

• Detecting relevant knowledge,

• Understanding functional relationships,

• Obtaining useful results for the activities performed by the organization.

All the above must be developed in a clear, inclusive and constantly validated process to be effective progress and impact in achieving the goal set to increase the level of understanding of the system and translate it into a practical way for the organization. The final result must have a comprehensive approach in order to use it in the ways considered necessary.

OBJECTIVES AND STRUCTURE

In this context, a need to create a methodological proposal to guide the process of:

• Formalizing the knowledge of the organization,

• Structuring an appropriate logical sequence,

• Treatment to achieve applicability of knowledge in the practical field of the organization is set.

This proposal must allow exploiting synergies produced by combining the information contained in the information systems of the company, with the knowledge from the experts of the organization, in order to substantially increase the level of understanding of the system studied under a holistic view of the entire system. This proposal is also focused on its application for complex systems, characterized by a large number of functional relationships, a low predictability to changes in key variables, a high interaction of different areas of the organization, a possible presence of industrial assets complex in technology and/or dimensions, among others. It's in this type of systems where proposals like the one desired give an essential support and allow facing a lot of tasks successfully.

The conditions that propitiate the development of such initiatives can be proactive within the organization, or they may also arise as a necessity came from other projects that require a high level of knowledge. Some of these projects can be:

• Changing operational criteria related to decision making,

• Redistribution available resource on the system,

• Improving reaction times to changes in the environment,

• Incorporating the risk factor in making operative decisions, • Among others.

All this looking for the improvement of key indicators of performance defined for the system. Furthermore, it is likely that as the level of understanding increases, changes to established indicators are done, either in its conceptual definition, the way of building it or the way of inferring the state of the system, making them even more representative.

It is important to know that when you want to increase the level of understanding of a particular system comes a series of difficulties, among which are:

• The various inner visions, often opposite to what is globally profitable for the business, which is reflected in local performance indicators (local optima), • Operational criteria and practices that oppose and hinder the collaborative work, and as a result, the overall results.

The problems on the implementation of the activities are usually identified but the assumptions of causes from each expert can be very different and even contradictory. The information required from the systems is disseminated in various databases, which does not assure you in any case the completeness and quality of the same. Actually, according to the experience of the working team, the database generally does not have the entire required field in order to have a correct analysis. Another frequent scenario is that there are organizations that are not even aware of all the information collected in different processes. This is why a methodological proposal of work that allows increasing the level of knowledge must have several features that ensure the compliance with its objective, among which stand:

• Based on a cyclical process with feedback, where both the expert knowledge and valuable information that can be extracted from information systems are strengthening. This cyclical process must have all the necessary validations and it will stop only when the level of understanding of the system that fulfills the objective is reached, • The results of this methodological proposal approach must be easily explained to those involved and they must contain the most relevant concept of the system, describing them both qualitatively and quantitatively, • The methodological proposal must include all the agents and information systems relevant for the system.

Being sure about this will only be possible when a satisfactory explanation of what happens in the system can be given.

METHODOLOGICAL PROPOSAL

To explain in a more friendly way the functioning of the methodological proposal, this will be divided into: the key stages that generate the cyclic process; and the step by step implementation process.

Key stages of the cyclic process

The methodology of work proposed consists of a virtuous cycle that is separated into four areas, generating a spiral progress as the level of understanding the system increases. This idea is based on the Spiral Model for Software Development [START_REF] Boehm | A Spiral Model of Software Development and Enhancement[END_REF], which consists of a series of iterations that groups a set of activities. As more operations are made, it is possible to increase the progress to the level of development of the objective desired. In the case of the methodological proposal, each repetition covers four areas. A brief description of every one of them is given below; understanding that in each one of them there will be qualitative and quantitative information:

• Definition of the system: Here the conceptual model based solely on the expert's knowledge is defined and the needs of knowledge and options to reach them are generated step by step.

• System Database: The main objective is to respond to all the principles in a qualitative and quantitative way based on historical information from the entire database necessary to incorporate.

• Expert Knowledge: Experts receive reports on the results of the postulates which are analyzed to obtain conclusions regarding the investigated concepts. Here is objectively defined which concepts are consolidated (unquestionable), which ones must be discarded and which ones must go deeper on further research.

• Conceptual Model: Here the consolidation of the knowledge generated is presented. It is constituted of several parts. A general scheme is easily understood; diagrams with more detail with descriptions, input and output of the key concepts, and quantitative information expressed in specialized reports associated with each concept.

The Figure 1 shown represents the ideas just described: The Figure 2 describes the same scheme incorporating activities to be performed:

Step by step implementation process

For a better understanding of the 3 step by step process implementation of the proposal, the following Figure 3 presents a diagram using of the notation of the Business Process Model and Notation method (BPMN) [START_REF]Object Management Group, Business Process Model and Notation[END_REF]. With this a higher level of detail is achieved and it is easy to understand this implementation process.

SYSTEM KNOWLEDGE LEVEL

Needs of knowledge and options to reach them are generated.

System database

El

Definition and consol idation ~ concepts .

=------

Selection of databases and generation of queries. Queries execution. Reports debugging. To be successful in the implementation of this methodological proposal is advisable to generate a specific project, which involves the definition of objectives, scope and objectives deadlines, together with the formation of a team responsible of the project. In addition, under the logic of the project it is much easier to quantify the needs of resources required. According to the experience of the authors in general the major resources required for such projects have to do with the hours of dedication of the members of the project and the external experts who are working. There could also be difficulties do evaluate the acquisition of new information systems, when the current system do not respond to the requirements of the organization, but such cases correspond to new independent projects. That is why the main resource requirements, has to do with the dedication of people to the project.

II Conceptual Model

Two key pillars have been added to the methodological proposal represented through BPM for understanding and usage of the same, one is referred to as "Responsible" agent and the other as "Participant" agent. The first one specifies the agent or agents that respond to the proper development of the activity. As a project, the main responsible will always keep that role, however, depending on the activity, other agents (Participants) will have a preponderant role either because of their knowledge and experience, because of the decisions making involved, or just because of the access to information they have. In the case of participant agents, it must be specified who is playing a major role as support or evaluator in the development of the activity, hence is essential his or her active participation.

Before starting the description of the process, three participant agents, the project team, the panel of experts and the Department of Information Technology (DTI) will be taken into consideration. This recommendation of three players involved arises from the experience in the application of the methodological proposal, but it may vary from case to case, being simpler or more complex depending on the situation. In addition, it also arises from the responsibilities in the activities of the process proposed and the work of support required. Anyways, it must be understood that this process is a guide which can be modified and adjusted according to the emerging needs. A brief description of each of these agents is given below:

Information
• Team Responsible of the Project: This team is made of the people the organization has given the task of increasing the level of knowledge of a given system. Some possible projects that require increasing the level of knowledge of the system are: systems modeling, improvement projects, failure analysis, etc. Whatever the origin of the project, the most relevant for the team is to define the system to be analyzed and scope in deepening the knowledge required to be generated. This is the key because it determines when the project has reached the expected result so there is no need to continue repeating the methodological proposal. It will also be essential in case that the level of knowledge required is not achieved as it will specify the grounds and constraints faced in this regard. This agent has a coordinating role between the different areas that interact in the system; therefore they must assure a fluent communication between the different agents. It is highly recommended that this team is made up of people who do not have a deep knowledge of the system being studied, because that includes an objective view and with a focus on the objectives sought. It is equivalent to the look of the moderator requested in the technical analysis of maintenance type Root Cause Analysis (RCA) or Reliability Centered Maintenance (RCM) [START_REF] Viveros | [END_REF].

• Expert Panel: The panel of experts must be defined by the organization and it must be composed of the people with relevant knowledge of the system under study who possess knowledge and experience for the definitions of the lines of study, research, definition of principles, understanding of the data existing in information systems and to validate the developments related to the conceptual model. They come from different areas of the organization, and may even be external to it, as in the case of external expert on specific issues. Each expert has his own perspective of the system, whether technical, management or practice; of day-to-day or analysis from pure data; from how it should operate at how it is done in practice; each of these perspectives involves people of different backgrounds, ages and experiences. Creativity and involvement is expected, since it depends heavily on them the degree of success of the project • IT Department: People with a high level of knowledge of the information systems managed by the organization in every area must participate. It is fundamental that people in this group have a global view of all sources of information and, in the case of unknowing a particular information system, to know with whom to interact in order to obtain the required. They must also possess the necessary skills to produce reports of varying complexity in their development, and a clear vision to show in a friendly way such reports in order to be easily understood by the rest of the team.

A brief description of each of the processes defined in BPM is shown, with special emphasis on the activities, participants and required information.

Process A, Definition of objectives and scope of the conceptual model:

In this activity, it must be specified what is sought with the conceptual model, which should involve responding to what for, so it is easier to determine when the target is achieved and that way the project can be finished. It is important to determine the system under study with its respective boundaries and areas involved. The work team, the expected deadlines of the project and the required resources are also established. The project team is who must ensure the proper behavior of this activity and the panel of experts defined by the organization must be actively involved in its development.

Process B, Definition base-situation of the system: Having objectively defined the system under study and the project objectives, especially linked to the desired level of knowledge of the system, the events that occur within the system are defined in the first instance. What is sought is to create the foundations of the conceptual model through the knowledge of the expert panel, determining the initial key concepts. Thus, a first overview of the associated processes, of agents that interact with the system, of technical specifications and of operating criteria relying on process flows, documentation and experience takes place. At this early stage of definition, the following research questions must be answered:

• What: It refers to the identification and description of the events occurring in the system.

• Who: It identifies the area, department or office that is responsible for carrying out each identified event.

• Why: It refers to the causes that boost the activation of activities.

• When: Knowing the causes which impel activities, it is described when these conditions and their characteristics are given.

• How: It aims to describe conditions and features under which the activity is performed, identifying resources and time mainly associated.

• What for: The implications of the events are set, describing the possible expected results both qualitative and quantitative.

• How Much: What will be measured, how it will be measured and how can this information be obtained from the available information systems must be established.

Once the base situation is defined and there is a consensus in the expert panel, this information must be documented in the repository of information named Conceptual Model System. The definition of the base situation corresponds to a first approach to the conceptual model, which is why it is expected to be composed of:

• A general scheme in which the main concepts are defined by the relationships between them and the system.

• Documents with information of each of the points discussed in the general treatment. Part of the information that these documents must contain are: description with more detail of each item; variables and key indicators of the concepts; sources of information related to the concepts; etc.

• Technical support documents when the information is sought in a more detailed way. Some of these documents are: drawing, process diagrams, technical specifications of equipment, etc.

Process C, Definition of the needs of knowledge and options to reach them:

On this stage the principles that will guide the process of the research of knowledge of the system must be formulated, in order to increase the level of understanding of it. The first main principle established in this investigation is that with the concepts defined in the scheme presented in the base-situation, the most significant part of the activities and events that occur within the system are being represented. This principle, which is done by default, must be constantly reviewed, because if at some point it is concluded that some of the concepts are needless or that other concepts must be incorporated, it would mean a significant change in the conceptual model, so it should be treated as soon as possible. Then a process of questioning all the concepts in the conceptual model must start. Some of these questions may target research in order to validate some features of the concept, while others may seek to deepen their level of understanding. Whatever the case, each of these questions must be represented as postulates for research, that is, they must be set in a purposeful way, linked to one or more concepts of the conceptual model explicitly, be specific, related to observable and measurable terms, and that presumably is deemed that itself has the means to answer them. With all the principles defined, the needs of knowledge are open in all the areas considered relevant.

Process D, Search of information sources to use and linking process:

In this activity the team from the Department of Information Technologies of the organization should be integrated, who will be responsible for the tasks of select, transform, debug, and link information sources required to respond to the principles proposed in the previous activity. The participation of the project team and the expert panel is essential to keep focus on the lines of research and the proposed principles. The tasks proposed that must be developed in this activity correspond to:

• Definition of relevant database: it responds to which are the sources of information used to work in order to meet the requirements.

• Transformation and debug of the original information: the conditions under which the information should be available to make it usable for study are set.

Regarding linking databases, the process by which databases are created with a greater number of relevant fields, being this database constructed from linking different sources of information, in which a more complete and centralized information system is obtained, with all fields that are deemed necessary focused on the principles and lines of research is established. There may be the need to generate a unified database able to represent the behavior of the entire system, which has the advantage of making all queries and reports from a single source. Another advantage is the elimination of duplicity and to speed up the responses times. However, in general the centralization of databases is not easy to achieve given the differences that occur in the various sources. One way of solving this difficulty is through applications that track data required from the different database. The Figure 4 represents the linking of different database through a logical procedure that keeps the integrity of the information.

Figure 4: Linking database through a documented procedure

It is essential in this activity that people with extensive knowledge of current information systems participate, because in the case of not having a global vision, there could be key sources of fundamental information that are not considered due to ignorance.

Process E, Query execution and reports with results generation: With the information system cleared, processed and linked, the execution of queries can be done. These queries will be the base information to respond the principles set and to increase the level of understanding of the system. The IT Department must be in charge of developing this task and it must ensure that the information given contains the proper level of detail in order to get the best response to the principles set. To make this task easy, it could be very useful to start with a general description of the results and proceed to increase the level of detail in the analysis done. In turn, graphs, tables and figures can be very useful.

Process F, Analysis and debugging reports:

With the reports from the above process, the panel of experts must discuss this information, trying two fundamental points: to debug the reports to reflect the concepts ideally pursued with a sufficient level of detail, and reports must express quantitatively and understandably friendly the answers to the principles expressed above. It is very likely that during the development of this activity new concerns and inconsistent information appear. This information must be presented and the respective justifications and their possible implications for the conceptual model must be sought in parallel. This may require additional efforts to develop the project.

Process G, Definition of new concepts, reports incorporation and modification of Conceptual Model:

The panel of experts must conclude about the reports and information released in the previous activity, concluding about the veracity of each postulate, expanding the description of each concept and associating to each concept quantitative information from reports released. The important on this stage is to conclude about the information that will be part of the conceptual model of the system. Finally, all the information, analysis, reports and generated dossiers are analyzed and integrated in the conceptual model, increasing the level of knowledge the system.

Process H, Decision making: With the current conceptual model, are the objectives achieved?:

With the updated conceptual model according to the new knowledge generated, the project team together with the expert panel, must decide whether there is enough knowledge according to the objectives of study of the system. This decision is directly related to the project scope, but in turn, it can be concluded that in order to achieve the objective of the project it is necessary to extend the scope of the system, covering new areas, adding more experts or people from the IT department in the organization or analyzing even other relevant processes among many others. It is also possible to conclude on the need to make major changes to the entire conceptual model, making a refocusing using a new perspective. On the other hand, it must be noted that this decision can be made in previous processes, by detecting the need for major changes, which would be very useful for the development of the process. This decision is very important because it can involve a considerable increase in time and resource requirement for the project, impacting directly in accomplishing the goals within the deadline. Process I, To define points where further investigation must be done: In case that, due to the previous decision, it is decided that the conceptual model still does not meet the requirements, the points of the analysis in which it must be worked and deepened to increase the level of knowledge the system must be defined. It is essential to always keep in mind the objective and scope of the project at this point. Then, you must return to the activity of defining principles and lines of research in order to repeat the process recursively until the objectives set for the project are achieved. Then it must be returned to the process C.

Process J, Documentation generation and delivery of the final results:

In the case that, due to the decision made above, it is considered that the conceptual model meets the level of depth required by the objectives and the scope of the project, final documents related to the work done must be established. The results obtained can be separated into two categories: those aimed at the objectives of the project and related valuable knowledge. Both are closely related, but it is important to make the distinction as the first category aim to meet the objective of the project, while the other one make up the conceptual model of the system that will be the source of knowledge that will guide the process of decision making and any analysis. It is in this conceptual model where the formalization of knowledge is done, structuring appropriate logical sequence and treatment to achieve applicability of knowledge in the practical field for the organization. In the next section will show what you expect to find in a conceptual model as proposed, its different sections, features and potential transforms it into an ideal tool to promote organizational learning

CONCEPTUAL MODEL

At the end of the processes described, it is expected to obtain a Conceptual Model as the main result of the work done. It is in this Conceptual Model where knowledge has been formalized, a logical sequence has been structured, and the proper treatment for the knowledge to be usable profitably for the organization has been given. It is expected that every decision making related to the system pass by the conceptual model system, being the basis for understanding the impact of decisions that will be taken. Thus the conceptual model will become a fundamental tool for the General Management, giving clear and precise fundamentals for the next steps to execute related to the system. On the other hand, it facilitates the induction of new members of the organization to understand the system more thoroughly and in less time, while favoring continuity in operating criteria and methodologies. Also it is the responsibility of the General Management to define who will have access to the conceptual model, establishing different levels of access, restrictions and privileges, and above all, who are responsible for update both the sources of information, such as the revision of the model itself, for determining when the conceptual model fail to represent the system, so detecting this will be essential to avoid making bad decisions based on an outdated model.

As mentioned before, the Conceptual Model consists of a series of elements, from general schemes that facilitate the understanding of the system through a general diagram, to more complex analysis generated by queries to information systems through technical information of processes and equipment, plans and reports, among other elements. Each Conceptual Model is particular for every case, so they may vary from case to case. However, what could be the basic elements of a Conceptual Model grouped in their main areas is shown below in the Figure 5, as an example: As it can be seen, the Conceptual Model brings together different perspectives of analysis, from qualitative and quantitative information, historical information and knowledge of experts, complex analysis and friendly conceptual visions to understand the concepts defined as key in a short period of time. All these features make the Conceptual Model a source of knowledge, identifying potential for improvement and detection of exceptional opportunities for the organization. A brief description of each group shown in the deliverable of the conceptual model is made below.

General Scheme: It corresponds to the visualization of the most relevant concepts identified for the system under study. It is wanted to be simple and to show in a friendly way the most important concepts and their interactions, so that anyone can quickly understand the approach that has research. It must also provide guidance for possible investigations developed and some of the factors to consider.

Lines of Research and Principles: With the key concepts defined and outlined, together with the experts of the organization, the relevant principles to increase the level of knowledge of the system and the lines of research to assist in the understanding of it can defined. To the extent that more operations of the methodology are performed, part of the principles will be solved becoming new concepts or strengthening the existing ones, while new more precise assumptions with refined knowledge involving will appear. On the side of the lines of research, a valuable knowledge is generated directly giving the possibility of detecting new lines of research or deepen the existing ones.

Descriptive Database: From the most relevant concepts expressed through the general scheme of the system, it can be possible to select databases of information systems in the organization that are useful for purposes of the conceptual model. It must be noted that the databases that come directly from the information system are identified without making changes in it, so to have a description of the concepts that arise directly from the data of the organization. The advantage of proceeding this way is to have a direct view of the concepts of interest from the information systems of the organization, which ought to be very easy to update as time passes.

Consulting Sources of Information: It is both, queries generated in computer language to the information systems and the use of any information source (digital or physical) that the organization has. These queries must be upgradable, either through the computer language or through defined specific procedures. The result of these consultations must be refined and with all fields to meet the analysis required by the conceptual model.

Reports, Spreadsheets, drawings with analysis released with experts: With the information gathered from the various sources of information known, the procedure is performed to obtain the desired knowledge. For this it is taken as a focus the principles and lines of research, and a response to these requirements are given. Here a number of documents such as presentations, spreadsheets, reports, drawings, technical information, among others groups. What Re search lines and Principles Global Analysis, conclusions y challenges matters is that all this generated information must be updatable either automatically or through a procedure, so it is feasible to do these tests at different times and verify the system behavior.

Synthesized global analysis, conclusions and challenges: These elements are of great importance because they carry all the knowledge generated and set it in a useful and friendly way to the organization. It corresponds to the synthesis of all the work done and gives definitions for the next steps, either to fulfill the objectives of the study of the system, or for more specific investigations of the defined concepts.

It is true that in the process of the methodological proposal there are some activities specially in charge of generating the Conceptual Model; however it is useful to appreciate how every activity contributes in this aspect. The Figure 6 helps to have this vision.

Figure 6: Direct relationship between the process of the methodological proposal and the deliverables of the conceptual model

Thus, it is possible to verify and better understand how each process of the methodological proposal helps to formalize knowledge, a logical structure and provide treatment to apply this knowledge in a practical way to the reality of the organization. From the Conceptual Model, countless initiatives may arise towards improving the efficiency and effectiveness of the organization. Some of these initiatives may target changes in the operational criteria, investment decisions in the production process, changes in the criteria of planning and programming activities, reorganization of different functional areas, changes in the criteria for subcontracting and bidding projects, among many others. In short, to increase the level of knowledge of a system gives the basis to make better decisions, to direct efforts where it is more appropriate and taking the proper measures to elicit the desired. The probability of success of all the initiatives of the organization should tend to increase. As next challenge, this proposal will be implemented in a real case, to check that the above is achievable in practical terms, and thus also gauge the potential of this type of initiative in the organization.

CONCLUSIONS

This paper presents a methodological proposal which aims to increase the organizational learning from the experts' knowledge of the organization and from the existing information systems. This objective responds to the need to:

• Formalize the knowledge of the organization,

• Structure an appropriate logical sequence • Make the treatment to achieve applicability of knowledge in the practical field of the organization -"""""""" The main feature of this proposal is that its purpose is to maintain an overview of the system, relying on both experts and information systems, achieving significant synergies and ensuring a proper conduct of the study. In addition, the presented iterative process is helpful, allowing progress orderly and comprehensively expanding the understanding of the system. It is essential to note that the results of this proposal are consolidated into a conceptual model composed of several partial results ranging from an overview, technical documents, and consultations to the information systems of the organization, reports with the main results, and qualitative and quantitative information, among others. In other words, it is possible to go through all the information considered relevant in a friendly, understandable and flexible way. This methodological proposal responds particularly to some conflicts detected by the authors in the field of asset management and maintenance and tries to support the process of definition of the main concepts, reach the desirable information and support a friendly structure to transfer knowledge to the whole organization. To implement this methodology, it is recommended to be under a project outline, a process consisting of several activities and people in charge. The reason why this is suggested is that a very necessary logic of objectives and scope is proposed for this type of study, in addition of defining key milestones and people in charge to ensure the proper development of the activities.

When applying the methodological proposal it is essential to correctly select the experts of the organization and the members of the IT department. These agents will be the ones who will provide knowledge and information that enable progress and objectives achievement. On the side of the project team, to have prior knowledge of the system is not needed, the most important work to be performed is to maintain an unbiased view of the development of the project and that directly points the goals set. In case of failure to maintain this position, you run the risk of deviating from the main goal or focusing the efforts on issues that are irrelevant to the project objectives.

When the project is run, it is possible that significant changes at various stages (iterations) may be carried out, which should be treated as part of the normal development of the project since the degree of knowledge of the system will always grow, which is rightly what is searched. It is also possible that the system is redefined repeatedly or that new experts are incorporated. The key of this process is that at each repetition the results are validated with the panel of experts.

Another important achievement is to unify knowledge and points of view of the different agents of the system. With this, a common language, a disciplined methodology of work and evidently the consensus between the parts involved in the project desired is achieved. Even if this goal is not achieved, some points of divergence will be identified, which is already very valuable to decide how to act under different scenarios.

Finally, with the conceptual model the objective of formalizing the knowledge of the organization, structuring an appropriate logical sequence and treatment to achieve applicability of knowledge in the practical field of the organization is reached, and gain greater advantage from the General Management, to all members of the organization, generating significant synergies and positive changes at all levels. From the conceptual model many initiatives may arise towards improving efficiency and effectiveness. Some of these initiatives may point to changes in operational policies, investment decisions in the production process, changes in the criteria for planning and scheduling activities, among many others.

BIBLIOGRAPHY

Figure 1 :

 1 Figure 1: Scheme of the development of Conceptual Model System

Figure 2 :

 2 Figure 2: Scheme of the development of Conceptual Model System with their activities

Figure 3 :

 3 Figure 3: Suggested process for developing the conceptual model using the notation BPMN

Figure 5 :

 5 Figure 5: Suggested elements of the Conceptual Model

  .. : ..,, ,,. ... . . . --........ ,_ --""' "" -• Qoe,y ~ and reportS with-

  + oef' rY:/:ird-.e"tinai ,-. .

  0


Aravind Rajan Ayagara
email: aravind-rajan.ayagara@univ-orleans.fr

André Langlet

Ridha Hambli

Experimental Characterization and FE Simulations of Ribs under Impact Loads

Introduction

Several authors have been studying the behaviour of ribs subjected to impact loads both in medical and biomechanical domains. The advances in numerical techniques and their advantages over expensive and painstaking experiments have led to the development of finite element human models. Despite their extensive use, these models have respective constraints. Knowing the realistic behaviour aids in the comprehension of the relation between injury criteria and fracture of ribs and indeed the severity of injury. Objectives 1. To develop an experimental methodology in order to accurately characterize the bone fracture induced by impacts to chest 2. To develop a three-dimensional finite element model of bone to simulate bone fracture and to predict its occurrence under specific dynamic environments This aids us in the comprehension of injury criteria and to predict the severity of the injury.

Conclusions

• Apparent density 𝜌 𝑎𝑝𝑝 as an independent variable for dependence of mechanical properties is sufficient for coherent results, given that it is correctly correlated to grey scale intensity 𝐶𝑇 𝑏𝑜𝑛𝑒 . • Elasto-viscoplastic constitutive law for trabecular bone produced promising results at a preliminary stage.

Damage Law

• An incremental, stress state dependent damage law was coupled externally to the constitutive law. It also considers damage accumulation (Eq.3) and instabilities (Eq.4) due to damage and also the effects of strain rate 𝜺 on fracture strain 𝜀 𝑓 .

Δ𝐷 = 𝑛 𝜀 𝑓 𝜂 𝐷 1-1/𝑛 Δ𝜀 𝑝 𝜎 = 𝜎 1 - 𝐷 -𝐷 𝑐𝑟𝑖𝑡 1 -𝐷 𝑐𝑟𝑖𝑡 𝑚
• Numerical rib sample was generated from DICOM images of HR-pQCT scans. The apparent density 𝜌 𝑎𝑝𝑝 was interpolated through a linear relation with 𝐶𝑇 𝑏𝑜𝑛𝑒 (Eq.1) as in [START_REF] Taylor | Determination of orthotropic bone elastic constants using FEA and modal analysis[END_REF]. 𝜌 𝑎𝑝𝑝 = 1.2907 𝐶𝑇 𝑏𝑜𝑛𝑒 • Mechanical properties were assigned through power law regression with 𝜌 𝑎𝑝𝑝 as independent variable. Respective equations were used with respect to porosity 𝑃 𝑜

Constitutive Law

• Both cortical and trabecular bone were assigned a modified elastic-plastic law to consider the effects of strain rate 𝜺 to have an elasto-viscoplastic behaviour through Cowper-Symmonds model (Eq.2).

𝜎 𝑦 𝑑 = 𝜎 𝑦 0 1 + 𝜺 𝐶 1 𝑃

• Where: 𝜎 𝑦 𝑑 is dynamic yield stress 𝜎 𝑦 0 is initial yield stress 𝐶 = 2.5 𝑎𝑛𝑑 𝑃 = 7.0 are Cowper-Symmonds parameters [START_REF] Li | Rib fractures under anterior-posterior dynamic loads: experimental and finite-element study[END_REF] (2)

•ResultsFigure- 3 :Figure- 1 :

 31 Figure-3 : Fracture Pattern

Figure- 2 :

 2 Figure-2 : Damage Evolution with respect to Force

Table - 1

 - : Mechanical properties of porcine rib constituents

	Property	Cortical bone	Trabecular bone
	Apparent Density 𝜌 𝑎𝑝𝑝	1690.090 kg. m -3	222.611 kg. m -3
	Young's Modulus 𝐸	10.46 GPa	0.378 GPa
	Tensile Yield 𝜎 𝑦 𝑇	75.36 MPa	2.096 MPa
			(3)
			(4)
			Forthcoming Research
			Implementation of current damage coupled constitutive law to generalized
			human thorax model and FE thorax model HUBYx of Radioss courtesy of
			Altair Hyperworks, to study and quantify the response of human thorax
			under different impact cases such as Automotive accidents and Ballistic
			impacts (Non-lethal projectiles).


Y Braidiz

A Polyakov
email: andrey.polyakov@inria.fr.

D Efimov
email: denis.efimov@inria.fr.

W Perruquetti
email: wilfrid.perruquetti@centralelille.fr.

On finite-time stability of homogeneous systems with multiplicative bounded function *

In this paper, we study the finite-time stability of a class of nonlinear systems ẋ = f (x) = H(x)b(x), where H is homogeneous and b is bounded. We define the homogeneous extension of the non-homogeneous function f and use this extension to prove that, under some conditions on b, if the system ẋ = f (x) is globally asymptotically stable, then it is finite-time stable. An example of global asymptotic stable system with some additional conditions is presented in the last section to illustrate the obtained results.

I. INTRODUCTION

Stability is one of the main research directions in the mathematical study of dynamical systems and there is a large number of papers and books that have been published on the subject (see [START_REF] Hahn | Theory and application of Lyapunov's direct method[END_REF]- [START_REF] Yoshizawa | Stability theory by Lyapunov's second method[END_REF] and references therein). In general, there are different rates of convergence.

Over the years, researchers mainly focused their attention on asymptotic and exponential stability: trajectories converge to an equilibrium state in "infinite time" [START_REF] Yoshizawa | Stability theory by Lyapunov's second method[END_REF], [START_REF] Earl | Theory of ordinary differential equations[END_REF]. However, in a lot of engineering domains, the time is a critical constraint which must be taken into account. Hence, in many applications, it is desirable to get the so called finite-time convergence of the solutions to an operating point. This possibility has been mentioned by Erugin 1951 and Zubov 1957, and studied by [START_REF] Roxin | On finite stability in control systems[END_REF], Korobov 1979 [6], [START_REF] Korobov | A general approach to synthesis problem[END_REF], many useful properties or finite-time stable (FTS) systems can be found in [START_REF] Sanjay | Finite-time stability of continuous autonomous systems[END_REF]- [START_REF] Perruquetti | Finitetime observers: application to secure communication[END_REF]. There are two methods to establish FTS: Lyapunov function approach or homogeneity.

Investigating FTS of dynamical systems by using Lyapunov function approach was introduced in several works (see [START_REF] Roxin | On finite stability in control systems[END_REF]- [START_REF] Bernuau | On homogeneity and its application in sliding mode control[END_REF]). Proving FTS of dynamical systems and differential inclusion using smooth or non-smooth Lyapunov function was investigated in [START_REF] Moulay | Finite time stability of differential inclusions[END_REF]. Some necessary and sufficient conditions to guarantee the FTS for autonomous scalar systems was presented in Haimo's paper [START_REF] Varda | Finite time controllers[END_REF]. This result was extended to n-dimensional systems in [START_REF] Moulay | Finite time stability of nonlinear systems[END_REF] through the use of Lyapunov functions as in [START_REF] Sanjay | Finite-time stability of continuous autonomous systems[END_REF]. Again, a necessary and sufficient conditions were developed for finite-time stabilization of systems like ẋ = f (x) + g(x)u (u is the input) [START_REF] Moulay | Finite time stability of nonlinear systems[END_REF].

We may also establish the FTS by using the so called homogeneity. This tool was used to prove different kinds of results. Particularly, in control theory, homogeneity has been introduced in order to investigate the stability and stabilization of dynamical systems [START_REF] Hahn | Stability of motion[END_REF], [START_REF] Hahn | Theory and application of Lyapunov's direct method[END_REF], [START_REF] Coron | Control and nonlinearity[END_REF].

FTS of homogeneous systems was studied by Zubov in the 1957 for standard homogeneity and extended by Bhat and Bernstein in [START_REF] Sanjay | Geometric homogeneity with applications to finite-time stability[END_REF] for weighted homogeneity (2005) and it also appeared with Rosier and Bacciotti in [START_REF] Bacciotti | Liapunov functions and stability in control theory[END_REF], where it has been proven that homogeneous systems of negative degree locally asymptotically stable (LAS) possess FTS property. This result was extended for differential inclusions (DIs) [START_REF] Bernuau | On an extension of homogeneity notion for differential inclusions[END_REF] together with Rosier's theorem [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF]. The research about FTS of homogeneous evolution systems was also presented in [START_REF] Polyakov | On homogeneous distributed parameter systems[END_REF] which uses a general definition of homogeneity in Banach and Hilbert spaces.

The novelty of this paper consists in a result about FTS of a class of a non-homogeneous systems of the form ẋ = H(x)b(x), with H is homogeneous and b satisfies boundedness conditions. We extend the result that was proved for a GAS homogeneous system with negative degree of homogeneity. In particular, we prove that, with some sufficient conditions on the function b, we may guarantee the global FTS (GFTS) of the GAS system ẋ = H(x)b(x).

To do so, we use a homogeneous extension which is given by a homogeneous differential inclusion. The main peculiarity of a homogeneous extension is that it can be defined in the cases when homogeneous approximations do not exist. Then, using the results that were proven for GAS homogeneous differential inclusion with a negative degree of homogeneity (see for more details [START_REF] Levant | Stability and robustness of homogeneous differential inclusions[END_REF]), we will prove the FTS of the original dynamical system.

The paper is organized as follows. After introducing some notations and definitions about finite-time stability and homogeneity in Section II, we define the homogeneous extension of a non-homogenous continuous function f : R n -→ R n , that we use, by giving a sufficient conditions on f, to prove the global FTS of a GAS system of the form ẋ = f (x) (in Section III and Section IV). In the last section we present one example which shows how to apply the obtained results. The aim of this paper is to investigate the GFTS of non-homogeneous differential equation using a constructed homogeneous differential inclusion and Lyapunov functions.

II. PRELIMINARIES A. Notations

• R + = {x ∈ R : x ≥ 0} , where R is the set of real numbers.

• | • | denotes the absolute value in R, • denotes the Euclidean norm in R n . • S = {x ∈ R n : x = 1} denotes the unit sphere in R n . B(x 0 , r) = {x ∈ R n : x -x 0 < r} denotes the open ball of radius r > 0 centered at a point x 0 . • M m,n is the set of all m×n-matrices over the field of real numbers, and it forms a vector space. When m = n we write M n instead of M n,n .

B. Stability properties and generalized homogeneity

In this subsection we give some definitions of homogeneity and FTS of a system:

ẋ(t) = f (x(t)), x(0) = x 0 ∈ R n (1)
where

x(t) ∈ R n is the state, f : R n -→ R n , f (0) = 0,
and denote the corresponding solution as x(t, x 0 ). 1) Finite-time stability: Now let us give the definition of FTS which will be investigated in this paper.

Definition 2.1: [START_REF] Sanjay | Finite-time stability of continuous autonomous systems[END_REF], [START_REF] Hong | Finite-time input-to-state stability and applications to finite-time control[END_REF] Assume that the system (1) possesses unique solutions in forward time for all initial conditions except the origin. Then the origin of the system (1) is FTS, iff there exists a nonempty open neighborhood of the origin V in R n such that:

• there exist a function α ∈ K such that for all x 0 ∈ V we have x(t, x 0 ) ≤ α( x 0 ), ∀t ≥ 0. • there exists a function T : V -→ R + such that if x 0 ∈ V then lim t-→T (x0)
x(t, x 0 ) = 0. T is called the settling-time of the system (1); • If the settling-time function T is bounded for all x ∈ V, we say that our dynamical system is fixed-time stable (FxTS). We say that the ball B is fixed-time stable for (1), if there exists 0 < T < +∞ such that x(t, x 0 ) ∈ B, ∀t ≥ T, ∀x 0 ∈ R n . In addition, if V = R n , then the origin is a globally FTS (GFTS).

The following proposition shows how to investigate the FTS by using Lyapunov theory.

Proposition 2.1: [START_REF] Moulay | Finite time stability and stabilization of a class of continuous systems[END_REF] Assume that the (1) has a unique solution in forward time outside the origin. The origin of this system is FTS with a continuous settling-time function at the origin if and only if there exists a real number c > 0

, α ∈ ]0, 1[ and a class CL ∞ Lyapunov function V : V -→ R + satisfying DV (x), f (x) ≤ -c[V (x)] α (2)
for all x ∈ V.

2) Homogeneity: In control theory, homogeneity simplifies qualitative analysis of nonlinear dynamic systems. So that, it allows a local properties (e.g., local stability) to be extended globally using a property of the solutions (homogeneity is a special kind of Lie symmetry). In order to define this notion, let us introduce the notion of dilation group.

Definition 2.2:

A map d : R -→ M n (R) is called a dilation in R n iff it satisfies • Group property: d(0) = I n , d(t+s) = d(t)d(s), t, s ∈ R. • Continuity property: d is continuous, i.e. ∀t > 0, ∀ε > 0, ∃γ > 0 : |s -t| < γ =⇒ d(s) -d(t) Mn ≤ ε. • Limit property: lim s-→-∞ d(s)x = 0 and lim s-→+∞ d(s)x = +∞ uniformly on the unite sphere. Definition 2.3: • The dilation d is monotone if d(s) Mn ≤ 1, ∀s ≤ 0 • The dilation d is strictly monotone in R n if ∃β > 0
such that d(s) Mn ≤ e βs , ∀s ≤ 0. Property 2.1: [START_REF] Polyakov | Quadratic-like stability of nonlinear homogeneous systems[END_REF], [START_REF] Polyakov | On homogeneous finite-time control for linear evolution equation in Hilbert space[END_REF] The matrix

G d ∈ M n defined as G d = lim s-→0 d(s) -I n
s is known as the generator of the group d(s) and satisfies the following properties:

d ds d(s) = G d d(s) = d(s)G d and d(s) = e G d s = +∞ i=0 s i G i d i! . Let S d (r) = {x ∈ R n : d(-ln(r))x = 1} , r > 0 is called the homogeneous sphere of radius r. If r = 1 we write S d instead of S d (1)
. Now we will defined the canonical homogeneous norm. Definition 2.4: A continuous function p : R n -→ R + is said to be d-homogeneous norm if p(x) -→ 0 as x -→ 0 and p(d(s)x) = e s p(x) > 0 for x ∈ R n \ {0} and s ∈ R.

For monotone dilations the homogeneous norm • d is called the canonical homogeneous norm

x d = e sx , where s x ∈ R : d(-s x )x = 1.
(3) Definition 2.5: A vector field f : R n -→ R n (resp., a function h : R n -→ R) is said to be dhomogeneous of degree ν ∈ R iff for all s ∈ R and all x ∈ R n we have e -νs d(-s)f (d(s)x) = f (x), (resp., e -νs h(d(s)) = h(x)).

Homogeneity property was introduced also for Banach spaces [START_REF] Polyakov | On homogeneous distributed parameter systems[END_REF], [START_REF] Polyakov | On homogeneous finite-time control for linear evolution equation in Hilbert space[END_REF], and it is given by a group of dilations in Banach or Hilbert space. The following lemma provides a useful comparaison between homogeneous functions.

Lemma 2.1: Suppose that V 1 and V 2 are continuous realvalued functions on R n , d-homogeneous of degrees l 1 > 0 and l 2 > 0, respectively, and V 1 is positive definite. Then, for every x ∈ R n we have

a 1 [V 1 (x)] l 2 l 1 ≤ V 2 (x) ≤ a 2 [V 1 (x)] l 2 l 1 , (4)
with a 1 = min {z: V1(z)=1}

V 2 (z) and a 2 = max {z: V1(z)=1}

V 2 (z).

The Lemma 2.1 has been proven in [START_REF] Sanjay | Geometric homogeneity with applications to finite-time stability[END_REF] for weighted homogeneous functions and we can use the same truck to prove it for d-homogeneous functions

V 1 and V 2 . Hence, because V 1 > 0 then for every x ∈ R n \ {0}, ∃z ∈ {y : V 1 (y) = 1} such that z = d(-ln(V 1 (x)))x.
In the next section we will use the Lyapunov theory to show some robust stability results. The existence of a homogeneous Lyapunov function for a GAS homogeneous system was provided by Zubov in 1957, Rosier [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF] and Polyakov [START_REF] Polyakov | Sliding mode control design using canonical homogeneous norm[END_REF], [START_REF] Polyakov | Quadratic-like stability of nonlinear homogeneous systems[END_REF] by using weighted and generalized homogeneity.

III. HOMOGENEOUS EXTENSION.

In this section, we define a homogeneous extension which allows some homogeneity methods to be applied for nonhomogeneous systems (see [START_REF] Polyakov | On homogeneous finite-time control for linear evolution equation in Hilbert space[END_REF]). This extension is given by a differential inclusion.

Differential inclusions appear as a model of dynamical systems which do not satisfy the classical assumptions of regularity. Regularization of the system (1) proposed by Filippov [START_REF] Fedorovič | Differential equations with discontinuous righthand sides: control systems[END_REF] consists in replacing (1) with a suitable differential inclusion

ẋ ∈ F (x), x ∈ R n , F : R n -→ 2 R n . ( 5
)
The notions of solution obtained in this way depend on the construction of the set valued map F. If the set valued map F (x) is upper semicontinuous, compact and convex valued, and locally bounded, then (5) has solutions. Definition 3.1: Homogeneous extension of a function f :

R n -→ R n is a set valued d-homogeneous map F : R n ⇒ R n , such that f (x) ∈ F (x) for any x ∈ R n and F (x) is bounded for every fixed x ∈ R n .
This extension appears, for example, as a result of Filippov regularization of a homogeneous discontinuous vector field (see [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF]). In this paper the set-valued map F is defined by

F (x) = s∈R {e -νs d(-s)f (d(s)x)}, x ∈ R n , ( 6
)
where d is a generalized dilation and ν ∈ R. F is the homogeneous extension (see [START_REF] Polyakov | On homogeneous finite-time control for linear evolution equation in Hilbert space[END_REF]) of the continuous function f. Indeed, this set valued map F is d-homogeneous in the following sense

e νs d(s)F (x) = F (d(s)x), s ∈ R, x ∈ R n . (7)
Another goal for which we define this extension is that there exist some functions which do not have a homogeneous approximation. Therefore, to study its stability and to evaluate its rate of convergence the homogeneous extension can be used. The investigation of this issue is presented in the rest of this section.

An examlpe of such a system is the following:

ẋ(t) = -2 + cos 1 x(t) x(t), t ≥ 0. (8)
The right-hand side of ( 8) does not have a homogeneous approximation, but it has a homogenous extension, which will allow us to study the behaviour of its solution: ẋ(t) ∈ F (x(t)) = -x(t) [START_REF] Hahn | Theory and application of Lyapunov's direct method[END_REF][START_REF] Bacciotti | Liapunov functions and stability in control theory[END_REF] , t ≥ 0. Remark 3.1: We mention that we do not need to prove the existence of a solution to the extended inclusion ẋ ∈ F (x), where F is given by (6), since f (x) ∈ F (x) and solution of ẋ = f (x) always exists if f is continuous: the solution of the system (1) belongs to the set of the solutions of the differential inclusion ẋ ∈ F (x). The idea of this paper is to study the FTS of the differential inclusion [START_REF] Earl | Theory of ordinary differential equations[END_REF], with F defined by [START_REF] Roxin | On finite stability in control systems[END_REF], by using the Lyapunov theory. If [START_REF] Earl | Theory of ordinary differential equations[END_REF], where F defined by ( 6), is GAS, then ( 1) is also GAS. It is interesting to know when the inverse result holds. Thus, we choose the function f such that the system (1) has a solution. The function f of the system (1) satisfies the following assumptions

H 1 : f : x ∈ R n -→ f (x) = H(x)b(x) ∈ R n . H 2 : H : R n -→ M n,m is continuous and d-homogeneous of degree ν (for every b ∈ R m the function x -→ H(x)b is d-homogeneous of degree ν, see below) and H(x) = 0 ⇔ x = 0. H 3 : b : R n -→ R m is bounded (∃M > 0, b(x) ≤ M, ∀x ∈ R n ) and b : R n \ {0} -→ R m is continuous. H 4 :
The system (1) is GAS.

In particular, the hypothesis H 4 implies that b(x) does not belong to the kernel of H(x), ∀x ∈ R n \ {0} . By the definition of the function f (x) = H(x)b(x) ∈ R n , since b is bounded, then the set F (x) is also bounded for any fixed x. This ensures that the supremum of F on S is finite. In the following theorem we give a sufficient condition which guarantee GAS of the differential inclusion ẋ ∈ F (x).

Theorem 3.1: Let

• the system (1) satisfies the assumptions H 1 , H 2 , H 3 and H 4 , • the dilation d is strictly monotone, • there exist a Lyapunov function V of the system (1) and

0 < β < 1, such that max x d=1 b(y) -b(x) ≤ -β max x d=1 DV (x), f (x) max x d =1 H(x) Mn,m max x d =1 DV (x) , (9)
∀y ∈ R n \ {0} .

Then the differential inclusion (5), with F is defined by (6), is GAS and there exists a d-homogeneous positive definite Lyapunov function W such that DW (x), h < 0, ∀h ∈ F (x), and x ∈ R n \ {0}. [START_REF] Perruquetti | Finitetime observers: application to secure communication[END_REF] All proofs are excluded due to space limitations. The following Remark treats the case when a nonlinear dynamical systems cannot satisfy the condition (9).

• max x d =1 b(x) ≤ max x d =1 b(y) -b(x) , ∀y ∈ R n \ {0}.
Then the system (1) with the assumptions H 1 , H 2 , H 3 and H 4 , will not satisfy the condition [START_REF] Sanjay | Geometric homogeneity with applications to finite-time stability[END_REF].

Next, we consider the simple case when the function b is a scalar function.

Corollary 3.1: If the vector fieled f of the system (1) is defined by f (x) = H(x)b(x), such that the function b : R n -→ R, is continuous, bounded and H : R n -→ R n is continuous, d-homogeneous with degree of homogeneity ν. We assume that the system (1) is GAS, and b(x) = 0 =⇒ x = 0. Then the differential inclusion (5), with F is defined by (6), is GAS.

IV. FINITE TIME STABILITY RESULTS

The following theorem shows the FTS of the system (1) when the degree of homogeneity of H is negative and ( 9) is satisfied.

Theorem 4.1: If

• the system (1) satisfies the assumptions H 1 , H 2 , H 3 and H 4 , • the dilation d is strictly monotone,

• the degree of homogeneity of H is negative (ν < 0) and (9) holds. Then the system (1) is GFTS. If the two first assumptions hold and the degree of homogeneity of H is positive (ν > 0). Then the system (1) is fixed-time convergent to a ball. Remark 4.1: • If the vector fieled f of the system (1) ensure the boundedness of the homogeneous extension F (x) for every fixed x ∈ R n , the dilation d is strictly monotone, the degree of homogeneity of F is negative and there exist a Lyapunov function V of the system (1) and 0 < β < 1, such that max

x d =1 e -ντ d(-τ )f (d(τ )x) -f (x) ≤ -β max x d =1 DV (x), f (x) max x d =1 DV (x) , (11)
∀τ ∈ R. Then the system (1) is GFTS. • If the degree of homogeneity of F is positive, then we may guarantee the fixed-time convergence of the system (1) to a ball.

• If max x d =1 f (x) ≤ max x d =1 e -ντ d(-τ )f (d(τ )x) -f (x) , ∀τ ∈ R.
Then the condition (11) will not hold. Corollary 4.1: If the vector fieled f of the system (1) is defined by f (x) = H(x)b(x), such that the function b : R n -→ R, is continuous, bounded and H : R n -→ R n is continuous, d-homogeneous with degree of homogeneity ν < 0. We assume that the system (1) is GAS, and b(x) = 0 =⇒ x = 0. Then the sytem (1) is GFTS.

Example 4.1: Let us consider the following example:

f : x ∈ R -→ -(1 + 0.5 cos( 1 x ))x 2 3 ∈ R. ( 12
)
Let F (x) =

2 . For a fixed x, F (x) is bounded. The system ẋ(t) = f (x(t)) is GAS, then there exists a Lyapunov function V fior the system (1). From the definition of f, one has

e τ d(-τ )f (d(τ )x) = -1 + 0.5 cos 1 d(τ )x x 2 3 . ( 13
)
The fact that 1 + 0.5 cos

1 d(τ )x 1 + 0.5 cos 1 x > 0, ∀x ∈ R * and ∀τ ∈ R, gives ∂V (x) ∂x e τ d(-τ )f (d(τ )x) = ∂V (x) ∂x f (x) e τ d(-τ )f (d(τ )x) f (x) = ∂V (x) ∂x f (x) -1 + 0.5 cos 1 d(τ )x x 2 3 -1 + 0.5 cos 1 x x 2 3 = ∂V (x) ∂x f (x) 1 + 0.5 cos 1 d(τ )x 1 + 0.5 cos 1 x < 0. ( 14
)
Which proves the GAS of the origin under the set valued F (x). This inequality and the fact that the degree of homogeneity ν = -1 imply that the system ẋ = f (x) is globally FTS.

V. ACADEMIC EXAMPLE

Let us illustrate this results by the following example, for α, β > 0

       ẋ = y -x 2 3 sign(x), ẏ = -αx 1 3 -β 2 -cos 1 x 2 + y 2 |y| 1 2 sign(y) (15)
Again, we can write the system (15), as the following

ż = H(z)b(z), z = x y , (16)
with

H(x, y) = y -x 2 3 sign(x) 0 -αx 1 3 -β|y| 1 2 sign(y) and b(x, y) =   1 2 -cos 1 √ x 2 +y 2   .
The function H is continuous and d-homogeneous of degree of homogeneity ν = -1, provided that the dilation d is defined as follows d(s) = e 3s 0 0 e 2s , which corespond to a weighted homogeneity. The function b is continuous and bounded. We use the Lyapunov function

V (x, y) = α 3 4 x 4 3 + 1 2 y 2 , (17)
) ∀z ∈ S d and ∀s ∈ R. This proves that the differential inclusion given by the homogenous extension (18) is GAS, and the degree of homogeneity ν = -1 < 0, then the theorem (3.1) implies that the origin of system (15) is FTS. The Figure 1 shows the solution of the system (16) with initial condition (3, 2) which reach the origin in finite time (less than t = 6 ).

VI. CONCLUSION

Finite-time stability of a non-homogeneous GAS system (homogenoeus system with multiplicative bounded function) is investigated using the homogeneous extension and Lyapunov approach. Specifically, we presented some sufficient conditions on the vector fields to guarantee the finite-time stability of dynamical systems which are not homogeneous. By using the results that were proved for a homogeneous systems [START_REF] Sanjay | Geometric homogeneity with applications to finite-time stability[END_REF] we showed the FTS of our system. Example from Section V demonstrates that we can use the homogeneous extension to study the FTS of dynamical systems, which do not admit a homgeneous approximation. The homogeneous extension is also useful for stabilizing in finite-time a nonhomogenous evolution equations. This can be considered as a possible direction for the future research.

Remark 3 . 2 :

 32 If the function b satisfies the condition

  s∈R {e -s d(-s)f (d(s)x)} be the homogeneous extension of the function f, with d(s) = e 3s . Then F (x) = -

Fig. 1 . 1 √ x 2 +y 2 |y| 3 2

 113 Fig. 1. The solutions of the system (15) with α = 0.5, β = 2 and the initial condition (x 0 , y 0 ) = (3, 6).

( 18 ) 1 √ e 6s x 2 +e 4s y 2 |y| 3 2

 1813 Again by using the homogeneous Lyapunov function V, one hasDV (z), H(z)b(d(s)z) = -α|x| -β 2 -cos < 0,

  Mm,n denotes the matrix norm induced by • , i.e. C p (R n , R) denotes the space of functions f : R

	A Mm,n = sup	Ax
	x∈S
	• A continuous function α : R + -→ R + belongs to
	the class K if α(0) = 0 and the function is strictly
	increasing. The function α : R + -→ R + belongs to the
	class K ∞ if α ∈ K and it is increasing to infinity.
	• The notation DV (x), f (x) stands for the directional
	derivative of a continuously differentiable function V
	with respect to the vector field f evaluated at point x.

•

• • n -→ R which have p continuous derivatives and C ∞ (R n , R) denotes the space of functions f : R n -→ R which are smooth. • we denote by CL 0 (E, F ) (respectively CL k (E, F )) the set of continuous functions on E, locally Lipschitz on E \ {0} with value in F (respectively the set of continuous functions on E, C k on E \ {0} with value in F ).

* This work was partially supported by ANR 15 CE23 0007 (Project Finite4SoS), the Government of Russian Federation (Grant 08-08) and the Ministry of Education and Science of Russian Federation (Project 14.Z50.31.0031).).