Daniel Noyes

François Pérès

MAINTENANCE STRATEGIES INTEGRATED INTO PRODUCTION MANAGEM ENT: TOW ARDS PERFORMANCE OPTIMISATION

Keywords: Production systems, maintenance strategies, per forma nce evaluation, stochastic variables, Markov models

An efficient management of the actual production systems oeeds a combined consider ation of the maintenance and production constraints to define the strategies to be im plemented. We propose a modelling form of these systems allowing this coupled analysis. Our presentation focuses on the non deterrninist aspects of the phenomenon which intervene on these systems.

INTRODUCTION

Any manufacturing plant can be divided into two groups usually distinct: production and maintenance departments. The production department is, obviously, directly influenced by many factors, sucb as machine capacity, loading process policies, machine breakdowns and maintenance operations. However, scheduling problems often assume ùninterrupted availability of machines such that jobs can be processed at any time during this ùninterrupted period. This assumption is seldom valid in reality but most of analytical models neglect the corresponding processes. Finally, when these phenomena are taken into account, breakdowns and mainten ance operations are often considered as "disturbing" events. So, pr o duction and main. tenancc are regarded as two aotagonist functions, each of them having different interests with respect to the working progress. Note that some works exploit this situation: preventive maintenance operations slow down the deterioration of machines wbile the whole production activities affects its characteristics [START_REF] Feichtinger | The Nash solution of a maintenance-production differentia .l game[END_REF].

A "contract" mu st be established between both departments. For example, when production requires a level A of availability, the corresponding maintenance latitude is A .This margin can be used by processing x % of corrective actions and l-x% of preventive actions (the x choice is based on cr iteria including economic aspects). A negot iation is needed with production to set and implement this strategy, production resources having to be accessible to carry out preventive maintenance operations.

Finally, a co-operating mode will help to define management and maintenance strategies, each of them integrating the constraints of the other and the ef f e cts of its implementation. An efficient belp when elaborating the corresponding strategies is to design and to engage hybrid models allowing to consider both points of view. These models will be used in predictive evaluat ion.

Our presentation deals with these aspects. The pap er is divided into three sections.

We first resume the analysis situation and we give the main characteristics whicb have to be considered. We present the worldoad diagrams in the second section and emphasize on distinct features of the representation. We illustrate this modelling and the corresponding analysis principles using an optimisation application.

1-SITIJATION

A simplified form of a "production mission" is the realisat ion of a manufacturing order which can be resumed as the manufacture of a certain amount of parts in an allotted tinte with a certain required quality level and fixed economic conditions [START_REF] Noyes | Objectifs de production et Fonction Maintenance : Modèles d'analyse[END_REF]. In fact, the firts three these mission attributes can be expressed as weel in tenns of cost It is often the final expression of mission accomplishment and evaluation of the reached ob j ectives. Only the first three attributes are directly accessible to production by analysis of physical measures. The q quantity and t delay attributes are generally linked with simple relationships (for example, production of q=300 parts in t=S days). Figure 1 shows an illustration of the q(t ) function.

�----------+--'--+-►

� rime ât The curve (a) represents the theoretical production. The curve (b) represents the practical considered function, using a weighting factor s. This security factor (s<l ) intervenes as a direct multiplier of q (q.s) or t ( t.1/s). This coefficient s enables to integrate the effects of possible risks in prevision. Estimation of s often results of know-how and experience of the operators. The grey curve represents the effective production. The end of the production order may be foreseen in a time period <i t taking into c-0nsideration the effects of some risks (and the incomplete knowledge of the system behaviour ) . Sorne .financial penalties may affect the service if a delay appears when c arryi.n g out the production order. These penalties may be directly linked with the current order or indirectly with the loss of future orders resulting from a customer unsatisfaction. Severa! works have been concemed wit h performance analysis of production systems. Only few of them have used models which consider explicitly the relations between production and maintenance departments (Paté-Comell, et ail., 1987) ... More often. these works deal with particular pr oblems and specific situations.

Sorne aspects have to be considered when defining models integrating botb production and maintenance point ofview.

-First, the mode! must allow to describe the effects of the main concurrent processes intervening on the system and influencing its behaviour. On the machine ground, these processes will express the effects of the loading strategy (resource engagement), running strategy (resource driving and failure process), maintenance strategy (preventive and corrective actions), ... For example, in j ust in time m, production constraint s relative to resource availability will be better specified when expressing: 1 ) the effective machine availability at the time it is required to treat the pr o duction order, 2 ) the probability to rely on the machine during the requirement period. -FinaUy, the mode! must take into accou nt the particular dependencies between production and maintenance (Lee, el al/., 1989). Bach de partment manages its ow n resources (material and human means), but maintenance resources are specifi c since they are applied on product ion ones which consequently depend on them. In other respects, human mcans of both functions may be shared (Total Productive Maintenance case for instance).

We present in the two next sections some mathematical tools and simulation methods to describe the effects of mai ntenance policy, machine unreliability, processing time variability, ratio of preventive maintenance time to processing time, ... and to develop useful measures of performance.

2-AN

A L YTICAL MODEL Among the tools aUowing an analytical modelling (probability theoty, stochastic proccsses, we present some developments bascd on markovian models representing the production systems. Our choice, dependent on the own characteristics of the considered processes, is linked to calculation capabilities of the state graphs and the aptitude of these charts to fi t the markovian hypothesis. We can find these rates in the simplified representation of a production mission (Figure 3). Let us recapitulate the mode! aptitude to integrate the situations to be analysed.

-State graph just expresses the competition between the main processes interacting on the system. The direct influence on the interactions bctween these processes can be expressed in the mode) (for instance, one can study the influence of the prevention actions by correlating the event rates of the corresponding samples according to the given dependencies.

-Most of structural characterist ics can also be easily integrated in the mode! through more details in the rate events (figure 4):

. addition of arcs bet ween states to represent other processes related to new events like failure due to preventive maintenance (y2) , . creation of new states to distinguish for example electrical and mechanical failures (states C2 and C3).

-The markovian hypothesis (process evolution considered as independent of the past of the process) is not always verified in production system treatrnent Sorne considered event distributions are not exponential. We noticed it is the case when dea]ing with failure rate (rather increasing with time) or repair rate (rather decreasing with time). To face this problem, we use the fi ctitious states method w h icb allows through introduction of some intcnnediate artificial states to simulate the various forms of distribution encountered (Laprie, 1975) (De Carvalho, 1991). Cost structures can be superposed on the mode) in order to evaluate the production/maintenance strategies but also to optimise the relevant parameters. The introduced costs can be connected to the events goveming the system (for instance: failure cost related to transition toward failed state) or be rime dependent (for instance: intervention cost per hour related to time spent in the state). The economical measures, representative of the maintained system running can thus be estimated from the mode! [START_REF] Pérès | Outils d'analyse de performance pour stratégies de maintenance dans les systèmes de production[END_REF].

Let us give an industrial example of this approach to optimise a strategy according an economical point of view.

We consider an elementary mechanical parts worlcshop. We focus on a specific unit downstream two production machines. The unit is a 5D machining centre having a specific palletizing system. Sometimes, this machine appears as the worlcshop bottleneck according to the effective load of the plant. lt has an increasing failure rate, justifying preventive operations. The analysis is intended to evaluate the frequency of the preventive actions.

CONCLUSION

The work presented in this paper concerns the product ion system performance as it appears in a plant by combining production and maintenance aspects. The problem approach using state graphs pr esents many advantages in building models for analysis. These models are well adapted to reproduce a situation shared by production and maintenance dcpartments. They allow to rcpresent the joint imponant processes : solicitation/response, failure/ repair, and inspection/revision and to express efficiently the dependencies effects between them.

The economical measures which can be obtained from these models represent a consistent help to guide th e production and maintenance managers in optimising strategies. Howcver, the rcsults established in the last section must be considered as an attempt to employ state graphs in the aim of optimisation. Wo. rk must now be twned toward the rcpresentation study of complex systems and, in particular, the correlation with plant production management

Figure 1

 1 Figure 1. q(t) function example.

-

  is the case of the failure and repair processes dependant on many factors such as: machine characteristics, operator exper ience,... lt is also the case of prev e nt ive maintenance actions in which the intervention time distribution and the effective duration of the actions are associated with non deterministic processes. This phenomena is particularly pronounced when one takes advantage of a corrective action to carry out a global inspection of the stopped resource. -The mode) must then integrate the time horizons specific of the identified processes. Preventive and corrective maintenance interventions are governed by processes with similar dynamics (frequency and duration of the events are roughly on the same order) but their effects are on two distinct time scales. C-Orrective actions have a real time schedule. Planning of preventive maintenance is based on a middle tenn period, the expected gain being moreover delayed until modifications of nature and frequency of the futures failures can be noticed. The actions of production result of a long term planning for the master production schedule, middle term planning for scheduling and real rime for the tasks in-progress.

  Figure 2. Basic workload diagrams of a production machine The transition rates between states correspond to the frequency of event occurrence making the system change of state. They can be expressed on severaJ ways: as a tirne unit function, as a pa rt number function, ... More often we consider the number of transitions per time unit. W e introduce in the model the following rates : solicitation rate (a), response

Figure 3 .

 3 Figure 3. Detailed view ofq(t) functionTwo transition rates, namely µ and 13 (repair rate and slope of q(t) or number of parts made per time unit), can at least influence the system evolution from a failure state or after a repair action. With regard to µ, the failure nature and the repair process characteristics have a direct influence on the down time (1/µ). Relating to 13, the repair quality can affects the slope value q(t) (Aj3=0: same performance rate than before failure, Al3<0: performance rate lower, .113>(): higher rate (resource amelioration after r e pair).The running rules ensuing a forward-looking estimation delay t, can modify the value of 13 (through programmed increase or decrease of the production pace leading to a variation Al}). The failure process (rate À) can be influe. nced by changes on the resources solicitation.Whe. n expressing the failure rate against the production rate : ,JI}, mission reliability R(t) can be expressed on other unit than rime as a function of the part number continuou sly produced, that is to say R(p). ConsequentJy, one can determine size batches to guarantee their continuous treatment (quality insurance requirement IS 9002-9003) or at least, be aware of the incurred risk of a mission interruption affecting the homogeneity of the in process batch treat mentThe model can represent in the proposed form the machine engagement according to a Kanban or MRP strategy through simple adaptation of the a and 13 rates, whioh can be associated to a single part production or a batch production. Merging W and R states leads to classical reliability models suited to bottleneck machines analysis in an OPT strategy or to the fl ow pushed machine analysis.

  Figure 4. Extended workload diagrams

Figure 5 .

 5 Figure 5. Workload mode! with fictitious states.

FigureFigure 6 .

 6 Figure 6 shows the F curves related to five cases in which cl .. lcu (cost unit), c2=1.5cu, c4 = lcu for each of the fivc cases and c3= 2u, Seu, Seu, l lcu and 14cu in the respective Sl, ... ,S5 simulations. The plots correspond to globo.! cost variation (or F values) against interval inspection y.As it is shown on the plots, the frcquency variations of the preventive actions evolve in a ratio of 1 to 4 relative to this data set According to the preventivc actions, the failure rate Â. often dccreases while the maintenance rate µ grows (Vesely, 1993), increasing the noticed effects.


Aïdin Sumic
email: asumic@enit.fr

Emna Amdouni
email: emna.amdouni@enit.fr

Thierry Vidal
email: tvidal@enit.fr

Mohamed-Hedi Karray
email: mkarray@enit.fr

Towards flexibility sharing in multi-agent dynamic planning: the case of the health crisis

Keywords: crisis management, flexibility, multi-agent system, decision making under uncertainty, negotiation

Planning problems in a crisis context are a highly uncertain environment where health facilities must cooperate in providing health services to their patients. We focus on the health crisis in France due to the COVID19 pandemic. In fact, the lack of appropriate scheduling tools, resources, and communication leads hospitals to be submerged by infected patients and forced to transfer them to other hospitals. In this work we aim to provide a global solution to such planning problems to improve the current French health system. We introduce a cooperative approach called OPPIC (Operational Planning Platform for Inter-healthcare Coordination). OPPIC is based on a decentralized system, where health facilities plan is dynamic, flexible, robust to uncertainty, and respond to goals and optimization criteria. This paper proposed a first planning model to OPPIC and provided a first way of negotiation between health facilities based on their plan's local and global flexibility.

INTRODUCTION

In general, the healthcare system planning must schedule a hospital task by allocating resources to each task and managing hospital goals. It should answer several patient needs, mainly operating, bed assignment and management, and nurse scheduling during a clinical routine or crisis (e.g., climate, nutrition, health, etc.). Delivering healthcare services during an emergency or a crisis is very challenging. For that reason, countries should be prepared to face any crisis. For instance, hospitals still facing several difficulties to allocate the needed resources to their patients during the COVID-19 pandemic also known as the corona virus pandemic. The COVID-19 was first identified in November 2019 at Wuhan and declared a pandemic by the WHO (World Health Organization) in March 2020. Indeed, the COVID-19 pointed out the weakness of all the existing worldwide healthcare systems. We will focus on the French case, one of the most affected countries where the problems due to COVID are common to those in other countries. In France, the COVID-19 caused several healthcare problems mainly related to the lack of available beds, the lack of human resources and material resources. Although it's possible to predict the pandemic's progression [START_REF] Kaplan | Aligning SARS-CoV-2 indicators via an epidemic model: application to hospital admissions and RNA detection in sewage sludge[END_REF], it is still hard to predict a hospital's needs regarding the availability of intensive care beds for their patients [START_REF] Aphp | Early indicators of intensive care unit bed requirement during the COVID-19 epidemic: A retrospective study in Ile-de-France region, France[END_REF]]. This has led hospitals to adopt new strategies such as transferring their patients to other national or international hospitals. For example, from March to April 2020, the French region of Grand-Est moved a total of 349 patients; among them, 164 patients were assigned to hospitals of other European countries, mainly Germany and Switzerland [START_REF] Habchi | Covid-19: les transferts internationaux de patients de réanimation, une solution pour les départements frontaliers[END_REF]]. The patients' assignment is a challenging task due to the lack of appropriate scheduling tools, for instance, tools to consult the plan of a hospital regarding the availability of its resources (e.g., beds, staff, ambulance, etc.). Some tools exist to help hospitals, such as the "Covid moi un lit" application that informs real-time about the availability of beds for any healthcare facility. Unfortunately, the current tools do not consider an online hospital's plan (e.g., patient transfer, beds reallocation, deprogram and reprogram operations, etc.); thus, they cannot propose a global solution for the overall healthcare system. In addition, the dynamic aspect of a crisis should also be considered by the scheduling tools to enable replanning of a current hospital plan regarding the fluctuations during a health crisis. For example, an infection of medical staff could reduce their availability to deliver health services to their patients.

Since 1970, several work have been proposed to improve planning in the healthcare field: a markovian model for hospital admission scheduling [START_REF] Kolesar | A Markovian model for hospital admission scheduling[END_REF]], a computational approach to patient flow logistics in hospital [START_REF] Hutzschenreuter | A computational approach to patient flow logistics in hospitals[END_REF]], a simulation of resources assignment in hospital [START_REF] Zhao | Modeling and Simulation of Patient Flow in Hospitals for Resource Utilization[END_REF], a simulation to optimize resource allocation and outpatient appointment scheduling [START_REF] Lin | Resource allocation and outpatient appointment scheduling using simulation optimization[END_REF], an interoperable operational emergency response system for large-scale situation [START_REF] Elmhadhbi | An Ontological Approach to Enhancing Information Sharing in Disaster Response[END_REF], and more [START_REF] Liang | Improvement of chemotherapy patient flow and scheduling in an outpatient oncology clinic[END_REF]], [START_REF] Meng | Analysis of patient waiting time governed by a generic maximum waiting time policy with general phase-type approximations[END_REF]]. These works aim to optimize resource allocation and patient waiting time in common or emergency situations but not in critical ones such as a health crisis. Moreover, they do not deal with planning problems in an uncertain environment during a health crisis. Indeed, such planning problems consider four main aspects to enable dynamic, robust, and flexible planning: -Time (e.g., starting or finishing time of an operation, duration of an operations, date of specific deadlines) -Resource (e.g., intensive care bed, medical machine, vehicle, medical staff) -Uncertainty (e.g., operation duration, resources availability, operation success, influx of patients, etc.) -Optimization criteria (e.g., saving lives as much as possible, the minimum duration of bed use) or to deal with uncertainty (e.g., integrating the maximum amount of flexibility in a schedule). Although these works do not consider such aspects, this does not mean that in planning literature, no solution exists to answer these planning problems. In addition, the newly adopted strategies that require coordination (e.g., patient transfer) bring another aspect to our planning problem, which is coordination between healthcare facilities. Therefore, if we represent a healthcare facility as an agent, we are faced with a multi-agent problem where, in a coordinated way, agents have to meet their goals and the other agent's goals. Furthermore, agents may not always be homogeneous, which means that agents may have the same goal but not the same definition; for example, agents do not have the same definition of flexibility. Therefore, they do not solve their goal in the same way. So, the more flexibility an agent allows to himself in his plan, the more it reduces its level of uncertainty. Still, the more he becomes a source of uncertainty for the others. Consequently, the problem of distributed optimal decision becomes complex and requires enhanced interoperability between the agents [START_REF] Behnke | Integrating Ontologies and Planning for Cognitive Systems[END_REF]. This is why, using an ontological approach can help to formalize better the information that the agents will have to exchange and reinforces the interpretation capacities of the agents to produce consistent reasoning [START_REF] Gayathri | Ontology based knowledge representation technique, domain modeling languages and planners for robotic path planning: A survey[END_REF], [START_REF] Marcus | The next decade in AI: four steps towards robust artificial intelligence[END_REF]], [START_REF] Traoré | Ontology for Healthcare Systems Modeling and Simulation[END_REF]].

The paper introduces an Operational Planning Platform for Inter-healthcare Coordination (called OPPIC) that implements a flexibility negotiation algorithm using Simple Temporal Network with Uncertainty (STNU) as a temporal planning model. [START_REF] Morris | Dynamic control of plans with temporal uncertainty[END_REF]. It is organized as follows: Section 1 presents an overview of planning and scheduling approaches under uncertainties. Then, Section 2 describes our proposed Operational Planning Platform for Inter-healthcare Coordination (OPPIC) system. Next, Section 3 details our proposed planning approach based on STNU's models. Finally, the conclusions and some future works.

AN OVERVIEW OF PLANNING AND SCHEDULING

This section focuses on planning techniques that ensure a dynamic, flexible, and robust plan to any type of uncertainty; multi-agent planning techniques that produce a global solution to a planning problem.

Planning under uncertainty

Planning means making decisions that are focused on the future. Uncertainty exists when several future developments are possible, but we cannot say with absolute certainty how the future will develop. Even if we cannot predict the future, it is possible to plan for it and focus on your planning [START_REF] Ghallab | Automated Planning: theory and practice[END_REF]]. The uncertainty in our collaborative planning problem might come from three main sources related to the planning model [START_REF] Martin | Composition flexible par planification automatique[END_REF]]: First, an agent's goals; an agent could not satisfy other agents' goals if their goals are not visible and clear for him. Second, the domain of the planning model; an activity duration or status (success or failure) might be undetermined and thus cause uncertainty. The third is the planning model's state, which defines if a model is observable, partially observable, or non-observable. Many techniques are proposed for planning under uncertainty [START_REF] Baki | Planification et ordonnancement probabilistes sous contraintes temporelles[END_REF]], such as conditional planning, probabilistic planning, temporal planning, and contingent and conformant planning. Conditional planning consists of creating a plan that considers all possibilities represented by a branch; a branch represents the observation too occurs during the execution to use it. Probabilistic planning procures a plan with a probability of success greater than a fixed threshold. Temporal planning creates a plan that takes into account constraints bound with time. Eventually, the contingent and conformant planning create a plan where a model is not fully observable where, respectively, one doesn't have action to observe it while the other can. These planning techniques can provide us with a robust plan against uncertainty. However, our need for a dynamic plan means that we need proactive techniques that belong to the temporal planning taxonomy.

In that context, we identified STNUs (Simple Temporal Network with Uncertainties) [START_REF] Morris | Dynamic control of plans with temporal uncertainty[END_REF]] as a suitable model to provide a dynamic, flexible, and robust plan. An STNU is a triplet (V, E, C), where V is a set of temporal points, E is a set of requirement links that refer to an STN link, and C is a set of contingent links. Each contingent link has the form (A, x, y, C), where A ∈ V, C ∈ V, and 0 > x > y > ∞. A is called the activation time-point, and C is the contingent time-point where the duration C -A is uncontrollable but lies in the interval [x, y]. An STNU is dynamically controllable if, regardless of the duration of all contingent links in an STNU, it is always possible to guarantee the success of a plan. In general, STNUs allow through time-points (start and end of the activity) to apply constraints between these time-points, which are intervals of possible values allowing to model the uncertainty and the flexibility.

Interleaving planning and execution in dynamic architectures

Bidot et al. [START_REF] Bidot | A theoretic and practical framework for scheduling in a stochastic environment[END_REF]] describe taxonomies of scheduling and planning techniques under uncertainty that are independent of any specific representation or reasoning technique. The authors distinguish in their work between three main taxonomy of techniques of "balancing schedule generation and execution". First, proactive approaches build a global solution at a generation time that will never be reconsidered at an execution time. A proactive approach considers uncertainty to produce more robust, more stable schedules or both than they would be without considering such knowledge. To provide a proactive approach, one may generate:

• A complete generic schedule insensitive to online perturbations by considering the worst-case scenario.

• A flexible schedule where a subset of decisions are made offline (i.e., generation time) and the rest online (i.e., execution time).

• A conditional schedule where everything is set but with alternative branches. The one is effectively chosen to depend on conditions that will only be observed at execution time.

Secondly, the revision techniques generate a complete schedule where decisions are made at generation time. Then, when the solution derives from the current observations at execution time, the decisions are revised and changed through online replanning. Lastly, the progressive techniques interleave scheduling and execution by solving the problem piece by piece. For example, each piece can correspond to the activities in a time window. The scheduled decisions in a plan are made periodically or when new information arrives, and once a decision is set, it can't be changed.

To summarize, several studies show that combining proactive, revision, or progressive techniques would improve results [START_REF] Shafaei | Workshop scheduling using practical (inaccurate) data Part 1: The performance of heuristic scheduling rules in a dynamic job shop environment using a rolling time horizon approach[END_REF], [START_REF] Chien | Using Iterative Repair to Improve the Responsiveness of Planning and Scheduling[END_REF]], [START_REF] Branke | Anticipation in dynamic optimization: The scheduling case[END_REF]. In this work, we need a dynamic, robust, and flexible plan where decisions must be made online; this plan belongs to proactive techniques that limit replanning. Thus, we believe that combining proactive and revision techniques would help answer our needs.

Multi-agent dynamic planning

In general, multi-agent planning (MAP) is used when an agent cannot solve a task by itself or when it is more efficient to cooperate with other agents. In addition, MAP problems can be classified according to their level of coupling. It's a measure of the number of interactions or coordination points between agents during the resolution of the task. In loosely coupled problems, each problem goal is likely to be solved by a single agent, while strongly coupled issues tend to require the cooperation of multiple agents. Therefore, it is necessary to ensure a satisfactory outcome in weakly and strongly coupled problems [START_REF] Torreño | A flexible coupling approach to multi-agent planning under incomplete information[END_REF]].

Fortunately, STNUs provide a first approach to planning in a multi-agent system with MaSTNUs [START_REF] Morris | Dynamic control of plans with temporal uncertainty[END_REF].

A MaSTNU is a quadruplet (A, V, E, C) with A a set of agents and (V, E, C) an STNU. In a MaSTNU, time-points are shared by agents, i.e., "for a time-point v in V, there exists a unique agent a in A that owns v, denoted owner(v) = a". Thus, the agent that owns v, an executable time-point, controls the execution of the event associated with v. If v is a contingent time-point, the owner of v is the unique agent supposed to observe the realization of v instantaneously. Time-points owned by other agents are not considered to be observed directly, but information about their realization can be obtained through external contingent links. MaSTNUs are used in centralized systems where agents share their time's points and synchronize to the same reference time-point. However, our cooperative planning problem supposes a system where agents have no information on other time-points and are not synchronized. Therefore MaSTNU does not match our needs.

Another way of using MAP is provided by Torenno et al., which combines single-agent planning technologies and a refinement-based methodology [START_REF] Torreño | A flexible coupling approach to multi-agent planning under incomplete information[END_REF]. Agents begin by sharing their initial information (state, goals, etc.). Then, based on a basic plan, each agent will initially propose one or more plans. Next, each of them will share its plan with the others to ensure that each agent gets a global view of each proposed plan. Finally, they choose together the best of all plans as the new baseline plan and repeat these steps until a complete and robust global solution is found. This technique gives excellent results in both loosely and strongly coupled problems. However, like MaSTNU, it focuses more on a centralized system where agents share all of their information. Furthermore, this technique needs a convergence we can't ensure since it's based on a recursive method.

Briand et al. work propose a cooperative approach in Job-Shop scheduling problems under uncertainty [START_REF] Briand | A Cooperative Approach for Job Shop Scheduling under Uncertainties[END_REF]. Compared to the first two approaches, this one is more oriented towards a decentralized system. Indeed, tasks are allocated to different machines and each machine is controlled by a decision center. Each decision center has its definition of flexibility and schedules the allocation of tasks to its machine while optimizing its makespan.

One of the primary keys to Briand's et al. work is a first approach to negotiating and renegotiating through decision centers. Moreover, the more flexibility a decision center gives itself, the more uncertainty it creates for the others. Hence, it is essential to maximize flexibility while minimizing inconsistency. It can be done as an optimization problem by solving a linear function with a linear program to give an acceptable but non-optimal plan. This approach cannot be reused in our cooperative problem because the Job-Shop context is based only on scheduling tasks on machines. However, we believe that using Briand's et al. idea to produce a decentralized MaSTNU network is a first way to fit our problem. Indeed, supposing a planning model wherein a decentralized MaSTNU, agents are not synchronized and do not share their time-points or only partially could fit our planning problem.

TOWARDS A COOPERATIVE APPROACH: OPPIC

The organization of the french healthcare system

In 2000, the French health system was considered the most globally efficient in terms of health care delivery and organization by the World Health Organization (WHO). However, it becomes inefficient to answer patients' health care demands during the COVID19 crisis. The first case of COVID-19 was detected in France on 24th January 2020, followed by the WHO's advice to help the French health system fight against the COVID-19 pandemic.

From the current French health system, we defined the most important actors that are involved in the strategic, tactical and operational process of the COVID-19 pandemic. Figure 1 describes these actor and their roles. It shows that hospitals, that are located in cities and town, must be at the operational level; take care of patients, deploy the required resources to the patient's need, transfer the patients, and respond to private and global objectives. Moreover, the regional health agency (RHA), that is located at the regional level, must be at the tactical level; coordinate all health facilities in their area, deploy external resources, reassign the distribution of human resources and constantly report at a higher level their area's situation. The health crisis center (HCC), at the national level, is composed of 40 specialists in their field (doctors, pharmacists, public health interns, managers and engineers). At the strategical level, they deploy and coordinate each RHA, monitor and analyze the health situation from the report of each RHA and set global goals. At the national level, the health minister implements the Government's policy in the field of health. At the strategical level, it defined new reforms and, with the HCC, created new strategies directions and can deploy army resources.

Disappointingly, the french healthcare system is not efficient in facing long-term crises such as the ongoing COVID-19 pandemic. We believe that a cooperative approach between health organizations to help them communicate and negotiate to fulfil their goals can face a long term crisis.

Our methodology

To handle with the collaborative planning problem, we need to answer different challenges. In that context, the work of [START_REF] Elmhadhbi | A Semantics-Based Common Operational Command System for Multiagency Disaster Response[END_REF] on disaster response helps us to define four main challenges to answer. The first one is the planning issues. We have to ensure to provide a flexible and robust plan thought replanning. Next, the resource assignment issues the need to allocate the needed resources at the right time. Then, the information issues defined the lack of information between users and the information uncertainty (e.g., validity, understanding, etc.). Lastly, the cooperation issues to share resources, flexibility and information among heterogeneous agents.

To deal with these challenges, we propose a common multi-agent architecture based on a knowledge graph, which is a semantic representation of data into a graphical model [START_REF] Hogan | Knowledge graphs[END_REF]], [START_REF] Horrocks | Ontologies and the semantic web[END_REF]]. The proposed architecture aims to formalize the representation, improve sharing, and automated reasoning for the involved The architecture shows four main modules:

• Module 1 refers to the users. A user refers to a healthcare facility with a plan and a planner. The planner considers the user goals, resources, and goals and resources of the decision center to provide a dynamic, robust, and flexible plan. Furthermore, the negotiation between agents is made with the help of the cooperative broker, which may notify him back when it's relevant to replanning the current plan.

• Module 2 refers to the decision center (DC). DC provides global goals and external resources to their users and also might be involved in the negotiation to make decisions. It might happen when users can't find a solution for their planning problem. Therefore, using each user's knowledge through the global ontology will choose which will be prioritized in the negotiation.

• Module 3 refers to the cooperative broker in charge of the negotiation between users. It helps consider an agent request to help him communicate and negotiate with the users involved in the request. Moreover, the notification module might notify an agent planner to replan the involved plan when necessary. For example, a patient transfer that will finish later than expected may need replanning.

• Module 4 refers to the global ontology that defines the planning problem's related concepts by containing the world's knowledge (e.g., resources, goals, etc.). This knowledge is used to help agents understand each other (e.g., goals, resources, plans, etc.) in the negotiation module to provide a solution to their requests.

In the next section, we propose a planning model to our architecture based on a decentralized MaSTNU system by representing each agent plan by an STNU.

OUR CONTRIBUTION: TOWARDS A PLANNING MODEL BASED ON STNU FOR OPPIC

Definition of our planning model

In Section 1, we presented the cooperative approach of Briand et al. [START_REF] Briand | A Cooperative Approach for Job Shop Scheduling under Uncertainties[END_REF]] and the centralized MaSTNU approach. Although both are not compatible, we propose a decentralized MaSTNUs model using both of them as references. Figure 3 show an interaction between hospital A and B plans using our planning model which contains:

• A set of agents, each agent has a plan defined by an STNU (V, E, C) not shared with the others

• A set of time-points that are shared among agents. When interactions between two activities occur and do not belong to the same agent, both agents contain their time-points in their respective STNU. On the other hand, unlike MaSTNUs, each agent has a reference time-point 0 unique to its STNU.

• An activity is defined by two time-points, i and j, where i is the start time-point, and j is the end time-point (e.g., 1 and 2 represent activity in figure 3). Moreover, an activity has constraints we define as: First, the duration constraint, the constraint that defines the temporal distance between i and j. Secondly are the external constraints, which are all other activity constraints. For example, in figure 3 the constraint between 2 and 3 is a temporal gap constraint: it means the temporal distance between these two activities must be within [0, 0]. In addition, an activity has an owner, which is the agent that controls its time-points.

• An agent has contingent or non-contingent activities. A contingent activity is either controlled by another agent, and therefore there is a sharing of time-points between these two agents, or belong to him, but he does not control it. For example, an operation which ending time cannot be predicted and controlled. In figure 3, the activity with the times points 1 and 2 is contingent for agent A (e.g., doted arc) but controllable for B because B owns this activity (e.g., plain arc). Now that we have described our system, we will present its main objectives. The first one is how to address the flexibility of an agent STNU; the second is how an agent may negotiate its flexibility to reach its goals. Either to regain its consistency or either to extend its flexibility.

Figure 1 .

 1 Figure 1. The organization of the french healthcare system during the COVID-19 pandemic

Figure 2 .

 2 Figure 2. The Operational Planning Platform for Inter-healthcare coordination (OPPIC) architecture

Figure 3 .

 3 Figure 3. Example of a decentralized MaSTNU system involving 2 agents' STNU

Table 1. Notation

X: an STNU S(M): the contingent activity source of M : any activity of X C: set of contingent constraints [ 1 , 2 , .., ] : all external constraints of ( ): a set of flexibilities [ 1 , 2 , .., ] F( ): the flexibility of E: set of controllable constraints [ 1 , 2 , .., ] (X): the flexibility of X ( ): the required flexibility of e ∈ E, where e ∈ (X): the flexibility degree of X : the required flexibility for M M: a cycle in X owner(c): the agent that control c with c ∈ C, where C ∈ X F(M): the flexibility of M (c): the flexibility owner(c) give to X ( ): the maximum duration of X the flexibility obtained from contingents constraints in M

3.2 Planning model: how to address flexibility?

First, table 1 defines notations that will be used in this section:

Definition 1 (Activities temporal distance) An activity in X an STNU X = (V, E, C), has a duration constraint defined by a lower bound and an upper bound :

In addition, an activity has external constraints, which are primitive relations between time-points. The most basic ones are relations between activities (e.g., activity A before B, B during A, etc.). Otherwise, we have primitive relation that represents a temporal gap between two activities (e.g., B start 5-time unit after A). Finally, it is also possible to have a constraint between an activity and a time-point (e.g., A must start before 2 pm). Respectively, we called these constraints: relationship constraints, gap constraints and time constraints Next, we distinguish between two categories of flexibility: first, the global flexibility of an agent STNU. Second, local flexibility is defined as the flexibility of activity.

Definition 2 (Base flexibility) A controllable activity's base flexibility is defined by the temporal distance between its lower bound and upper bound. For a contingent it's equivalent to 0:

The base flexibility (Equation 2) of does not consider its external constraints, which can reduce its base flexibility. An external constraint can act on the time-points of , which will reduce its base flexibility (see Table 2 example 1). Therefore if has more than one external constraint, it leads to different possible flexibilities for noted ( ).

Definition 3 (Local flexibility)

The flexibility of is the minimum flexibility contained in ( ), a set of all possible flexibility of . The constraint propagation of the dynamic controllability checking algorithm already gives these hidden flexibilities:

Definition 4 (Global flexibility) The flexibility of an agent's STNU is the sum of the flexibility of its activities, and the sum of the possible flexibilities given by its gap constraints noted . Indeed, gap constraints between activities can provide flexibility (e.g., see figure 3: the gap constraint between 0 and 1 give a flexibility of 5):

Definition 5 (Flexibility degree) The flexibility degree of an agent STNU X can be defined by its global flexibility, and its STNU makespan noted :

The definition of an agent's STNU flexibility degree also defines an agent's flexibility degree. The more flexibility an agent has, the higher the degree of flexibility. Thus, the more flexibility it potentially has to give. Therefore, comparing agents by their STNU's flexibility degree is possible.

Planning model: how to negotiate the dynamic controllability of an STNU?

An agent must negotiate when its STNU is no more dynamically controllable (DC) due to some disturbances or when the activity's flexibility is too low and hence needs to be negotiated. This paper will only study the first case, how an STNU can negotiate its DC. We need to identify for any not met constraint, all contingent activities to negotiate flexibility to resolve the not met constraint.

How to get all contingent activities for negotiation?

Property 2 (STNU) A contingent activity (i.e., activity) is the source of the non-DC of an STNU X. An STNU may have more than one source, which means that the non-DC of an STNU can be caused by more than one contingent activity.

Proof:

The definition of the dynamic controllability of an STNU is: "An STNU is DC if and only if, regardless of the duration of all contingent activities, it's always possible to guarantee the success of a plan". Thus, X is non-DC if the duration of at least one contingent activity is too short or too long to satisfy all constraints of X.

Suppose an STNU X containing one constraint not satisfied . There exist a cycle activities M ∈ X where S(M) ∈ M is the contingent source of the not meet constraint. This cycle contains all activities to negotiate for the not meet constraint. Consequently, the more X has not met constraints, the more cycle can be found. Therefore, the agent must negotiate his consistency on each cycle to regain consistency on X. The following section will present the first approach to negotiate consistency on a cycle by negotiating flexibility from all contingent activities in the cycle.

How to regain consistency on a cycle ?

First, it is important to clarify that negotiating on S(M) may not be the best way to regain consistency. In the case of Table 2 example 2, it's more appropriate to negotiate with the agents owning 2 and 3 than the one owning 1 . Therefore, it is essential to know the flexibility of each contingent activity, which their owner can give before negotiating with them.

Next, suppose M a cycle and k the flexibility needed to regain consistency. The cycle may contains contingent activities, controllable activities or both. So, in M we can get flexibility from contingents and controllable activities. Therefore, we define 1 the flexibility gained from controllable activities in M. Then, to get only the flexibility that M must negotiate on contingent activities, noted 2 , we subtract 1 to k : 2 = k -1 . Hence, this leads us to a new definition of k:

In contrast, giving k flexibility to M means that M is consistent but F(M) = 0 ≡ ∀ e ∈ M, F(e) = 0. However, the agent might need some flexibility on the controllable activities of M. Therefore, we define the flexibility needed for M where F(M) ≥ with ≥ 0 (Table 1):

Definition 6 (k) If M needs some flexibility, then the agent must negotiate more flexibility on contingent activities to gives the controllable activities the flexibility M need . As a result, we can redefine k by:

Now, we will focus on contingent activities and their flexibility. We previously defined the flexibility of a contingent to 0. Still, if another agent owns c, then for this agent, c is controllable and might have some flexibility on this agent's STNU: F(c)≥ 0. Thus, we define ( ) , the flexibility the owner of c can give to the agent's STNU where c ∈ M. In addition, we define , the sum of each contingent flexibility given by their owner considered as the flexibility obtained from contingent activities in M:

Consequently, M can regain consistency if and only if the condition ≥ ( 2 + ) is satisfied. Then, the agent selects the best candidates from contingent activities by ordering them by their flexibility (see Table 2 example 2). Next, the agent negotiates with the best candidates owners to get the required flexibility k. ( 3 ) = 8 and S(M) = 1 with 1 , 2 and 3 in M The best candidates order is : ) holds, then the agent negotiates with the best candidates to reclaim consistency on the cycle. Unfortunately, the algorithm is incomplete because the Negotiation-Cycle algorithm do not consider the case where the condition ≥ ( 2 + ) is false. How does an agent negotiate consistency when the flexibility given by contingent activities in M is not enough? This question is still a problem unresolved. Furthermore, we still need to provide an answer to how an agent negotiate flexibility on controllable activities when its STNU is DC.

CONCLUSION

In summary, providing balanced planning between different health institutions coordinating during health crisis time is very challenging. In such a situation, each hospital schedule plans to take charge of its patients and try at the same time to answer others' requests. In this work, we presented a collaborative architecture for a cooperative planning problem under uncertainty in health crisis time. Moreover, an overview of planning and scheduling techniques allowed us to provide a first planning model for the proposed architecture. This model is based on a MaSTNU decentralized system and provides a first definition of flexibility and negotiation through the Negotiation-Cycle algorithm. We believe that the planning model should be completed by incorporating other temporal planning techniques. Furthermore, we still have some issues to resolve in the negotiation process. Indeed, our current work does not consider the second case of negotiation: regain flexibility on activities when the STNU is DC. Moreover, the Negotiation-Cycle algorithm does not consider the case where contingent activities flexibility of a cycle M is insufficient. In addition, we will use alternatives or complementary models to provide a stochastic aspect to the temporal model to enhance our ability to handle uncertainties. This work is in progress; the different components of the proposed architecture should be investigated to be developed or readapted from state of the art as, for example, the global ontology that will ensure the semantic interoperability between the different stakeholders.


Frédéric Rotella

Javier Francisco

Mounir Carrillo

Ayadi

POLYNOMIAL CONTROLLER DESIGN BASED ON FLATNESS

By the use of fiatness the problem of pole placement, which consists in imposing closed loop system dynamics can be rclatcd to tracking. Polynomial controllers for finitc dimcnsional linear systems can then be designed with very natural choices for high level parameters design. This design leads to a Bezout equation which is independent of the closed loop dynamics but depends only on the system model.

INTRODUCTION

For finite-dimensional linear systems, a well-known control design technique is con stituted by polynomial two-degrees-of-freedom controllers [START_REF] Astrom | Computer Controlled Systems, Theory and Design[END_REF][START_REF] Franklin | Digital Control of Dynamic Systems[END_REF][START_REF] Kucera | Analysis and Design of Discrete Linear Control Systems[END_REF], which have heen introduced forty years ago by [START_REF] Horowitz | Synthesis of Feedback Systems[END_REF]. Whatever the chosen design method, this powerful method is based on pole placement and presents one deficiency: it needs to know a priori wherc to place ail the poles of the closed loop system. Following [1]: "the key issue is to choose the closed loop poles. This choice requires considera.ble insight . . . ". This can be done, for instance, through LQR design, but the problem is then replaced by the correct choice of the wcighting matrices in the cost fonctions.

In order to overcome the drawback of this design technique, it will be seen, in the following, that the use of a new method for system control, namely with a flatness point of view, enlightens on the choice of the high level parameters and brings physical meanings to obtain a clear guideline for polynomial pole placement design. Following [START_REF] Fliess | Flatness and defect of nonlinear systems: introductory theory and applications[END_REF][START_REF] Fliess | A Lie-Backlund approach to equiv alence and flatness of nonlinear system[END_REF], flatness is a very interesting property of processes to design a control, specially for trajectory planning and tracking for nonlinear systems.

The paper is organized as follows. Section 2 is devoted to survey very quickly, the design of polynomial controllers. Section 3 resumes the fiatness property and the control design implied for a fiat system. At the end of Section 3, a methodology for the control of fiat systems is proposed. The implication of method on finite dimensional linear systems is given in Section 4. This point of view leads to propose a flatness-based two-degrees-of-freedom controller which is realized in Section 5. Section 6 is devoted to the rejection of a static perturbation which can be seen as a complement to the previously designed control. In Section 7, an example of a RST controller applied to a second order linear system is presented.

In the following, for n EN, the following notations will be used, u < n ) (t) = d:�Yl = p"u(t), where p denotes the differential operator, and the paper will be developed, for the sake of shortness, for SISO linear systems, but all the results can be adapted to MIMO linear systems.

POLYNOMIAL CO TROLLERS

This section offers a short description of the design principles of the polynomial two degrees-of-freedom controllers for linear systems. More details are given in [START_REF] Astrom | Computer Controlled Systems, Theory and Design[END_REF][START_REF] Franklin | Digital Control of Dynamic Systems[END_REF][START_REF] Kucera | Analysis and Design of Discrete Linear Control Systems[END_REF] and the references therein, and in the following these controllers will be denoted as RST controllers [START_REF] Landau | Identification et Commande des Processus[END_REF],.

Consider the finite-dimensional SISO linear system described by the input-output model:

Ay = Bu, ( 1
)
where y and u are the output and• control signals, A is monic and A and B are coprime polynomials.

For (1), the RST (two-degrees-of-freedom) controller [START_REF] Astrom | Computer Controlled Systems, Theory and Design[END_REF] is given by:

Ru= -Sy +Tr, ( 2)
where r is the reference to track, and R, S and T are polynomials in the considered operator. These polynomials are given by the following rules: R and S are solutions of the Bezout equation:

P = AR+BS, (3)
where the roots of the polynomial P are constituted by the desired closed loop and observer poles, and S and R are monic; T is given by the desired closed loop transfer such that:

PB m =TBA m . ( 4
)
When all these conditions are fulfilled, the closed loop behavior is obtained:

(5 )

Sorne remar ks for the design:

(i) It has been used, for the choice of T, the point of view developed in [START_REF] Astrom | Computer Controlled Systems, Theory and Design[END_REF], where (B m , A m ) was a model-to-follow, but it can be also chosen the proposed one in [START_REF] Landau | Identification et Commande des Processus[END_REF], where r is given by: [START_REF] Fliess | Linéarisation par bouclage dyna mique et transformées de Lie-Backlund[END_REF] where (B m , A m ) defines a trajectory-to-follow or a trajectorj• generator of r(t). In this last point of view, T is designed such that:

TB=P. ( 7
)
(ii) For the implementation, the RST controller (2) must be written in the proper operator (p-1 ) which leads to write the RST control as: [START_REF] Fliess | A Lie-Backlund approach to equiv alence and flatness of nonlinear system[END_REF] with R*(O) = 1.

As the major point is to choose the desired poles, we will see that the flatness point of view can fruitfully help us.

SHORT SURVEY ON FLATNESS

The fiat property, which has been introduced recently (5, 6, 7) for continuous-time nonlinear systems, leads to interesting points of view for control design. In the fol lowing, a short review about flatness of systems and the application of this property to design a controller will be given. The interested reader may find more details in the quoted literature and the references therein.

A system described by:

x < 1 > = J(x, u), ( 9
)
where x is the state vector of dimension n, and u is the control signal, possesscs the fiatness property ( or is fiat) if there exist a variable z:

_ h( (1)
(a) )

Z-

x,u,u , ... ,u , where a EN, two fonctions A(-)and B(-), and an integer /3 such that:

x = A(z, ... , z <J1> ), u = B(z, ... 'z UH l ) ). (10 ) (11)
The selected output z is called a fiat output and, obviously, there is no uniqueness. But, as it has been observed on numerous examplcs, the fiat output has a simple and physical meaning. R.oughly speaking, the implications of flatness are of very importance in several ways for control. For motion planning, by imposing a desired trajectory on the fiat output, the necessary control to generate the trajectory, can be obtained explicitly (without any integration of the differential equations). The desired trajectory, zd(t), must be (/3 + 1)-times continuously differentiable. For feedback control which only ensures a good stabilization around the desired motion zd(t).

All these points, which have been formalized through the Lie-Backlund equiv alence of systems in [START_REF] Fliess | Linéarisation par bouclage dyna mique et transformées de Lie-Backlund[END_REF][START_REF] Fliess | A Lie-Backlund approach to equiv alence and flatness of nonlinear system[END_REF], lead to propose a nonlinear feedback which ensures a stabilized tracking of a desired motion for the fiat output. This methodology has been applied on many industrial processes as it has been shown previously, for in stance, on magnetics bearings [START_REF] Lévine | A nonlinear approach to the control of magnetic bearings[END_REF], chemical reactors [START_REF] Rothfuf3 | Flatness based control of a nonlinear chemical reactor[END_REF], cranes or flight control [START_REF] Lévine | Are there new industrial perspectives in the control of mechanic:al systems?[END_REF] or turning process [START_REF] Rotella | Flatness ba.sed control of a turning process[END_REF][START_REF] Rotella | Flatness approa.ch for the numerical control of a turning process[END_REF], among many other examples.

The main objective of the flatness based controller is to obtain the asymptotic tracking of a desired trajectory and this can be ensured through the following steps:

(i) Motion planning: it consists in the design of a trajectory defined by zd(t), which must be differentiable at the order ({3 + 1).

(ii) Motion tracking: by the control:

{3 v = zY J + 1 \t ) + L k i (Z à i )(t) -z ( i ) ( t )), (12) i=O
where the ki ensure that the polyn�mial K(p) = r/3+ 1 + Ef=o kip i is Hurwitz, the complete control is then as follows:

-• {3 u B(z, ... , z ( /3 ) , z<j3+I) (t) + L ki(Zài} (t) -z ( i } (t))) i=O <}(z, ... , z { /3), K(p) Zd( t )),
which leads to the asymptotic tracking of the desired trajectory.

(

) 13
otice, in the one hand, that the information needed by this control can be obtained through observers, and a major advantage of this controller with respect to other nonlinear strategies is that it overcomes the problems generated by non stable zeros dynamics [START_REF] Isi<lori | Nonlinear Coutrol Systems[END_REF][START_REF] Nijmeijer | Nonlinear Dynamical Control Systems[END_REF]. In the other hand, if the output of ( 9) is given by y = g(x, u) then from [START_REF] Franklin | Digital Control of Dynamic Systems[END_REF] it can be related to the fiat output by: y = f(z, ... , z C13 +1 > ). [START_REF] Kailath | Lincar Systems[END_REF] This relationship leads to a trajectory for the output deduced from the designed trajectory for the fiat output, namely, Yd = f(zd, ... , zf+ 1 >). If this trajectory is not admissible for the output, the key is to design a piecewise trajectory where some conditions for smoothness are verified on the cutting points, but keeping in mind that the relationship ( 14) is available between these points.

We will see in the next part that the design of the fiat output trajectory will be a guideline, in a linear framework, for the poles choice of a RST controller.

IMPLICATION FOR LINEAR SYSTEMS:

TOWARDS RST CO TROLLERS Despite the fact that flatness has been firstly developed for nonlinear systems, it has been applied to finite-dimensional linear systems [START_REF] Bitaud | A flatness based control synthesis of linear sys tems and application to windshield wipers[END_REF][START_REF] Fliess | Regulation of non minimum pha.se out puts: a fiatness approach[END_REF] and extended for infinite dimensional ones [START_REF] Fliess | Controllability and observability of linear delay systems: an algebraic approach[END_REF]. It will be seen, in this section, that applying the guideline induced by a flatness based control to a linear system leads to express it in a natural RSTform.

The previous methodology will be applied now to a linear lumped parameter SISO system defined by the transfer:

A(p ) y (t) = B(p) u(t), ( 15
)
where the notations have been previously defined but with:

A(p) =p n + H J2 a iP { =p n + A*(p), B(p) = J2 b iP \ (16) i=0 i=0
From coprimeness, it has been shown in [START_REF] Bitaud | A flatness based control synthesis of linear sys tems and application to windshield wipers[END_REF], [START_REF] Fliess | A Lie-Backlund approach to equiv alence and flatness of nonlinear system[END_REF], that this system is flat with a flat output defined by:

z(t) = N(p)y(t) + D(p)u(t), ( 17
)
where N(p) and D(p) are the polynomial solutions of the following Bezout equation:

N(p)B(p) + D(p)A(p) = l. ( 18
)
Due to coprimeness, existence of N(p) and D(p) are guaranteed and the minimum degree solution is, for n > 1, such that degiV = n -1 and degD = n -2.

The explicit expressions of the output y(t) and the control u(t) are given by:

u(t) = A(p) z(t), y(t) = B(p) z(t), (19)
which allows to relate the flat output of a linear system to the partial state defined by [START_REF] Kailath | Lincar Systems[END_REF]. Following the step (ii) of the methodology, the control is given by:

u(t) = v(t) + A*(p)z(t), (20)
where:

v(t) = ->>(.) + £ *,(*«(«) -*<0( t )). ( 21
)
t= 0

and by introducing the polynomials:

n -l K(p) = P n + £ fa* =p n + K*(p), ( 22
)
i=0
the control u(t) is given by:

u(t) = K(p) z d (t) + [A*(p) -K*(p)} z(t). ( 23
)
Taking into account that z(t) = N(p) y(t) + D(p) u(t), then it can be written:

u(t) = K(p) z d (t) + [A*(p) -K*(p)} [N(p) y(t) + D(p) u(t)}, ( 24
)
which leads to:

[1 -[A*(p) -K*(p)} D(p)} u(t) = K(p) z d (t) + [A*(p) -K*(p)} N(p) y(t). (25)

This appears as a RST controller form with:

R(p) S( p )

1 -[A*(p) -K* (p) ] D(p), -[ A • ( p ) -K* ( p ) ] N ( p ), ( 26
) ( 2 7)
with the différence that here the trajectory to follow is directly integrated to the controller with the term K(p) zd(t). An important property of this controller can be also deduced, due to the fact that P =AR+ BS. From the previous definitions of

R(p) and S(p), and with the help of N (p) B(p) + D(p) A(p) = 1, and A*(p)-K*(p)

= A(p) -K(p), it folJows that:

A (p) R(p ) + B(p)S(p) = K ( p ).

(28) From (28), it is then obtained that the closed loop poles for. the proposed RST controller are those dcsigned for the tracking of the desired fiat output trajectory. The choice of these poles is then enlightened. But as:

d e g (1 -[A* -K*] D) = de g ([A* -K*] N ) -1, (29)
it is not realizable. The realization of this controller will be the subject of the next part.

REALIZATION

To implement the control [START_REF] Rotella | Flatness approa.ch for the numerical control of a turning process[END_REF], it can be used an observer of the vector Z =

[ z(t) . . . z(n-1 >(t) ]

T which is the state vector of the controllable Luenberger realization of u(t) = A(p) z(t), y(t) = B( p ) z(t), namely: where:

A= z< 1 > = AZ + Bu, y =CZ, 1 0 ,B= 1 0 -On-1 1 C = [ bo b1 • • • bn-1 ] .
A full-order observer of Z is given by:

z(l) = (A-rC).Î +Bu+ ry, ( 30
) (31) (32)
where ris chosen such that the eigenvalues of F = A -rc are with negative real part. This leads to: z = (pl -F ) -1 ( Bu + ry ) .

(33)

By introducing a= [ ao ai • • • On-1 ] and k = [ ko k1 • • • kn-1 ] ,
the control (23) is implemented by:

u(t) = K(p) zd(t) + (a -k)Z(t), ( 34
)
as in [START_REF] Fliess | Regulation of non minimum pha.se out puts: a fiatness approach[END_REF] . But, in this solution the difficulty is the choice of the observer poles. To overcome this point the enlightening ideas suggested in [START_REF] Fliess | Sur des pensers nouveaux faisons des vers anciens[END_REF] and applied in [START_REF] Marque | [END_REF] can be used. In the one hand, from [START_REF] Kailath | Lincar Systems[END_REF] :

y= o(A,c)Z + M(A,B,c)U, where Y = [ y y(l ) y (n-l) ] T , U = [ u u<1) O ( A,C
) is the observability matrix:

and M(A , B , c) is given by:

0 CB M(A,D,C) = CAB C CA CB 0 0 CA 0 -2 B CAB CB (35) (36)
(37)

From this equation , and due to the fact that A(p) and B(p) are coprime, thus rankO ( A , C ) = n, it becomes:

z = O ëi , c) {Y -M ( A , B , c ) U } . ( 38
)
As the first component of Z is z(t), it can be seen that the first line gives the fiat output expressed in terms of the derivatives of y (t) and u ( t ) . Namely , a solution of the Bezout identity ( 18 )

is obtained , with h = [ 1 0 • • • 0 ] : N(p) = hO cJ , c ) d iag { 1 ,p, ... ,P n -l } , ( 39
) [ p -í " " 1 ) p -( " -2 ) ••• p " 1 1 ] T , (48) p -í " -1 ) p-(«-2) . . . p -i
By denoting R* (p-1 ) = 1 + Q* (p-1 ), this control can be written in the RST form: A second remark can be done here. Namely, it follows also:

where h is previously defined. Thus:

z(t) y An-1 0 -1 M - (A,C) (A,B,C) n-1 + LA i -ln p -iu i=l p-(n-1) 1.t p-(n-2) u p-l u N*(p-1 )y(t) +D*(p-1 )u(t),
which defines the fiat output in terms of the proper operator p-1 .

DISTURBANCE REJECTION

In order to reject a static perturbation, an integral action must be added to the model. The proposed methodology is then applied to the following augmented model:

.z(l) = ÀZ + Ëü, y=éz,

(

where Z = ( z . . . z ( n ) ) T and ü(t) = pu(t). The matrices À, iJ and ë are given by: À= 0 1

0 1 0 -ao -an-1 bn-1 0 ] . B= 0 0 1 (56)
The correspondiog RST controller is obtained from equations ( 4 7) and ( 46). How ever, as the degree of the polynomial S* becomes in this case n -1 and the degrec of R* is n, it then follows a simplification of the operator p with the integrator. This will have as a consequence that the controller will not be able to reject a constant disturbance added to the output of the model. To overcome the simplification problem which appears in the case of the aug mented model, it is proposed to replace in the equation ( 44) the operator p by a. new operator p + a which is denoted 1r (a> 0). By applying the same methodology as before, the following s: and Q ; expressions are obtained: where R; ( 1r-1 ) = 1 + Q;(1r-1 ). The a parameter is chosen with respect to the bandwidth of the considered model.

This strategy is illustrated in the next section on a second order continuous-timc model.

EXAMPLE

Let be considered the system defined by A(p) = p 2 + p + 1 and B(p) = p + 1, then the Bezout equation ( 18) admits the least degree solution:

N(p) = -p, D (p) = 1 ,
which leads to the fiat output z (t) = u(t) -py(t). The control is then:

u(t) = K(p) zd(t ) + [( 1 -k1)P + 1 -ko] z(t ), (60) (61)
where K(p) = p 2 +k 1 p+k 0 defines the error dynamics. From u(t) = (p 2 + p + 1) z(t) and y(t) = (p + 1) z(t), we have:

Z(t) = [ � l !1 ] Z(t) + [ � ] u(t), y( t) = [ 1 1 ] Z( t). ( 62
)
The chosen closed loop poles are -4.2 ± 4.28j and lea.d to the following RST controller in the realizable form (50):

n•( p -1) s•(p-1) K( p )

In order to cope with a static perturbation, added to the system output, an integral action must be integrated into the modcl and one obtains the augmented model:

( ) p + 1 -(t) yt =-----u p(p 2 + P+ 1 ) (64)
where ü(t) = pu(t) and .A(p) = pA(p). The RST controller corresponding to this mo<lel is given by: fl

where the chosen closed loop poles are: -4.2 ± 4.28i, -15. The trajectory to follow by the output is Yd(t) = 1. To achievc this purpose, the desired fiat output must have the following expression:

zd(t) = 1 --yexp(-t), (66)
which satisfies the derivability condition for any constant -y. It can be noted here that the obtained controller does not reject the a.dded constant perturbation as shown in Figure 1 (for 'Y= 1).

By applying the same methodology replacing the operator p by the operator 1r as proposed previously, the state representation becomes:

1r i(t) = A a i(t) + Ëü(t), y(t) = ë i(t), ( 67
)
where Z(t) = ( z(t) z( 1 >(t) z( 2 >(t) ) The matrices O a and Ma are given by: Da = ( iz4-) , Ma = ( -�Ë _ 0_ �B-) .

CA! CAaB

Also notice that the following relation remains valid:

À(p-1).n:c1r-1) + n(p-1 )s;c1r-1 ) It can be noted that the rejection dynamics depend on the choice of the a pa rameter. A good disturbance rejection has been obtained for a= 1. The numerical expressions for the polynomials R: and s; are: R;(1r-1 ) = 1-22.41r-1 , s:c1r-1 ) = 161 + 356.61r-1 + 22.41r-2 , (73) which leads to a realizable RST which does not simplify the integral action.

= 1 + k2p-2 + k1p-1 + kop-2 = p-3 K(p).
The simulation results are displayed in Figure 2.

CONCLUSION

This paper showed that the use of a flatness point of vicw allows a simplification in the design of high levcl parameters of RST controllcrs. The main feature of the fiatness approach for RST controller design is to avoid the problem of the closed loop poles choice which are constituted of the observer poles and those obtained with a state feedback [START_REF] Astrom | Computer Controlled Systems, Theory and Design[END_REF]. Now the design is focused in the choice of the trajectory Zd to follow and the tracking dynamics with K(p).

In the case where a constant output perturbation, for instance, is to be rejected, an integral action must be added in R. This can be achieved by forcing the presence of an integrator in the open loop transfer fonction as shown in the equation (64).

Then the proposed method can be applied aga.in using a. new operator 1r = p + a instead of the usual operator p.

These developments were done in a continuous-time framework, but are trans posable for discrete-time systems [START_REF] Rotella | Digital flatness-based robust controller applied to a thermal process[END_REF]. ln this case, the robustness of the proposed digital controller based on flatness, by introducing fixed polynomials HR and Hs as proposed in [START_REF] Landau | Adaptive Contrai[END_REF], were treated for the fiat discrete-time systems.

1 )

 1 , from (5 0) and (19), we get: R* ( p-A(p) z(t) = K(p) zd(t) -S* ( p-1 ) B(p ) z(t ) .

  some manipulations, we deduce R*(p-1 ) A(p)+S*(p-1 ) B(p) = K(p), and if we notice that A(p) and K(p) are of the same degree, the expression A(p-1 ) R* (p-1 ) + B(p-1 )S*(p-1 ), can be written as: (52) which gives the rclationship of the poles of the RST controllcr with the tracking dynamics.

  wherc  a represcnts the matrix À + al n and fI = [ 1r-n 1r-1 1 ] T andft• = [ 1r-n • • • 1r-1 ] T . The contrai ü(t) is then obtained from:

TFig. 1 .

 1 Fig. 1. Tracking in the case of a = O.

Fig. 2 .

 2 Fig. 2. Tracking in the case of a= 1.

(

  Received March 14, 2002.)