"From A300 to A350: technical and organisational innovation trajectory of Airbus"

Keywords: Airbus, industrial organization, innovation, modularization, outsourcing, pivot-firm

The aim of this paper is to show that Airbus's success can be attributed to two types of factors. First, each new aircraft model has produced a technological breakthrough in the design and manufacturing of aircraft. Secondly, Airbus's capacity to evolve their industrial organisation model in keeping with technological transformations. In this trajectory, modularization and outsourcing policies have played major roles. In particular they play a major role in the emergence of new actors the pivot-firms. These firms have a critical position in management of technical and organizational interfaces between the architectintegrator and the firms participating in the design and production of aircrafts.

Introduction

The factors which explain the Airbus's commercial success are various and diverse. Some of authors attributes Airbus's competitive advantage to the quality of organization and management of the supply chain. For Rose-Andersson et al. (2009), the evolution of the different forms of organisation in aeronautical supply chain is an efficient response to the evolution of markets and competitors. Similarly, [START_REF] Nolan Peter | The Challenge of Globalization for Large Chinese Firms[END_REF] argue that modern aircraft have become so complex that the competitive advantage resides in the capacity of aircraft manufacturers to coordinate the supply chain and to integrate the various components of the aircraft. According to [START_REF] Cagliano Raffaela | Lean, Agile and Traditional Supply: How Do They Impact Manufacturing Performance?[END_REF] or [START_REF] Oliver | Conceptualising aerospace outsourcing: Airbus UK and the lean supply approach[END_REF], it is in fact the rationalization of the production process as a whole that accounts for Airbus's performance. For other authors (Rose-Andersson et al. 2009), an evolutionary approach shows that it is the Risk-Sharing Partnerships that are at the origin of the creativity and innovation involved in the aircraft production process. Acha, Stefano and Prencipe (2007) support that learning and forms of dissemination of knowledge are essential in the success of aeronautical programs. In order to underline Airbus's success market, [START_REF] Campos | On the Competition between Airbus and Boeing[END_REF] highlights the family strategy through A320 and A330 ranges, while M. Givon and Rietveld (2009) show the comparative advantage of having a fleet of modular aircraft.

The point in common between all these explanations of Airbus's competitive advantage is the leading role they attribute to the dynamics of innovation in a complex product industry. Their other similarity is that they mostly look at what is occurring « outside » the firm, and they do not relate technical innovation to organisational innovation.

In this paper we will take a look at the technical and organizational dynamics that are behind Airbus' industrial strategies. The adopted point of view is : the embracing of a status of architect-integrator of aeronautics systems (and not as an aeronautics manufacturer), the increase in the modular decomposition of the airplane building process, the emergence of pivot-firms as well as the innovationsin particular those related to the composite materials, the integrated modular avionics and the embedded systemsmake a profound reorganization of industrial model of the European aircraft manufacturer. The purpose of the article is not to propose an explanatory model with all the factors.

The aim of this paper is to focus one two types of factors. First, each new aircraft model has produced a technological breakthrough in the design and manufacturing of aircraft. Secondly, Airbus's capacity to evolve their industrial organisation model in keeping with the technological transformations. Without indulging in excessive technological determinism, it can be said that the successive adoption of technologies has generated profound modifications in the industrial organisation of the aircraft manufacturer. They have also completely reconfigured the relationships between the different participants in the aircraft manufacturing process. More precisely, the positive relationship between technological choices and organisational forms is concretely observable through Airbus's programmes. Airbus's technological expansion can be construed as, for each new programme, a break with the old technological paradigm as well as the adoption of a new organisation of production. According to us, it is this technological and organisational dynamic that contributes substantially to the commercial success of Airbus.

Two factors seem to us to be crucial in this organisational model and its evolution: firstly, high modularisation and its corollary, high integration of aeronautical systems, and secondly, the politics of refocusing and outsourcing. These characteristics are admittedly shared by other manufacturers, but, according to us, they are in more pronounced in Airbus's case. Today, the reinforcement of these mutations announces the emergence of a new industrial organisational model: "aircraft manufacturers" are progressively becoming "architect-integrators" of aeronautical systems [START_REF] Cagli Ayse | Gestion stratégique de la supply chain et firme-pivot dans le secteur aéronautique[END_REF]. It is this historical, technical and organisational trajectory which we will analyse.

The main idea underlying this analysis is that there are positive correlations between technology (and innovation) and the organisational forms that produce complex products. More generally, our basic hypothesis is that technological changes require organisational set-ups that are compatible with the nature of these changes. Therefore, we feel it is relevant to approach these issues from the point of view of the industrial model and inter-firm relations (Zuliani, 2016).

There will be three stages to our analysis. First, we will present the spatial distribution model of Airbus's activities between the different actors (countries) that participate in aircraft manufacturing. We will note that, from its origin, the organisational model is widely spatially distributed between different production sites and different players. We will also see that the division of labour between the different partners (German, French, and British) has changed very little over time. Then, we will analyse the technological trajectory of the different programmes. We will see that the technological breakthroughs successively introduced by each new aircraft have consequences on the organisational model of the firm. Finally, we will look at the modularisation and the integration of systems. We will see that a reinforcement of these two characteristics of aircraft manufacturing leads to a pronounced redistribution of the roles of the firms involved in the aircraft manufacturing process. This redistribution manifests itself by the emergence and development of pivot firms.

2

Airbus's European spatial organisation activities.

Several theoretical and empirical frameworks account for the clustering and anchoring factors in aeronautics (Agrawel and Cockburn 2003;[START_REF] Cooke Phil | Proximity and Procurement: A study of Agglomeration in the Welsh Aerospace Industry[END_REF][START_REF] Niosi Jorge | Anchor tenants and regional innovation systems: the aircraft industry[END_REF]Kechidi and Talbot 2010). In the case of Airbus, the industrial organization is based on a division of labour at national, European and international levels founded on the main abilities of each production site [START_REF] Kechidi Med | Coordination inter-entreprises et relations de sous-traitance: le cas d'Aérospatiale[END_REF][START_REF] Talbot Damien | Institutional dynamics and localized inter-firm relations: the case of Aérospatiale and its subcontractors in Toulouse[END_REF]. The specialization of the sites has not been significantly modified since creation of the consortium in 1970. The production cycle of an Airbus is done within the boundaries of a quadruple division of labour:

 A European division of labour between the four national companies that are involved in EADS, United Kingdom, Spain, Germany and France. There are in Europe 16 sites of production which participate directly at division of work.

 A national distribution of labour orchestrated by the national company. In the case of Airbus France, production is divided among four production sites (Toulouse, Nantes, Méaulte, Saint-Nazaire).

 A dispatching of tasks among the different plants within the same site. For instance, for the Toulouse site, tasks are divided among the four plants of the site (Blagnac, Colomiers, Saint-Martin du Touch and Saint-Eloi).

 A division of labour between Airbus and a vast network of subcontracting firms located in France and abroad.

For distribution of workload on a European level, the most common organization is to build the wings in the United Kingdom, the tail units in Spain, the fuselage in Germany and the cockpit and the central section of the fuselage in France. The final assembly is done either in Toulouse or in Hamburg. The specialization per site didn't change since the beginning of Airbus project.

Puerto Real Gouvernail

Gosselies (Belgique) Bec de voilure

Stade

Dérives verticales

Nevertheless, that specialization is not perfectly transverse to all the programs. Indeed, depending on the model of the plane, the division of labour may vary. Thus, the wings of the A300, A310, A330 and A340 that are made in Broughton are first routed to Bremen in order to add equipment before being redirected to Toulouse where they are assembled, while the wings of the A380 do not transit through Germany. The A320s are assembled in Toulouse while the other planes of that family (the A318, A319 and A321) are assembled in Hamburg. This duplication of the assembly sites reflects the regular resurgence of the confrontations between the European industrial partners, the German partner wanting to take care of the very symbolic act of doing the final assembly of some Airbus aircraft, for national prestige reasons but also because the teams based in Hamburg master the technical and organizational skills needed for such tasks.

Like a giant puzzle that needs to be put together, the assembly of an Airbus generates numerous inter-sites flows. Even if they introduce a more functional organization and a stronger horizontal integration, the current reorganizations do not fundamentally question the current distribution of industrial tasks, at least on a midterm basis. The expected configuration over the next few years is most probably the end of the industrial duplications with a specialization of the Toulouse and Hamburg assembly sites depending on the planes: in Toulouse the long haul aircraft and jumbo jets (A330-340, A380 and A350) and in Hamburg the A320 and its future successor.

However, the impact is very noticeable when it comes to the nature, the density and the content of the relationships between Airbus and the firms that participate directly or indirectly to the construction of the aircraft. That impact seems to generate the industrial and organizational conditions necessary to create a new organizational model of aeronautical activities. Before underlying its main characteristics, let's take a look at the evolution of the main programs, since it seems to us that the trajectory of innovation and of the industrial model is linked with the technical and organizational trajectories. These trajectories illustrate the industrial evolution of Airbus and the affirmation of market logic. This market logic replaced, during the 80s, an arsenal logic in which public intervention was very strong [START_REF] Muller Pierre | Logique d'Etat, logique de marché[END_REF]. While an aircraft used to be considered successful it is was technically impressive, a good aircraft is now an aircraft that sells. From a symbolic standpoint, we have gone from a technical success but a commercial failure (Concorde) to a commercial and technical success (Airbus).

3

The innovation paradigm: "all up front", "all electric", "all composite"

In order to understand what is happening nowadays in this sector, it is necessary to take into account the background of technological innovations that have marked the evolution of the Airbus programs. At every stage, these innovations have constituted a paradigm break in the sense that they introduced innovations and new product architectures. Without indulging in excessive technological determinism, it can be said that the successive adoption of technologies has generated profound modifications in the industrial organisation of the aircraft manufacturer but also in its relationships with the subcontracting firms. If the increase in complexity in aeronautics is nowadays mainly observed in avionics and embedded systems, it can also be found in the whole design and production of the aircraft from the beginning of the first programs.

The A300 was the first long-range twin-engine jet with a large fuselage. It also unveiled the "all up front" control cabin. This innovation enabled two-man aircraft handling and introduced a new approach in the design of the product. The A300 started the "a good aircraft is an aircraft that sells" era. In terms of the outsourcing process, during this phase Airbus manufactured most of the aircraft components internally. It was a classical subcontracting situation where firms functioned as external workshops supplying components according to the detailed design provided by the manufacturer.

In 1984, the introduction on the A320 of fly-by-wire control as well as a new cockpit design generated an authentic revolution in that area. Electrical control and auto flight unveiled the era of the mass arrival of electronics and embedded systems. In this case as well, this had an effect on Airbus's industrial organization. The whole architecture of the airplane was partially revisited. The mastery of electrical systems became a fundamental specific asset.

In 1987, the A318/A319/A320/A321 and A330/A340 families introduced a quasi-standardization of the cockpit equipment. Aircraft belonging to the same family came to have the same instrument panel, the same attitude control procedures, the same avionics and almost the same systems. These similar configurations allowed the same crew to fly all the aircraft belonging to one family. Maintenance and operating costs were also reduced.

The technological breakthroughs made with the A380 were notably innovations linked with the size of the airplane and Integrated Modular Avionics (IMA). The size and the weight of the airplane have required the development of electric and hydraulic sub-systems for transporting energy that do not generate large weights nor important head pressure losses. The A380 sped up the era of the "all electronic airplane". This concept replaced the hydraulic central systems that are linked though complex circuits by electro-hydraulic systems that are dedicated to each piece of equipment. This innovation enabled important weight gains and a reduction of the production and maintenance costs.

The other innovation introduced in the A380 was the integrated modular avionics. Until the A380 program, the avionics systems were made of a range of linked numerical devices (calculators), each dedicated to only one function. Because this meant a multiplication of devices, it generated massive loads and costs. The IMA adopted by the A380 (and by the B 777 and 787) meant abandoning that principle of a dedicated resource in order to use one architecture for different applications. This electronic organization favoured a large upgradability of the software functionalities since it allowed for elements to be changed without needing to change the computer science architecture of the aircraft.

If there was a major change with the introduction of more electric technologies in aircraft, "the integration of systems and the standardization of avionics could represent a much bigger change, which would allow a better management of the fleets within the companies and a better valuation of them"2 .

The benefits of this kind of technology are clear. These are typically technologies that improve the performances of the aircraft while reducing the manufacturing and maintenance costs as well as the weight of the planes. From an industrial standpoint, going from the "one calculator for each function" logic to a "one calculator for various functions" logic reduces the number of stakeholders in the manufacturing of the electronic modules and allows the delegation of the coordination function to a sole actor, a pivot-firm, that will have to organize the subcontracting tasks.

Although the A350 adopts technologies that have already been tested on previous programs, among others the A380 and the A400M, it pushed further the "all electric" and "all composite" concepts. The percentage of composite materials thus went up to 52% compared to 38% on the aborted version of the A350, while the amount of aluminium and lithium stayed at 20%. This evolution toward composite materials was another vehicle of the mutations that affected the aircraft manufacturer and their partners. The presence of composite materials in the Airbus programs has not ceased to increase since the A300. Thus, while the level of composite equipment was of 5% on the A300-600, it reached 25% on the A380. This brief "technical history" of the successive Airbus programs clearly shows that the deployment of each new aircraft model was accompanied with important technical innovations. This seems to illustrate Lawrence and Thornton's (2005) point that the success of an aircraft depends crucially on its novel characteristics in comparison with competitor's models.

Clearly, the "New Airbus" announced by Power 8 Plan in 2007 and Power 8+ Plan in 2008 aims to refocus the activities on the core competencies of the firm and reconfigure all relations with supply chain partnerships . These competencies are at the heart of the profession of aircraft manufacturer and include the design of the global architecture of the aircraft and the cabin, the integration of systems, the assembly of components and the customisation and testing of equipment. Other activities, that involved technology that is completely mastered or common, are outsourced. By strengthening its status as an architect-assembler, Airbus positions itself upward and downstream of the value chain. All the activities that are considered strategic from the technological, industrial and market standpoints are undertaken internally. This trend translates into an important increase in outsourcing non-strategic activities. This outsourcing is thus based on the creation around the firm of a stable network of partners with complementary activities in which lasting relationships are contracted and partnership links are created with the objective of improving performance while sharing risk.

The observed trend, particularly in the last few years, shows an increasing focus on Airbus's core business and an emphasis on outsourcing policy3 . More than this, it demonstrates a real transformation in the status of aircraft manufacturers. Airbus and Boeing and other aircraft manufacturers radically changed their way of building and developing new aircraft [START_REF] Amesse Fernand | Issues on partnering: evidences from subcontracting in aeronautics[END_REF][START_REF] Destefani | A look at the Boeing's outsourcing strategy[END_REF][START_REF] Kechidi Med | Les relations verticales dans l'industrie aéronautique : le cas de la sous-traitance d'Airbus France[END_REF][START_REF] Cagli Ayse | Gestion stratégique de la supply chain et firme-pivot dans le secteur aéronautique[END_REF]). In the past, aircraft manufacturers designed, conceived and built their planes mainly internally. Nowadays, they increasingly defer the design and production of whole sections to partners. Therefore, they go from being an "aeronautics manufacturer" to being an "architect-integrator of aeronautics systems". In the context of this evolution, the modularization of aeronautical systems has played a central role.

4

Modularity: technical, organisational and cognitive dimensions

The literature on modularity is based on H. Simon's theories on the decomposability of complex systems (1962). This conception of modularity, called modularity of the product or technical modularity [START_REF] Ulrich | The role of product architecture in the manufacturing firm[END_REF]Cohendet et al. 2005) is firstly a tool aimed at reducing complexity. "Modular architecture proves to greatly reduce the complexity of the complex systems by proposing a decomposition into autonomous subsystems, as it is possible to develop and pre-assemble them separately, linked on to another by relatively stable interfaces" (Frigant 2005, p.5).

Without limiting it to this aspect, product decomposability is a strategy to reduce the production complexity and rationalization. A modular object is then a complex product composed of subassemblies, produced independently of one another but that can be linked together to form a coherent system, stabilized by standard interfaces.

Through decomposability, the modularization of complex products enables to reduce the complexity of technical objects. It also enables to rationalize the organization of production processes and the hierarchical structure, as recalled by H. Simon through his parable about watchmakers named Tempus and Hora . were two watchmakers renowned for the quality of the watches they would design. They had such a good reputation that they were often interrupted during their work by clients who wanted to place an order. The more famous they were becoming, the more calls they would receive in their respective work-shop. However, Hora's activity prospered while Tempus went bankrupt. The explanation of these two fates is the following: each watch designed by both matchmakers involved 1000 components. Hora designed his watches in a way that "he would combine ten elementary components into small subassemblies, and then he would combine ten subassemblies into larger subassemblies, and these in turn could be combined to make a complete watch" (Simon 1962, p. 470). By contrast, Tempus did not decompose into stable and homogenous subassemblies. Every time the phone would ring, he would abandon his current assembly which would immediately fall apart. After each interruption, Tempus would start again the entire design process. Considering a reasonable interruption probability for both men, H. Simon shows that in the end, it takes 4000 time more time to Tempus than Hora to finish assembling a watch (Simon 1962, p. 470).

The perspective developed by Sanchez and Mahoney (1996) is rather different; it is indeed not focused on the product. They developed the idea according to which there are positive correlations between the evolution of complex products development processes and the organizational forms that create them. The modularity-product perspective is coupled with an organizational perspective in which "modularity is presented like a specific structure in terms of coordination and labour division that particularly aims at minimizing transaction costs" (Cohendet et al. 2005, p. 122).

the works of Langlois ( 2002), Baldwin and Clark (2000) and Brusoni et al. ( 2001), both perspectives of technical and organizational modularity are completed by a strong cognitive dimension. This dimension means that the knowledge underlying the products (design, production, and assembly) falls within the modules' combination and assembly. In other words, modularization is also a modularization of bodies of knowledge that give rise to the products.

In fact, Brusoni and Prencipe (2001, p.184) questioned this productorganization-knowledge trilogy when they recalled that the literature on modularity was based on three basic premises: "(i) there exists a positive correlation between product, organizational and knowledge modularity ; (ii) modular product architectures enable increasing specialization, both within and across companies ; and (iii) modular product architectures allow for coordination to be achieved with minimum managerial effort".

From this point of view, modularity represents the end of integrated firms and the emergence of particular organizational forms; which come within the theories of inter-organizational links, such as the "producers networks" (Langlois and Robertson, 1992), "modular networks of production" (Sturgeon, 2002), "loosely coupled networks" [START_REF] Brusoni Stéfano | Unpacking the Black Box of Modularity: Technologies, Products and Organizations[END_REF]. Also, modularity becomes a management strategy for the supply chain and the inter-firms relationships. 5 Modularity and system integration: the emergence of pivot-firms Modularization is not a new phenomenon in the aeronautics industry [START_REF] Araujo Luis | Managing interfaces with suppliers[END_REF][START_REF] Amesse Fernand | Issues on partnering: evidences from subcontracting in aeronautics[END_REF][START_REF] Virginia | Exploring The Miracle: Strategy And Management Of The Knowledge Base In The Aeronautics Industry[END_REF]. Recent changes announce a deepening of this type of organization as well as a strong redistribution of the roles of the firms involved in the aircraft production process.

A modular structure involves a product architecture decomposable into modules connected to each other by more or less standardized interfaces (Baldwin et Clark 2000). The architect firm designs the general architecture of the product and manages the interfaces between the different modules during the integration of these modules [START_REF] Ulrich | The role of product architecture in the manufacturing firm[END_REF][START_REF] Bresnahan Timothy | Technological Competition and the Structure of the Computing Industry[END_REF][START_REF] Vincent | Vanishing Hand versus Systems Integrators. Une revue de la littérature sur l'impact organisationnel de la modularité[END_REF]. The architect-integrator therefore occupies a strategic position all along the value chain, but mostly intervenes in the upstream and downstream stages.

Furthermore, a modular product architecture facilitates incremental innovation, either by adding functionalities to product components, or by transforming one of these components. A modular product can therefore regularly respond to modifications in demand (R. Langlois and P. Robertson (1992)). Similarly, it speeds up the development of variations on a same design. This is typically illustrated by the A320, 321,319 and 318 families of aircraft.

Modularization is based on the technical and cognitive division of the production processes [START_REF] Ulrich | The role of product architecture in the manufacturing firm[END_REF][START_REF] Baldwin Carlisse | Design rules: The power of modularity[END_REF]. From then on, the processes that generate the products are such that the organization of the firms active in those processes must be rethought [START_REF] Langlois Richard | The vanishing hand: the changing dynamics of industrial capitalism[END_REF][START_REF] Vincent | Vanishing Hand versus Systems Integrators. Une revue de la littérature sur l'impact organisationnel de la modularité[END_REF]. In a modular organization, the architect delegates the design and the production of modules and components of the end product to specialized firms [START_REF] Ulrich | The role of product architecture in the manufacturing firm[END_REF]Sanchez and Mahoney 1996;Brusoni and Principe 2001;[START_REF] Ulrich Karl | Product Design and Development[END_REF]. Its role as a supervisor is to control the manufacturing of sub-units and to ensure the compatibility of the interfaces between modules.

The increasingly modular organization at Airbus mainly involves delegating to specialized firmspivot-firms or "Hub firms" (Jarillo 1988)increasingly important components, for instance the whole aero structure, embedded systems, the landing gear. The bigger the size of the units and sub-units, the lesser the coordination duty of the architect, as the number of interfaces to control decreases along with the number of modules that need to be assembled. The economic advantage of modularization is also clear in terms of the cost decrease through the reduction in the number of direct partners. The pivot-firm can then play the role of architect firm for the units that it is in charge of. It articulates the technical and organizational competencies of the other participants (Kechidi 2008[START_REF] Cagli Ayse | Gestion stratégique de la supply chain et firme-pivot dans le secteur aéronautique[END_REF][START_REF] Gilly Jean-Pierre | Firms Hub and the Dynamics of Regional Innovation: Case Studies of Thales and Liebherr in Toulouse[END_REF]. Technical dimension it's a modular decomposition. Complexity indicates interrelationship, interaction and inter-connectivity of elements within a system and between a system and its environment (Mitleton-Kelly 2003). A complex system can be described as "one made up of a large number of parts that interact in a non simple way" (Simon 1962, p.195). In such system, the performance of the whole is threatened even if a single part does not function properly. Consequently, designing complex systems turns out to be complicated. As a solution, [START_REF] Simon Herbert | The architecture Of Complexity[END_REF] suggests reducing the number of distinct elements by dividing the whole system into subsystems which in turn are broken down into smaller components and so on. If this decomposition assures one-to-one mapping between components and functions, it enables components decoupling and interface standardization, and thus is considered as "modular" [START_REF] Ulrich | The role of product architecture in the manufacturing firm[END_REF]. Even though the complex products are more and more modularized, which should allow, by definition, a decoupled organization to develop these products ; we witness on the contrary more and more collaborative relationships between the system integrators and the suppliers (Howard & Squire 2007). This seemingly contradictory situation can be explained by the fact that firms pursue different strategies in regards to modularizing their products on the one hand, and outsourcing parts of these products on the other hand. More precisely, while a firm can decide to modularize its product by pursuing marketing, production, financial or technological strategies, only the two latter can lead to outsourcing practices. In this regard, modularization of complex products is primarily driven by concerns about production efficiency whereas the reason why modules are outsourced is rather related to financial and/or technological concerns. In case of complex product development process, problems have to be resolved in a top-down principle (Ulrich & Eppinger 2003). The system-level design phase during which the complex product is decomposed into subsystems and these into many components becomes critical. The architecture of the entire product is defined during this phase. Here we refer to "synthesis" activities which require a thorough understanding of higher-level problems about the product and the relevant development process (Brusoni 2005). Hence, the underlying knowledge base is considerably wide including the whole array of technological problems involved in the product development. Once the interfaces across the subsystems are explicitly described and the functionality requirements concerning each subsystem are identified, the detail design of components can begin. The detail design phase corresponds to the development of internal parameters which should meet the physical and functional interfaces described in the integral design phase. It covers "analysis" activities where engineers explore specific problems which are framed during the synthesis phase (Brusoni 2005). The knowledge base they use is rather specialized in a specific matter. At the end of this phase, detailed production drawings for each subsystem are tested and frozen. Subsystems can be manufactured according to the drawings describing every detail about the materials, techniques and tools to be used. Thus, the knowledge transferred to the manufacturing phase is considerably explicit and specialized for each subsystem and component. The manufacturing phase is followed by an integration phase during which manufactured subsystems and components are assembled into the final product with respect to the product architecture defined in the integral design phase. In sum, complex product development process implies evolution of knowledge bases created and integrated along the phases. The relevant knowledge for each phase requires different level of interactions between the participants.

The relationship between the architect firm and the network of firms it structures depends of the nature of the organisational interface [START_REF] Araujo Luis | Managing interfaces with suppliers[END_REF][START_REF] Nellore Rajesh | The impact of supplier visions on product development[END_REF][START_REF] Virginia | Exploring The Miracle: Strategy And Management Of The Knowledge Base In The Aeronautics Industry[END_REF]. T Araujo, Dubois and Gadde (1999) distinguish four kinds of interfaces: [START_REF] Araujo Luis | Managing interfaces with suppliers[END_REF] propose a categorization of "supplier interfaces" on the basis of the extent of resources shared between the prime contractors and their suppliers. Here, we take up and analyze this categorization in relation to the knowledge bases shared and integrated during the complex product development process. Since the breadth and depth of knowledge bases vary depending on the phases of product development process (Ulrich & Eppinger 2003;Brusoni 2005), the extent of knowledge shared and integrated by firms varies also depending on the phase the supplier is active in [START_REF] Parker | Timing and extent of supplier integration in new product development: a contingency approach[END_REF]Wynstra et al. 2010). That's why, we propose to distinguish the four types of supplier interfaces [START_REF] Araujo Luis | Managing interfaces with suppliers[END_REF], on the basis of timing of suppliers' involvement:

-Interactive interfaces are to be established when the supplier is involved during the integral design phase. By sharing mutually their knowledge bases, the prime contractor and the supplier discuss different technical problems in order to develop the best solutions about forms, materials, etc. The relevant knowledge base is characterized by broad higher-level problems, and includes mostly system-dependent and complex knowledge. The relevant suppliers refer to "zero-level suppliers" which interact extremely tightly with integrators and thus are difficult to be separated from the integrators on productive level.  When the supplier gets involved in the detailed development phase of a component, translation interfaces are established. More precisely, the prime contractor determines the critical specifications about the functionality of components and transfers them to suppliers. The latter then develops the component on its own in order to meet these specifications. Hence it translates the functional description of the component into a design and manufacturing context. The extent of knowledge sharing is not as large as within the interactive interfaces. Moreover, the knowledge base involved within this phase is less system-dependant than the one needed by the integral design activities. However it is still not particularly explicit, which makes the interactions quasitight.  Specified interfaces appear when the supplier gets involved during the manufacturing phase of a component by taking into account the specifications provided by the prime contractor. Prime contractor prescribes in explicit detail all the information regarding the characteristics of the component and how it is to be manufactured. The supplier then follows these directions and produces the component. Since the transferred knowledge is quite explicit and specialized on the component, the coordination between firms can be easily supported by loosely coupled relations.  The last type of interface appears when the supplier provides standardized components to be integrated into the final product with no or little modification. This involvement is characterized by standardized interfaces. The supplier develops its own products independently from any specifications, and the prime contractor, like a regular buyer, chooses through "catalogues" or "off-the-shelf" the parts. Since the firms do not share at all any resources, the coordination is decoupled.

The pivot-firm articulates the translation and interactive interfaces. It develops combinatorial skills. It has the capacity to mobilize a set of internal and external resources in order to participate in the design and production take-over of a major technical sub-unit of the final product.

In configuration taking place, a firm-pivot in charge of aerostructures or avionics systems, must meet the criteria of size, financial and technological capabilities to share industrial and financial risks. The optimal configuration of subcontractors network consist in a large companies in the first level in charge of all technical and industrial homogeneous module (aero structures, electrical wiring, embedded systems ...) related with a wide network of suppliers and subcontractors. The main characteristic of the firm-pivot is to able to share the risks according to the formula, "the equipment is paid when the aircraft is sold."

In this topic, a pivot-firm is a firm that:

-Has specific competencies. These specific skills involve a homogeneous set of knowledge and know-how (avionics, aero structures, engine nacelles…), linked with the design and/or the production of major modules or of important components for the final product.

-Has combinative competencies. These skills are technical and organizational. They are said to be combinatorial because they put their holders in the position to manage the interfaces with the other participants in the design and production process, in other words the architect-integrator and the subcontractors at the lower levels.

-Participates in the co-specification of the products. In particular in an industry of complex goods, this capacity is a direct consequence of the former two. From an organizational standpoint, co-development and co-specification generates structures that encourage decreases in time frames and costs. The economic advantage of modularizationcombined with outsourcingis, again, clear in terms of reducing the transaction and contract management costs through the decrease in the number of direct partners.

-Plays the role of an architect-firm for the units that it is in charge of. The relationship between the architect-firm and the network of firms that it structures depends on the nature of the modular interfaces (Figure 1).

Figure 1: Interfaces, competencies and pivot-firms On that basis, the "optimal" configuration for a subcontracting network would include big pivot-firms for homogeneous technical-industrial sets (aerostructues, wiring harness, embedded systems…), other smaller pivot-firms and finally suppliers and sub-contractors. The characteristic of the pivot-firms would be to share the risks according to the principle that 'the equipment is paid for only if the aircraft is sold". [START_REF] Cagli Ayse | Gestion stratégique de la supply chain et firme-pivot dans le secteur aéronautique[END_REF] illustrate the interactions between the pivot-firms and Airbus and Boeing with examples. For instance, the Safran group, through its different subsidiaries, positions itself on a high number of strategic technical modules and components in the production of an aircraft. Through Techspace Aero, Messier-Dowti and Messier-Buggati, Safran participates in the design and production of aircrafts' engines, wheels, brakes and landing gears. Earlier in the paper, we have seen that the components come within the strategic segmentation of Airbus' supply chain (Figure 2). As a pivot actor, Safran has mainly business relationships with Airbus, even if it participates to the subcontracting chain of Boeing (48 business relationships with Airbus and 24 with Boeing). Besides, if some subsidiaries of the group (Labinal, Messier Buggati and Messier-Dowty) are Boeing global partners, only Airbus considers the entire group as one of its major subcontractor. This relation with Airbus can be explained by the geographic proximity with the group's subsidiaries and by how long they have had relationships ( Kechidi and Talbot 2010; [START_REF] Gilly Jean-Pierre | Firms Hub and the Dynamics of Regional Innovation: Case Studies of Thales and Liebherr in Toulouse[END_REF].

Conclusion

Aircraft has become an increasingly complex technical object. This complexity has generated changes in the organization of aircraft manufacturing. The case of Airbus is significant in this evolution. In comparison with Boeing, it is clearly Airbus that has been able to draw the most advantages from the modularization trend of the past few years. According to us, the dynamic of technical and organisation innovation, as well as the industrial model that generated it, have played a large part in the success of Airbus on the civil aviation world markets. The industrial organisation which has progressively been put in place is characterised by:

• a refocusing of activities on the core of Airbus's business. With Power 8, Airbus has accelerated this refocusing. It has evolved from a status of "aircraft manufacturer" to the status of "architect-integrator". The position of architect-integrator, in the upstream stage (design of the aircraft) and the downstream stage (assembly of the components and liaising with airlines) of the production and commercialisation process is strategic because it enables them to control the entirety of the value chain. This control means that the technologically strategic or profitable components continue to be manufactured internally.

• the development of modularisation in aircraft manufacturing. A real technical evolution, it has been accompanied by important organisational mutations in Airbus as well in its partners.

• the outsourcing of activities deemed to be non-strategic. Outsourcing is not a new phenomenon in the politics of industrial acquisitions. The novelty is that it concerns new activities and/or involves larger volumes. As a consequence of the development of modularisation, the outsourced "technological packages" now represent systems and entire technical sets.

• the reinforcement of the role of Airbus's major partners. These firms, that produce important "technological packages", in turn play the role of architectintegrators for the sets they produce. They have the specific capacity to articulate the vertical relationships between the architect-integrator and the other (inferior) levels of sub-contracting or of supplying components and equipment. More than simple "tier one" or Risk Sharing Partnerships, these firms become essential partners in the design and manufacturing of the aircraft.

The strong outsourcing as well the international fragmentation of the production process are however important limitations on this organisational model. The difficulties encountered by Airbus for the assembly of the A380 and by Boeing for the B787 reflect the limitations of outsourcing and modularisation. The recomposition of the industrial puzzle poses real problems with regard to the management of the different technical and organisational interfaces. These problems directly affect the profitability of the projects. Indeed, according to the Seattle Times of 18 December 2010, the cost of the 787 has increased from 5 billion dollars to between 12 and 18 billion dollars.

Boeing acquiring Vought Aircraft's share of Global Aeronautica can be interpreted as a return to the internalisation of a part of the construction of the aerostructure. Similarly, Airbus keeping the manufacturing of the central box and the wings of the A350 internal is clearly a sign of the breaks being put on outsourcing. The elements which in the past pushed forward the performance of this production model seem today to be holding it back.

Figure

  Figure 1 Logistics of an Airplane assembled in Toulouse

Figure 2 :

 2 Figure 2 : The new supply chain partnerships organisation

Figure 3 :

 3 Figure 3: Safran Group as a pivot-firm

A first version of this paper was published in International Journal ofTechnology and Globalisation (2013, Vol.7, N°1/2)

Emmanuel Grave, vice-president of the aeronautics division of Thales, Air & Cosmos, November 24 th 2006.

For example, Boeing subcontracts more than 70% of the equipment of B787. Airbus considers a similar approach for A350 by subcontracting 50% of the tasks linked with the aero structure to external firms.


Luana Batista Goulart

Stefan Tzortzis

Isabelle Séguy

Gérald Quatrehomme

HAL is

INTRAPOPULATION GROWTH VARIATION ANALYSIS USING FEMUR LENGTH: THE CASE OF MARTIGUES PLAGUE VICTIMS (SOUTH OF FRANCE -1720-1721)

Luana Batista-Goulart 1 , Stéfan Tzortzis 2, Isabelle Séguy 3 and Gérald Quatrehomme 4

1 Uni v e r si t é Côt e d'A zur , CN R S , CE P A M, F r a nce 2 UMR 7 2 6 8 A D E S A MU, CN R S , E F S , F r a nce 3 Inst i t u t N a t i ona l d'É t u de s D é mogr a p hi q ues (IN E D ), F -7 5 020 P a r i s; Uni v e r si t é Côt e d'A zur , CN R S , CE P A M, F r a nce 4 Uni v e r si t é Côt e d'A zu r , Inst i t ut Uni v e r si tai r e d'A nt hr op ologi e mé di co -lé ga le , F a cult é de Mé de ci ne ; CE P A M, F r a nce Contact: luana.batista-goulart@cepam.cnrs.fr • Individual's living conditions influence his/her stature by 20% (Koepke, 2016).

• Bioarchaeological context: stature analysis has been used to evaluate past populations' living conditions. Aims:

(1) evaluate whether all groups, divided according to age, gender and location at the archaeological site, were equally affected by these famines;

(2) investigate how differently men and women suffered from them;

(3) link age classes with starvation episodes in that region.

Results

(1) Variables: sex, inhumation place and femur length (FAMD) No pattern of stature variation linked to the inhumation place was observed.

(2) Variables: sex, age class and femur length (FAMD)

• Significant difference between the mean of groups 1 and 3 (Student's ttest) o Group 1: women (almost only) from age classes young-mature, mature and mature-old. o Group 3: women from age classes young and old.

• Variation in women stature: the youngest and the oldest are taller than the others.

• Variation of mean among male groups (2 and 4) is not statistically significant.

• Considering both genders, the average statures of different age groups do not differ in a statistically significant manner (ANOVA test ).

(3) Famines Victims of 1720-22 plague were born between 1702 and 1625 (90% between 1702 and 1655) 1695 -1702 -born period of individuals classed as "young"; 1684 -1710 -several episodes of starvation; Uterine life and development period during the penury; 1625 -1655 -born period of individuals classed as "old"; Both classified in the groups with higher statures (3 and 4).

Results of the FAMD considering sex, age class and femur length.

Discussion

Gender inequality hypothesis: women's stature correlated with age, while male's stature are homogeneous.

• Our analysis suggests that women were the most affected by changes in life conditions. "Osteological paradox" (Wood, 1992):

• The youngest and the oldest were the most vulnerable during the penury period:

• Younger: growing and development period;

• Older: vulnerable because of advanced age. • Event though, they had higher stature than the other age group and survived the famines.

• We hypothesize that:

• they survived because they had better life conditions than their contemporaries;

• differential life conditions made them taller than the other age group.

Methods

• Femur length analysis

We directly examine femur length, instead of predicted statures, to avoid modelling errors.

• Factorial Analysis of Mixed Data (FAMD) Variables: sex, age class, inhumation place and femur length.

• Investigation of historical sources.

Future perspectives:

• Examine the occurrence of physiological stress markers in the bones and teeth and also the variation in dental microwear to verify the consistency of these results.

• Comparison with other populations from the same region and/or period.

Material•

  In this work we analyse the skeletons of 95 adults: o victims of the Great Plague of Marseilles (1720-1722); o exhumed from mass graves located at the city of Martigues (Provence -France) (Tzortzis, 2009); o suffered several starvation episodes before the epidemic (Séguy, 2017). References: Koepke, N. 2016. The Biological Standard of Living on Europe from the Late Iron Age to the Little Ice Age. In The Oxford Handbook of Economics and Human Biology, eds. John Komlos and Inas R. Kelly. Oxford: Oxford University Press. Séguy I. 2017. De Charybde en Sylla : les Provençaux à l'épreuve des calamités (fin XVIIe-début XVIIIe siècle). In : Gil Bellis, Elisabeth Brown, Philippe Cordazzo, Virginie De Luca et Alain Parant (eds), Les populations vulnérables (actes du colloque de la CUDEP, Conférence universitaire de démographie et d'étude des populations, Aix-en-Provence, 28-31 mai 2013), CUDEP. 530-546 Tzortzis S., Signoli M. 2009. Les tranchées des Capucins de Ferrières (Martigues, Bouches-du-Rhône, France). Un charnier de l'épidémie de peste de 1720-1722 en Provence. C. R. Palevol, 8 (2009), 749-760. Wood, J. et al. 1992. The Osteological Paradox. Current Anthropology 33(4): 343-70.


R J Vazquez Guerra

J F Marquez Rubio

D F Novella Rodríguez

B Del Muro Cuéllar

D Cortés Rodríguez

Control of unstable delayed recycling systems

Keywords: Linear Systems, time-delay, unstable, recycling

This work deals with the stabilization of recycling systems with unstable dynamics at both, the direct path and the recycling paths as well as different timedelays for each path. The main contribution of this work is to provide a stabilizing strategy to the mentioned class of systems using an observer based control approach. Stability results concerning stable recycling systems or partially unstable recycling system can be found in literature. However, to the best of our knowledge, there is not stability results when unstable dynamics are considered in both paths of the recycling process. The provided necessary and sufficient stability conditions of the proposed stabilizing controllers are obtained from a frequency domain analysis.

Introduction

Recycling systems are commonly found in chemical industry, for instance, in a typical plant formed by reactor/separator process, where reactants are recycled back to the reactor [START_REF] Luyben | Plantwide process control[END_REF]. This kind of systems reuses the energy and the partially processed matter increasing the efficiency of the overall process. In recycling systems, a partial feedback of the process output to the input yields a model with positive feedback, which can give rise to some undesirable effects. In [START_REF] Luyben | Temperature control of autorefrigetator reactor[END_REF] the effects of recycle path on dynamics process and their implications to plant-wide control were studied. In [START_REF] Taiwo | The design of robust control systems for plants with recycle[END_REF], robust control for recycling plants was discussed and a scheme of recycle compensation to eliminate the effects of the recycle was proposed. Similar approaches were employed in [START_REF] Scali | Performance of control systems based on recycle compensators in integrated plants[END_REF][START_REF] Lakshminarayanan | Empirical modelling of processes with recycle: some insights via case studies[END_REF][START_REF] Kwok | Seasonal model based control of processes with recycle dynamics[END_REF].

When in addition to the recycle path a time-delay is present, exponential terms appear in the direct and the recycling paths. In a state space representation, recycling systems with time-delay can be related to systems with delays in the input and the state variables. In [START_REF] Muro-Cuéllar | Model approximation for Dead-Time Recycling Systems[END_REF], an approximated model to represent recycling systems by using discrete-time approach was proposed. Such approximated models can be used for stability analysis or control design [START_REF] Scali | Performance of control systems based on recycle compensators in integrated plants[END_REF][START_REF] Samyudia | Gap metric based control of processes with recycle systems[END_REF][START_REF] Astrom | A new Smith predictor for controlling a process with an integrator and long dead time[END_REF][START_REF] Maza-Casas | On the state prediction of linear systems with time-delays in the input and the state[END_REF].

In general, unstable delayed systems are more difficult to control than stable delayed systems. For instance, the original Smith Predictor cannot be used in the case of delayed unstable processes. However, many works have proposed modifications to the original Smith Predictor in order to deal with unstable systems, see for instance [START_REF] Sanz | A generalized smith predictor for unstable time-delay SISO systems[END_REF]. If a recycle is added in such unstable delayed system would lead to a more difficult (although interesting) problem. Existing Smith Predictor based schemes cannot directly be applied to this kind of systems. This is, because the open-loop recycling system is not just a system with one time-delay in the direct path but it is a system with an internal delay term, i.e., an open-loop infinite dimensional system. To tackle this problem, we consider here recycling systems composed of a plant with one unstable pole, several stable poles and a delay term in the direct path and the same type of delayed unstable subsystem (with possibly different delay magnitude) in the recycling path.

The present work extends the results presented in [START_REF] Marquez-Rubio | Observer scheme for linear recycling systems with time delays[END_REF][START_REF] Marquez-Rubio | Control of delayed recycling systems with unstable first order forward loop[END_REF] where recycling systems are restricted to be unstable first order plants in the direct path. A first attempt to extend these results was presented in [START_REF] Marquez-Rubio | Control of delayed recycling systems with an unstable pole at forward loop[END_REF], dealing with a high order plant in the direct path with one unstable pole and a stable recycling subsystem. Results developed in [START_REF] Marquez-Rubio | Control of delayed recycling systems with an unstable pole at forward loop[END_REF] do not achieve step tracking or step disturbance rejection.

There are stability results to stable recycling systems or partially unstable recycling system. However, to the best of our knowledge there is not stability results when unstable dynamics are considered in both paths of the recycling process. In this work, a control scheme is proposed in order to stabilize and control the delayed recycling processes previously depicted. The proposed controller allows to achieve the closed-loop stability. Then, necessary and sufficient conditions are given for the existence of the proposed stabilizing scheme, which are obtained from a frequency domain approach. It is important to note that, the problem of the stabilization and control of delayed unstable plants even without recycle path is not completely solved, at least from the classical controllers P, PI, PD and PID viewpoint. For instance, recent works deal with the stabilization and control of delayed systems with one [START_REF] Silva | PID controllers for time-delay systems[END_REF][START_REF] Lee | Stabilization of all-pole unstable delay processes by simple controllers[END_REF][START_REF] Hernández Pérez | PID for the stabilization of high-order unstable delayed systems with possible complex conjugate poles[END_REF] and two [START_REF] Novella-Rodríguez | PD/PID controller for delayed systems with two unstable poles: a frequency domain approach[END_REF] unstable pole(s).

The main contribution of the work is that the stability step into the proposed observer based scheme considers the existence of the recycling path, this feature is not considered in some works, see for instance [2]-[6], [START_REF] Marquez-Rubio | Observer scheme for linear recycling systems with time delays[END_REF]- [START_REF] Silva | PID controllers for time-delay systems[END_REF], [START_REF] Bamimore | Recycle compensator facilitates rapid parameterization of proportional integral/derivative (PI/D) controllers for open-loop unstable recycle processes[END_REF]. These mentioned works intend to eliminate the recycling dynamic in order to design a controller considering only to the direct path. Thus, when modelling uncertainties appears into the recycling path, the removal stage preserves the characteristic of the nominal system and some troubles may arises with respect to closed-loop stability/robustness. The proposed methodology in this manuscript does not intend to cancel recycling dynamics. Instead, under the knowledge of a special internal variable system, a control strategy is designed taking into account the whole system consisting in direct and recycling paths. This property allows to improve robustness issues considerably and this feature is precisely the key point to consider unstable dynamics at recycling path. However, the use of the proposed control scheme is not limited to unstable dynamics at recycling, i.e., the proposed result can also be used in stable recycling systems.

The outline of the paper is as follows. In Section 2 the problem is formulated and the class of systems considered is stated. The general idea of the solution is also outlined in this section, namely the need of an observer strategy. In Section 3 some preliminary results are presented. The Section 4 presents the main results. Some numerical examples are described in Section 5. Finally Section 6 presents some conclusions.

Problem formulation

Consider the class of recycling system (see Figure 1) described

[ ]       = ) ( ) ( ) ( ) ( ) ( ) ( s Y s U s G s G s G s Y r d d , (1)
with, The transfer function of the recycling system (1) is given by,

s s d e s D e s G s G 1 1 ) ( ) ( ) ( 1 1 τ τ α - - = = (2a) s s r e s D e s G s G 2 2 ) ( ) ( ) (
s s e s D s D e s D s U s Y ) ( 2 1 2 2 1 1 ) ( ) ( ) ( ) ( ) ( τ τ τ αβ α + - - - = . ( 3
)
Note that exponential terms appear explicitly in the numerator and the denominator of (3). Stability of (3) is determined by the roots of its characteristic quasi-polynomial,

s e s D s D s Q ) ( 2 1 2 1 ) ( ) ( ) ( τ τ αβ + - - = . ( 4
)
The transcendental term in Q(s) causes an infinite number of roots. For this kind of plants it is not an easy task to conclude something about the dynamical behaviour (stability for instance) even in the case of uncontrolled plant. Obviously, the related transfer function when the system is controlled with an output feedback becomes more complicated involving more than one transcendental term.

Let us to describe some initial ideas. With reference to Figure 1, if the signal ω 1 (s) were measured, then we could set,

) ( )) ( ) ( ( ) ( 1 s C s s R s U ω - = , (5)
obtaining the system shown in Figure 2, where R(s) is the input reference and C(s) is the controller. Then, it would be possible to design the controller, C(s), such that the closed-loop system is stable. In fact, the main aims of this work can be summarized as follows: i) Finding the stability conditions of the proposed feedback shown in Figure 2.

ii) Since ω 1 (s) is assumed as an unmeasured internal system signal, an observer scheme to estimate this variable is developed in the Section 4.

In the next section some preliminary stability results are presented. 3 Preliminary stability results

Consider the system,

s p e b a b a b a a s s U s Y τ γ - + + + - = ) )...( )( )( ( ) ( ) ( 2 1 , (6)
a, b i ∈ ℝ, with a, b i > 0 ∀i = 1, 2, ..., p;

∑ = > - m i i b a 1 0 1 1
and the static control law,

k s Y s R s U )) ( ) ( ( ) ( - = , (7)
with R(s) as the input reference. The following result provides a stability condition when a proportional controller is used for unstable delayed systems including one unstable pole, p stable poles and a time-delay.

Lemma 1 [START_REF] Lee | Stabilization of all-pole unstable delay processes by simple controllers[END_REF][START_REF] Pedraza-Beltrán | Stabilization of High Order Systems with Delay Using a Predictor Schema[END_REF] There exists a gain k such that the closed-loop system,

s p s e k b s b s b s a s e k s R s Y τ τ γ γ - - + + + + - = ) )...( )( )( ( ) ( ) ( 2 1 , ( 8
)
is stable if and only if

∑ = - < p i i b a 1 1 1 τ . ( 9
)
From condition [START_REF] Astrom | A new Smith predictor for controlling a process with an integrator and long dead time[END_REF], the size of the delay to achieve the closed-loop stability is restricted by the position of the open-loop unstable pole and the stable poles in the real axis of the plane s.

Let us now find the set of stabilizing gains k related to Lemma 1 is obtained. The phase crossover frequency ω c is given by,

∑ = =         -       + - m i i c c c b a 1 0 arctan arctan ω ω τω , ( 10
)
where ω c > 0. Then, the stabilizing gain k is within the interval of,

∏ ∏ = = < < m i i m i i M b a k b a 1 1 γ γ , (11)
with

∏ =                 +               + = m i i c c b a M 1 2 2 1 1 ω ω .
In what follows the stability condition of a particular output injection schema is provided. This proposed schema will be used later to design the proposed observer strategy.

Consider the recycling system given by ( 1)- [START_REF] Luyben | Temperature control of autorefrigetator reactor[END_REF]. A state space representation of G 1 (s) as

(A d , B d , C d ) is given by, [ ] 1 0 0 0 , 0 0 0 0 , 0 0 0 0 1 0 0 0 1 0 0 0 2 1           =                 =                 - - - = d d m d C B b b b a A α . ( 12
)
A state space representation of G 2 (s) as (A r , B r , C r ) with,

and the injection vectors defined by [ ]

) 1 ( 1 ) 2 ( 1 ) 1 ( 1 1 + - - - = m l l l L  , [ ] T l L 0 0 ) 1 ( 2 2  - = .
Therefore, a state space representation of the closed-loop system ) (

) ( s R s Y
related to the control scheme presented in Figure 3 can be written as,

) ( ) ( ) ( ) ( 1 1 τ θ - + - + = t r B t x A t x A t x  ( 14
) ) ( ) ( t x C t y = (15) with       - = r d r z A C B L A A ) ( 0 2 ,       - = 0 0 0 1 r d C B A       = 0 d B B , T d C C       = 0 , [ ] T r d t x t x t x ) ( ) ( ) ( = , d d z C L A A 1 - =
, and

=

such that the closed-loop transfer function,

) ( ) ( s R s Y
or equivalently ( 14)-( 15) is stable if and only if, given by ( 12) and the partial output injection through 1 L . Then, 1 L should be computed such that, { }

∑ = - < n i i d c 1 1 1 θ (16)
1 2 1 1 ,..., , ) ( + - - - = + - m d d z z z C L A sI σ (17)
where ) ( * σ represents the eigenvalues of * , and -z 1 , -z 2 , ..., -z m+1 are the relocated stable poles of G 1 (s). i.e., A z = A d -L 1 C d and it can be written as,

                - - - - = +1 3 2 1 0 0 0 0 1 0 0 0 1 0 0 0 m z z z z z A         .
Consider also that the state space representation of ) ( 2 s G given by ( 13) yielding to the closedloop representation given by ( 14)- [START_REF] Silva | PID controllers for time-delay systems[END_REF]. Notice that the term

d r C B L ) ( 2 -
in equation ( 14) is a free term due to the gain 2 L is the control parameter. From ( 14)-( 15), it is possible to obtain the closed-loop transfer function ) (

) ( s R s Y which results, s s e B e A A sI C s R s Y 1 1 1 ) ( ) ( ) ( τ θ - - - - - = , ( 18
)
where I is the identity matrix. The closed-loop stability depends on the characteristic equation associated to the the transfer function [START_REF] Novella-Rodríguez | PD/PID controller for delayed systems with two unstable poles: a frequency domain approach[END_REF], which is given by,

0 ) det( 1 = - - -s e A A sI θ or equivalently, 0 ) )...( )( )( )...( )( )( ( 1 2 1 2 1 = + + + + + + + - - + s m n e K z s z s z s d s d s d s c s θ αβ (19)
where

) 1 ( 2 1 - - = l K
, and

) 1 ( 2-
l is a free static gain. From ( 19), the closed-loop stability of the proposed output injection is reduced to a system of the form [START_REF] Samyudia | Gap metric based control of processes with recycle systems[END_REF], where its stability properties are analyzed in Lemma 1. In this way the associated stability condition is obtained as,

∑ ∑ = + = - - < n i m i i i z d c 1 1 1 1 1 1 θ (20)
However, the stability condition ( 20) can be improved by choosing 1

L such that the relocated poles of

) ( 1 s G , { } p z z z - - - ,..., , 2 1 are sufficiently far from the origin i.e., ∞ → - - - p z z z ,..., , 2 1
. Giving as result that condition [START_REF] Pedraza-Beltrán | Stabilization of High Order Systems with Delay Using a Predictor Schema[END_REF] becomes

∑ = - < n i i d c 1 1 1 θ .
Hence the result follows.■ Lemma 3. [START_REF] Xiang | Stabilization of second-order unstable delay processes by simple controllers[END_REF][START_REF] Kharitonov | Static output feedback stabilization: necessary conditions for multiple delay controllers, Automatic Control[END_REF] Given the open-loop transfer function

) (s G defined by: s e s D s N s G τ - = ) ( ) ( ) ( , (21)
a necessary condition for the closed-loop stability (when a unitary feedback is considered) is that the polynomial H(s):

s s v m m e e s D s ds d s H τ τ ) ) ( ( ) ( 1 1 + + = , ( 22
)
has all its zeros in the open left half plane, where m is the degree of ) (s N and v is the system type, defined by the order of the integer in the open-loop transfer function.

Main Results

Observer strategy

In order to estimate the signal ) ( 1 s ω , in Figure 1, we propose the observer-predictor depicted in Figure 4. The convergence properties are established in the following result.

Theorem 1 Consider the observer scheme shown in Figure 4, assuming G r (s) and G d (s) defined by [START_REF] Luyben | Temperature control of autorefrigetator reactor[END_REF]. Proof. A state space representation of the observer-predictor scheme shown in Figure 4 is,

There exists a vector

[ ] T L L L 2 1 = such that [ ] 0 ) ( ) ( lim 1 1 = - ∞ → t t t ω ω (23)
) ( ) ( ) ( ) ( 1 1 τ θ - + - + = t Bu t x A t Ax t x  (25) ) ( ) ( t Cx t y = (26) with, 2 1 τ τ θ + = , [ ] T r d r d t x t x t x t x t x ) ( ) ( ) ( ) ( ) ( = , [ ] T t y t y t y ) ( ) ( ) ( = , [ ] T d d B B B 0 0 = ,             - - - = r d d d r d d d d r d r d A C L C L C B C L C L A C L A C B A A 2 2 2 1 1 0 0 0 0 0 0 0 ,             = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 r d r d C B C B A ,       = 0 0 0 0 0 0 d d C C C , where 2 ) ( + + ∈ n m R t x is the state vector, R t u ∈ ) ( is the input, 2 ) ( R t y ∈ is the output, 0 1 ≥ τ and 0 2 ≥ τ are the constant time-delays in the system. ) 1 ( ) 1 ( + + ∈ m x m d R A , 1 ) 1 ( x m d R B + ∈
, and

) 1 ( 1 + ∈ m x d R C
are matrices and vectors parameters that correspond to the direct path in the process, G 1 (s) is represented by, [ ]

1 0 0 0 , 0 0 0 0 , 0 0 0 0 1 0 0 0 1 0 0 0 2 1           =                 =                 - - - = d d m d C B b b b a A α , (27) and ) 1 (
) 1 ( + + ∈ n x n r R A , 1 ) 1 ( x n r R B + ∈
, and

) 1 ( 1 + ∈ n x r R C
are matrices and vectors parameters that corresponds to the recycling path in the process,

) ( 2 s G is represented by, [ ] β           0 0 0 , 0 0 0 0 1 , 0 0 0 0 1 0 0 0 1 0 0 0 2 1 =                 =                 - - - = r r n r C B d d d c A , ) ( ˆt x is the estimation of ) (t x .
From Figure 4, we are strictly interested in the convergence of

) ( ) ( ) (
) ( 0 0 0 ) ( ) ( 0 ) ( 2 1 θ -       - +       - - = t e C B t

=

. Notice that the term 28) is a free term due to the gain 2 L is the control parameter. On the other hand, consider the closed-loop system analyzed in Lemma 2, given by ( 14)- [START_REF] Silva | PID controllers for time-delay systems[END_REF]. In this way, comparing (28) and ( 14) it is clear that Lemma 2 can be applied to system (28). Hence the result of the theorem follows.■

d r C B L ) ( 2 - in (
To find the observer parameters, the following procedure is suggested. L can be obtained using the Ackerman formulation.

Procedure 1 1. Verify the necessary and sufficient condition in

3. By applying the results of Lemma 1 to the case of our interest, the parameter

) 1 ( 2- l can be obtained from ∏ ∏ ∏ ∏ + = = + = - = + < < + 1 1 1 1 1 ) 1 ( 2 1 1 1 m i i n i i m i i n i i z d M c l z d c αβ αβ (30) with . 1 1 1 1 1 1 2 2 2 2 2 2 ∏ ∏ = + =         +         +         + = n i m i i c i c c z d c M ω ω ω
and ω c is obtained by solving, where ω c > 0.

∑ ∑ + = = =         -         -       + - 1 1 1 0 arctan arctan arctan m i i c n i i c c c z d c ω ω ω θω , ( 31
)
where ω c > 0.

Proposed Control Strategy

In order to implement the ideas developed in Section 2 let us asume that an estimation of the signal ω 1 (t) is taken from the observer designed in Subsection 4.1. In this way, in the following developments the stability properties of the closed-loop system shown in Figure 2 are analyzed. The following result presents the stability properties for the closed-loop system shown in Figure 2 when a Proportional-Derivative (PD) controller is considered. A frequency domain analysis is employed. Later, a Proportional-Integral-Derivative (PID) control is also considered to stabilize the closed-loop system.

Theorem 2 Consider the feedback control shown in Figure 2. Then, the delayed recycling system given by ( 1) can be stabilized with a PD controller given by,

C(s) = K P + K D s, ( 32
) or C(s) = k p (k d s -1) + 1, ( 33
)
with k p , k d > 0, if and only if, ∑ ∑ ∑ ∑ = = = = + + + - - - + < n i n i i m i i i m i i d b c a d b c a 1 1 2 1 2 2 2 1 1 1 1 1 1 1 1 1 θ (34) with θ = τ 1 + τ 2.
Proof. Sufficiency The closed-loop transfer function of the system shown in Figure 2 is given by, (

)

. ) ( ) ( ) ( 1 1 ) ( ) ( ) ( ) ( s G s G s C s G s C s R s Y r d d - - = (35)
and the associated closed-loop characteristic equation is,

1 + C ¯(s)G d (s)G r (s) = 0 (36)
with C ¯(s)=C(s)-1. The characteristic equation (36) can also be obtained from the auxiliary closed-loop system consisting in an open-loop transfer function

) ( ) ( ) ( ) ( s G s G s C s Q r d =
, with an unitary feedback. In this way, the stability conditions of the proposed auxiliary closed-loop system are equivalent to the required stability conditions for the characteristic equation (36).

The Nyquist stability criterion states that for a given open-loop system ) (s Q , there is a stabilizing proportional control k such that the closed-loop system is stable if and only if 0 = N +P , where P is the number of unstable roots in ) (s

Q
and N is the number of clockwise round trips to the point (-1, 0j

) in the open-loop ( )

) (s Q Nyquist diagram.
It can be seen that for our particular case

) ( ) ( ) ( ) ( s G s G s C s Q r d
= , P = 2 and therefore N should satisfy N = -2, i.e., in order to achieve closed-loop stability the Nyquist plot should encircle twice the critical point (-1, 0j) in counter-clockwise direction. Therefore, the selection of the derivative should be proposed to obtain a Nyquist trajectory such that the closedloop system is stable, in this way a desirable behaviour of the trajectory should be as it is shown in Figure 5 or Figure 6.

For the sake of simplicity on the frequency domain analysis, let use an alternative representation of the PD controller given by,

C ¯(s) = k p (k d s -1), (37)
with k p , k d > 0. Notice that (37) is a non-minimum phase controller. The selection of this controller structure allows to us to find the best stability conditions in terms of the delay value. Let us assume that the condition (34) holds, it could be selected a k d gain for the PD controller such that,

ε θ + - - - + = ∑ ∑ = = n i i m i i d d b c a k 1 1 1 1 1 1 , ( 38
)
with ε being a sufficient small positive constant.

In order to satisfy the Nyquist stability criteria a frequency domain analysis of the system

) ( ) ( ) ( ) ( ω ω ω ω j G j G j C j Q r d =
is required and performed in what follows. Thus, the phase equation of the system is, ( ) It can be seen that the phase begins in Φ Q (0) = -π and its trajectory follows a negative direction, this could be demonstrated considering the derivative of the phase, , 1

. arctan arctan arctan arctan arctan ) ( 1 1 π ωθ ω ω ω ω ω ω - - -         -         -       +       = Φ ∑ ∑ = = d m i n i i i Q k d b c a (39)
) ( 2 2 1 2 2 1 2 2 2 2 2 2 θ ω ω ω ω ω ω ω - + - + - + - + + + = Φ ∑ ∑ = = d d n i i i m i i i Q k k d d b b c c a a d d (40)
for frequencies in the neighborhood of ω ≈ 0,

. 0 1 1 1 1 ) ( 1 1 0 < - - - - + = Φ ∑ ∑ = = ≈ θ ω ω ω d n i i m i i Q k d b c a d d (41)
From (41), the starting angle of the system depends on the value of ϵ. It can be seen that for an 0 → ε , the initial direction of the phase is slightly negative, which yields a crossover frequency ω c 1 → 0, where ω c 1 is the first frequency ω c1 > 0 such that Φ Q (ω c 1 ) = -π. Moreover, there is a frequency ac s = ω such that Φ Q (ω s ) > -π, this can be found numerically.

Additionally, analyzing the phase derivative for high frequencies, namely ω ≈ ∞, we can see that its trajectory is decreasing for high frequencies. In order to complete the analysis let us consider the magnitude equation given by, ,

1 1 1 1 ) 1 ( ) ( 1 1 2 2 2 2 2 2 2 2 2 2 ∏ ∏ = =         +         +         +         + + = m i n i i i d Q d b c a k K M ω ω ω ω ω ω (42) with ∏ ∏ = = = m i n i i i p d b ac k K 1 1
1 1 α and .

αβ α =

For small frequencies, the magnitude has a positive direction, which is easily concluded from:

. 0 1 1 1 1 ) ( 1 2 1 2 2 2 2 0 > - - - - ≈ ∑ ∑ = = ≈ n i i m i i d Q d b c a k d dM ω ω ω
Since the open-loop system Q(jω) is strictly proper, there is a decreasing magnitude for large frequencies. As it can be observed, the selection of k ¯d results in a system with a positive gain margin, for a positive crossover frequency ω c 1 → 0, and a positive phase margin occurring at ω m > 0. Then, it can be concluded that the system G d (s)G r (s) can be stabilized with a PD controller (37) choosing a derivative gain according to (38); and equivalently the recycling system (1) can be stabilized with a PD controller (32) in the control loop shown in Figure 2.

Necessity. Taking into account Lemma 3, the stability of the closed-loop system with a PD controller requires that the polynomial:

H Q (s) = 0 + 1 s + 2 s 2 + • • •+ n+1 s n+1 + n+2 s n
has all its zeros in the open left half plane, since the higher order coefficient:

∏ ∏ = = + + = n i i m i i n m d b ac 1 1 2 2 1 1
θ γ is a positive constant, a necessary condition for the stabilizability is that the independent term

∑∑ ∑∑ ∑ ∑ ∑ ∑ - = > - = > = = = = + +         +       + - +         - - + + = 1 1 1 1 1 1 1 1 2 0 , 1 1 1 1 1 1 1 1 1 1 1 n i n i j j i m i m i j j i n i i m i i n i i m i i d d b b d b c a ac c a d b θ θ γ
is positive. Solving for the delay term we have:

∑ ∑ ∑ ∑ = = = = + + + - - - + < n i n i i m i i i m i i d b c a d b c a 1 1 2 1 2 2 2 1 1 1 1 1 1 1 1 1 θ . ■
Remark 1 An adequate choice of the derivative gain k d assures a Nyquist path with two encirclements in counter-clockwise direction, namely the system can be stabilized. In order to obtain a closed-loop stable system, the proportional gain k p should be selected such that:

, ) ( ) ( 1 1 1 1 1 cq n i i m i i p c n i i m i i F d b ac k F d b ac ω α ω α ∏ ∏ ∏ ∏ = = = = < < ( 43
)
where q = 0 for F (ω c2 ) < F (ω c0 ) < F (ω c1 ), q = 2 for F (ω c0 ) < F (ω c2 ) < F (ω c1 ), , 1 1 1 1 ) 1 ( ) ( 1 1 2 * 2 2 * 2 2 * 2 2 * 2 * 2 2 * ∏ ∏ = =         +         +         +         + + = m i n i i c i c c c c d c d b c a k F ω ω ω ω ω ω (44)
ω c0 = 0, ω c 1 and ω c2 are denominated the first and second positive crossover frequency such that Φ Q (ω c 1 ) = -π, respectively.

The frequency domain analysis used in the proof of Theorem 2 can be applied to the case of the PID controller obtaining the same stability condition. In this way, the following result is stated. 2. Then, the delayed recycling system given by ( 1) can be stabilized with a PID controller given by,

Corollary 1 Consider the feedback control shown in Figure

, ) ( s K s K K s C D I p + + = (45) or 1 1 ) ( +       + - = s k s k k s C i d p , ( 46
)
(with k p , k i , k d > 0) if and only if the condition (34) is satisfied.

Remark 2 An adequate selection of the derivative gain k d and the integral term k i in the PID controller given by (46) assures a Nyquist path with two counter-clockwise direction, i.e., the closed-loop stability warranty.

In order to obtain a closed-loop stable system, the proportional gain k p should be selected such that,

, ) ( ) ( 1 1 1 1 1 cr n i i m i i p c n i i m i i d b ac k d b ac ω γ α ω γ α ∏ ∏ ∏ ∏ = = = = < < ( 47
)
where r = 2 for γ(ω c3

) < γ(ω c1 ) < γ(ω c2 ), r = 3 for γ(ω c1 ) < γ(ω c3 ) < γ(ω c2 ), . 1 1 1 1 ) ( ) ( 1 1 2 * 2 2 * 2 2 * 2 2 * 2 * 2 * 2 2 * 2 * ∏ ∏ = =         +         +         +         + + + - = m i n i i c i c c c c c i d c c d b c a k k ω ω ω ω ω ω ω ω γ (48)
where ω c1 ,ω c2 ,ω c3 > 0 are denominated the first and second, third crossover frequencies respectively, such that Φ(ω) = -π, where . arctan arctan arctan arctan arctan ) ( 11

π ωθ ω ω ω ω ω ω ω - -       - -         -         -       +       = Φ ∑ ∑ = = i d m i n i i i k k d b c a (49)
A simple procedure to determine the PD/PID controller parameters is proposed.

Procedure 2

1. Identify the plant parameters a, c, bi, di, α ¯, θ.

2. If the stability condition (34) holds, continue with step 3; otherwise, a PD or PID controller can not stabilize the recycling delayed process [START_REF] Luyben | Plantwide process control[END_REF] in the control scheme shown in Figure 2.

3. Compute the minimum stabilizing derivative gain kd for the PD/PID controller (given by ( 33)/( 46)) according to (38).

4. For the controller (33)/(46), search for a maximum k d such that the phase equation Φ Q (ω c 1 ) =-π (given by ( 39)), for some ω c1 > 0 ω c 1 > 0 where ω m > ω c 1 > 0 and ω m is the frequency such that M(ω m ) = 1.

5.

If a PD controller is required, select a stabilizing kd from the obtained interval in Steps 3-4 and compute the stabilizing kp interval using (43).

6. If a PID controller is required, select a stabilizing k d from the obtained interval in Steps 3-4 and obtain the maximal k i (k i max ), starting from k i = 0 such that Φ(ω) > -π (Φ(ω) given by ( 49)) for some ω > 0. Then, select a k i into the interval 0 < k i < k i max and with the selected k d and k i compute the stabilizing k p interval using (47).

Remark 3 Notice that a particular case of the condition (34) in Theorem 2 can be obtained for the case of two unstable poles, i.e.

. 1 1 1 1 2 2 c a c a + - + < θ (50)
In this way, since it is assumed θ > 0 implies

0 1 1 1 1 2 2 > + - + c a c a
which from a graphical viewpoint satisfies the properties of a right triangle as well as the inequality triangle

        + > + 2 2 1 1 1 1 c a c a
, see Figure 7.

Proposed observer based control

To implement the results presented in Sections 4.1 and 4.2 through an observer based controller, Figure 8 shows the complete scheme and the following result is presented in order to assure the stability condition of the whole proposed strategy.

Corollary 2 Consider the observer-based control depicted in Figure 8. Then, there exists a controller, C(s), PD/PID such that the closed-loop system is stable if and only if,

∑ ∑ ∑ ∑ = = = = + + + - - - + < n i n i i m i i i m i i d b c a d b c a 1 1 2 1 2 2 2 1 1 1 1 1 1 1 1 1 θ (51) with θ = τ 1 + τ 2.
∑ ∑ + = = - - < 1 1 1 1 1 1 m i i n i i z d c θ (52) for θ = τ 1 + τ 2 .
On the other hand, by Theorem 1 and Corollary 1 it is possible to find a controller PD/PID such that the closed-loop system is stable if and only if,

∑ ∑ ∑ ∑ = = = = + + + - - - + < n i n i i m i i i m i i d b c a d b c a 1 1 2 1 2 2 2 1 1 1 1 1 1 1 1 1 θ (53) with θ = τ 1 + τ 2 .
Then we can conclude that the closed-loop system of Figure 8 is stable if and only if the most restrictive condition between the conditions (52) and ( 53) is satisfied, i.e., the condition (53).■

The proposed control strategy can be also extended to a class of systems that considers unstable dynamic at direct path but only stable dynamics at the recycling path. The following result establish how to apply the corresponding extension. , where stable poles could be complex conjugate. Then, there exists a controller, C(s), PD/PID such that the closed-loop system is stable if and only if,

∑ ∑ ∑ ∑ = = = = + + + - - - - < n i n i n i m i i n i m i i d b a d b a 1 1 2 2 1 2 2 1 2 1 1 1 2 1 1 1 ω ξ ω ξ θ (55)
Proof. Consider the observer scheme shown in Figure 8. An adequate estimation ω ˆ1(s), of the signal ω 1 (s) is always assured since the parameter 2 L can be performed via a Nyquist plot. On the other hand, by Theorem 2 in [START_REF] Hernández Pérez | PID for the stabilization of high-order unstable delayed systems with possible complex conjugate poles[END_REF] it is possible to find a controller PD/PID such that the closed-loop system is stable if and only if,

∑ ∑ ∑ ∑ = = = = + + + - - - - < n i n i n i m i i n i m i i d b a d b a 1 1 2 2 1 2 2 1 2 1 1 1 2 1 1 1 ω ξ ω ξ θ with θ = τ 1 + τ 2 .
Then we can conclude that the closed-loop system of Figure 8 is stable if and only if the most restrictive condition between the convergence condition and controller condition (55) is satisfied. In such case, we have (55).■

Numerical Results

Let us start with a theoretical and academic example that illustrstes all the potentiallity of the results here exposed. Later, a more realistic example borrowed from the literature is presented.

Figure 11 shows the closed-loop of the controlled system. It can be seen that the transitory behaviour of the output presents an initial undershoot and oscillations in the opposite direction with respect to the final stationary response. It is important to recall that we are dealing with a very complex delayed system with two unstable poles. The interconnection among the plant, the observer as well as the controller induces such characteristic response. In a practical situation, the inverse response could be bounded to zero, it depends on the nature of the variable. It can also be seen that if the delay size is not well characterized (for instance +20% of error), the stability is preserved but oscillations are present in the output signal.

Now, if a PID controller is considered to stabilize the plant given by (54), the stabilizing parameters plane k d -k i is shown in Figure 12, which is obtained using Step 6 from Procedure 1. Notice that for each point k d -k i into the stabilizing region, a k p interval should be computed using (47). In this way some stabilizing triplet k p -k i -k d are presented in Figure 13. Example 2. Consider the reactor-separator recycle plant presented in [START_REF] Xiang | Stabilization of second-order unstable delay processes by simple controllers[END_REF], time delays at direct and recycling paths are included. Thus an open-loop state space representation is, Step reference adjustment were used in order to obtain unitary state staionary value. Figure 14 shows a comparison of the output performance under nominal conditions, the proposed observer-based control in this manuscript is denoted as POBC. Closedloop robustness with respect to parametric modelling uncertainties into system (56a)-( 56b) is illustrated in what follows. Consider the recycling system under additive bounded uncertainties, given by, of uncertainty, respectively. As Figure 15 shows, under these conditions, DCS is unstable while POBC remains stable. Moreover, Figure 15

) ( ) ( ) ( ) ( ) ( 2 2 1 1 t u B t x A t x A t x A t x + - + - + = τ τ  , ( 57a)
) ( ) ( 1 1 τ - = t x C t y , (57b) where, [ ] T C T t x t x t x ) ( ) ( ) ( = ,             - - - ∆ - + - - - = s ps AS p p k V F V F k C c H c V UA V F V F A ) 1 ( 0 0 ) ( ) 1 ( λ ρ ρ λ .         =         ∆ - =       - = 0 , 0 0 ) ( 0 , 0 0 0 2 1 p s p ps AS c V UA B k c H A k C A ρ ρ , ) /

Conclusions

The analysis developed through this work gives a stabilization strategy for a particular kind of recycling system. The proposed stabilization scheme allows the use two different controllers, i.e., a Proportional-Derivative controller and a Proportional-Integral-Derivative controller with a proposed observer strategy. Explicit stability conditions of the proposed controller as well as the observer design are obtained in terms of the system model parameters. Even when a classical PID controller has the characteristic of step tracking reference and step disturbance rejection, in this work the used PID controller does not satisfies the mentioned properties and in this case only assures closed-loop 23 stability, perhaps this is the consequence of dealing with a complex class of recycling systems, where the two paths consider unstable dynamics. The importance of this work arises from the fact that the proposed strategy simplify the stability problem with two different time-delay terms to a problem with only one delay term. It should be highlighted that the proposed control scheme can be applied to the case of the stable recycling system i.e., G 1 (s) and/or G 2 (s) being stable subsystems. As a future work it should be interesting to extend the proposed control strategy to a class of systems without limit in the number of unstable poles in the transfer functions of the recycling system, G 1 (s) and G 2 (s). Also, the analysis of the time-delays varying case can be performed, taking into account the proposed control scheme.

  where D 1 (s) = (sa)(s + b 1 )...(s + b m ) and D 2 (s) = (sc)(s + d 1 )...(s + d n ). G d (s) and G r (s) are transfer functions of the direct and the recycling paths, respectively; τ 1 , τ 2 ≥ 0 are the timedelays associated to G d (s) and G r (s). a, b i , c, d j ∈ ℝ + with i = 1, 2, ..., m and j = 1, 2, ..., n, i.e., G d (s) and G r (s) are unstable. U (s) and Y (s) are the input and the output signals, respectively.

Figure 1 :

 1 Figure 1: A process with recycle

Figure 2 :

 2 Figure 2: Recycling delayed process with the proposed control law (5)

Lemma 2 .

 2 Consider the recycling system given by (1)-(2), and the output injection strategy shown in Figure3. There exists a vector

Figure 3 :

 3 Figure 3: Output injection schema.

  if and only if

Figure 4 :

 4 Figure 4: Proposed observer-predictor schema.

Figure 5 :

 5 Figure 5: Desired stable Nyquist trajectory of the system with two open-loop unstable poles.

Figure 6 :

 6 Figure 6: Desired stable Nyquist trajectory of the system with two open-loop unstable poles.

Figure 7 :

 7 Figure 7: Graphical interpretation of condition (50).

Figure 8 :

 8 Figure 8: Proposed observer based control schema.

Corollary 3 .

 3 Consider the observer-based control depicted in Figure8 with

Figure 10 :

 10 Figure 10: Output estimation error, e y (t) = y ˆ(t) -y(t), for Example 1.

Figure 11 :

 11 Figure 11: Output response for Example 1 using a PD controller.Figure 11 shows the closed-loop of the controlled system. It can be seen that the transitory behaviour of the output presents an initial undershoot and oscillations in the opposite direction with respect to the final stationary response. It is important to recall that we are dealing with a very complex delayed system with two unstable poles. The interconnection among the plant, the observer as well as the controller induces such characteristic response. In a practical situation, the inverse response could be bounded to zero, it depends on the nature of the variable. It can also be seen that if the delay size is not well characterized (for instance +20% of error), the stability is preserved but oscillations are present in the output signal.Now, if a PID controller is considered to stabilize the plant given by (54), the stabilizing parameters plane k d -k i is shown in Figure12, which is obtained using Step 6 from Procedure 1. Notice that for each point k d -k i into the stabilizing region, a k p interval should be computed using(47). In this way some stabilizing triplet k p -k i -k d are presented inFigure 13.

Figure 12 :

 12 Figure 12: Stabilizing k d -k i region for the PID controller (33) for Example 1.

Figure 13 :

 13 Figure 13: Stabilizing k pk dk i region for the PID controller (33) for Example 1.

Figure 14 :.

 14 Figure 14: Nominal performance comparison of Example 2.

  the nominal case is set. Therefore a ∆ is a deviation on the

Figure 15 :

 15 Figure 15: Output performance for uncertainties at direct loop ) (s G d . nominal parameter, same criteria is used for the other two defined uncertainties. In Figure 15, output is shown under uncertainties into direct loop ) (s G d . This is, deviations 1357 . 0 = ∆ a and 02 . 0 1 = ∆ τ are considered, which corresponds to % 20 + and % 13 +of uncertainty, respectively. As Figure15shows, under these conditions, DCS is unstable while POBC remains stable. Moreover, Figure15shows the output performance when POBC is submitted to uncertainties of % 95 + ( 6357 . 0 = ∆ a ) and % 13 + ( 02 . 0 1 = ∆ τ) with the corresponding parameter. Therefore,

Figure 16 :

 16 Figure 16: Output performance for uncertainties at direct loop ) (s G r .

Example 1 .

 1 Consider a recycling process with the transfer function

		G	d	(	s	)	=	(	s	-	)( 4 . 0	s 0 . 5 12 )( +	s	+	) 20	e	-	2 . 0	s	,	(56a)
		r G	(	s	)	=	(	s	-	)( 8 . 0	s 0 . 5 9 )( +	s	+	) 11	e	-	1 . 0	s	,	(56b)
	First, the observer design is regarded, hence the Procedure 1 is followed. The existence of the observer strategy is
	assured due to the condition (24) in Theorem 1 is satisfied, namely θ < 1.0480. The relocated poles of G 1 are
	proposed at {-8 -9 -10} i.e., z 1 = 8, z 2 = 9 and z 3 = 10. Taking into account the desired relocated poles previously
	proposed, (29) is solved to L 1 , obtaining	L				[			18		96 . 104		6 . 4	] T

Table 1 . Parameters of system in Example 2.

 1

	k	s	=	k	0	exp(	-	E	a	s RT	and	ps k =	k	s	exp(	E	a	/	2 s RT	)	.

Taking into account Corollary 3, the condition (

5

) is satisfied. From Step 2 of Procedure 1, it is proposed to relocate the

  shows the output performance when POBC is submitted to uncertainties of

	+	% 95	(	∆ a	=	6357 . 0	) and	+	% 13	(	∆ τ	1 =	02 . 0	) with the corresponding parameter. Therefore,
	POBC is robustly stable with the deviations		∆ a	∈	[	6357 . 0 , 0	]	and	∆ τ	1 ∈	[	, 0	. 0	] 02	. Figure 16 shows
	the output performance for uncertainties into the recycling path	) G r . Thus, a compraison of the (s
	output is considered with uncertainties of	+	% 222	(	∆ τ	2 =	. 0	22	) and	+	% 333	(	∆ τ	2 =	0	.	42	) in the
	recycling delay-time. From Figure 16, POBC is robustly stable with	∆ τ	2 ∈	[	, 0	0	.	] 42	, while for the
	case	∆ τ	2 =	0	.	42	DCS is unstable.

 -=. Using the Step 3 of Procedure 1 the interval of the observer parameter

Secondly, the design of the controller C(s) is derived. To this end, Procedure 2 is considered to design a PD controller. Figure 9 shows the complete stabilizing region k p -k d for the controller given by ( 33). In order to illustrate the behaviour of the proposed control strategy, some simulations are performed, Figure 10