John Levesque
email: john.levesque@minesparis.psl.eu

Cedric Dalmasso

Sophie Hooge

INNOVATIVE HRM DESIGNERS: THE DESIGN REGIMES OF HUMAN RESOURCE MANAGEMENT IN FRENCH INDUSTRIAL HISTORY

INTRODUCTION

The topic of Strategic Human Resource Management (SHRM)'s impact on a firm's innovativeness (or its contribution to innovation, as it is often phrased) has been an ongoing subject of academic debate for the past 2 decades, for both the HRM and the innovation management communities. Indeed, if the purpose of SHRM is to allow "the choice, alignment, and integration of an organization's HRM system so that its human capital resources most effectively contribute to strategic business objectives." (Kaufman, 2015, p.404), then it becomes quickly relevant to understand in what ways the HRM systems contribute to a firm's innovative activities. This type of questioning has produced several works on topics such as 1) how specific HRM strategies, practices or tools directly or indirectly affect a firm's capability to innovate, through its workforce, whether it be employees, managers, or professionals from other support functions; 2) whether the HRM function, characterized by its actors, themselves innovate, to provide the firm with new strategies, practices or tools; 3) how HRM professionals help the firm to respond to external innovations that disrupt its organization and threaten its core activity.

It is not coincidental that the same period has been characterized by a profound shift in the context in which firms, particularly large industrial ones, have been operating. Today, the disruptive effects of exogenous breakthrough innovations are no longer an isolated or ephemeral phenomenon: digital transformation, for instance, has become a reality for most industries, creating observable impacts across all sectors of activity, as well as the functions that drive them. This context of intensive innovation, which imposes an acceleration of the pace and intensity of innovation [START_REF] Christensen | Six Keys to Creating New-Growth Businesses[END_REF][START_REF] Hatchuel | Strategy as Innovative Design: An Emerging Perspective[END_REF][START_REF] Midler | Project Management for Intensive Innovation Based Strategies : New Challenges for the 21st Century[END_REF][START_REF] Phelps | Mass Flourishing: How Grassroots Innovation Created Jobs, Challenge, and Change[END_REF], implies being able to establish an ambidextrous approach to the firm's activities and steer continuous exploration activities [START_REF] March | Exploration and Exploitation in Organizational Learning[END_REF] to renew "dynamic capabilities" simultaneously [START_REF] Teece | Explicating Dynamic Capabilities: The Nature and Microfoundations of (Sustainable) Enterprise Performance[END_REF]. This current context impacts the entire organization of firms, thereby generating important repercussions on their employees. As a result, Human Resources Departments today are faced with problems that call into question the sustainability of their operations and, by extension, of the firms they support: the actors of HRM find themselves having to deal with new challenges, such as accompanying and/or preventing the accelerated mutation of strategic skills, managing the loss or appearance of knowledge and expertise, as well as recruiting or implementing training programs in the face of unknown futures, or assessing the value of the work of exploring innovative project teams [START_REF] Wright | A Research Revolution in SHRM: New Challenges and New Research Directions[END_REF]. At the SHRM level, this brings both practitioners and researchers to wonder how to ensure an alignment between the firm's goals and its available human resources if its strategies keep changing in real time and their employees' competences (sometimes even highly specialized ones) are being made less relevant by exogenous innovations. Yet, this new context is far from being the first transformative episode to challenge HRM systems and practices: on the contrary, the HR function has a rich history of evolution and diversification when it comes to its mechanisms.

The present paper is built on the theory that HRM actors have long been unrecognized designer collectives, who have regularly mobilized their resources and organized creative processes to introduce new managerial solutions, in the form of innovative processes, structures and tools. To test this theory, a longitudinal qualitative study was performed, using the conceptual framework on design regimes to identify collective design phenomena within the evolution of the HRM function throughout industrial French history. The main source of historical data was obtained from Jean Fombonne's seminal work "Personnel & HRM: the affirmation of the Personnel function in [industrial] firms (France, 1830-1990" 1 . The article starts by presenting a review of the literature on HRM contributions to a firm' innovative activities and highlight the enduring absence of a framework to describe the "design activity" expected from SHRM actors. Subsequently, the research question will be presented, and the following longitudinal study will rely on the conceptual framework of design regimes to analyze the historical evolution of the HRM function in French industrial firms. This approach will aim to confirm the hypothesis that HRM actors have historically demonstrated collective design activities that mirror those of industrial engineers, albeit in a less formal way. The core managerial implication of this work is that HRM actors, can build on this history of informal design activity to institutionalize HRM design practices and empower SHRM actors to create better dynamic alignments in intensely innovative situations.

LITERATURE REVIEW The complexities of defining HRM

From an academic point of view, the concept of HRM is polysemous [START_REF] Haslinda | Evolving Terms of Human Resource Management and Development[END_REF], which may be seen as desirable by those who would like to avoid compartmentalizing the discipline [START_REF] Wang | The Dilemma of Defining International Human Resource Development[END_REF], but which also leads to numerous debates on what constitutes this concept and how to appropriately name it. We also note the growing emergence of new terminology, such People Management, which seems to indicate the need for a new definition, one which offers "a wider, more contextual, more multi-layered approach founded on the long-term needs of all relevant stakeholders" [START_REF] Beer | Back to the Future: Implications for the Field of HRM of the Multistakeholder Perspective Proposed 30 Years Ago[END_REF]. Given the multiplicity of definitions and forms that the HRM function can take, it is not surprising to observe, depending on the case, varying levels of visibility or the scope of activities in different Human Resources Departments. The difficulty in defining the concept of HRM is potentially not recent and could even be a characteristic of this activity. However, in today's context of intensive innovation, this lack of consensus can complicate any exercise that seeks to rethink HRM mechanisms to develop the right management tools and processes, for the purpose of supporting a collective group's innovation capabilities [START_REF] O'connor | Major Innovation as a Dynamic Capability: A Systems Approach*[END_REF][START_REF] Hooge | Is Intrapreneurship Scalable? The Challenge Of Managing A Massive Internal Startup Call[END_REF][START_REF] Hooge | La valeur de l'inconnu en entreprise : modélisation des stratégies, outils et dynamiques collectives de la performance de l'innovation intensive[END_REF].

The fragmentation along research communities

While the implications of intensive innovation for HRM, as a corporate function, are regularly addressed in the field of innovation management sciences, the existing literature generally focuses more on broader issues: one part of the academic work aims to define the dynamics of professions (expertise, work or collective dynamics) [START_REF] Garcias | Paradoxical Tensions in Learning Processes: Exploration, Exploitation and Exploitative Learning[END_REF][START_REF] Dalmasso | Simuler les dynamiques organisationnelles pour concevoir l'organisation. Comment retisser les liens entre stratégie[END_REF], while the other focuses on identifying skills perceived as innovative (creativity, feedback, agile method, etc. ), that need to be acquired by employees and fall under the HRM subfield of Human Resources Development (HRD) [START_REF] Arouna-Hardouin | Dynamique de constitution des compétences d'innovation dans une FMN[END_REF].

For the more limited works that focus specifically on HRM contribution to innovation, they usually fall under either traditional HRM research or Innovation Management and, unsurprisingly, they tend to exhibit different understandings of the situations observed. There seems to be a tendency, on the side of innovation research to view HRM issues as "fragmented and instrumental approach" where HRM practices are presented as a toolkit that supports innovation at two levels: at the organizational level and at the level of specific innovation stages. On the other side, there seems to be a tendency for HRM research to consider innovation as a rather static phenomenon, thereby not taking into consideration the dynamic and changing processes inherent to innovative activities (de [START_REF] De Leede | Innovation and HRM: Towards an Integrated Framework[END_REF]. This is made evident in certain cases of collaboration between HRM practitioners and Innovation actors (such as IT implementation teams) seem to speak two different languages and fail to create an efficient collaboration to bring a project to fruition [START_REF] Bondarouk | HR Contribution to It Innovation Implementation: Results of Three Case Studies[END_REF]. If there have already been several attempts to bridge the gap between these two traditions and offer an integrated model of innovation & HRM (de Leede & Looise, 2005;[START_REF] Looise | Creativity and Innovation Management[END_REF], they have mostly confirmed the notion that HRM strategies and practices are intrinsically linked when it comes to contributing to innovative activities.

The multiplicity of expectations handed down to HRM

In many articles, there is often the idea, mainly in the results and conclusions, that HRM actors are expected to change or come up with solutions to be more relevant to innovation actors. This can take many forms such as 1) building the firm's identity to ensure employee membership [START_REF] Schoemaker | Identity in Flexible Organizations: Experiences in Dutch Organizations[END_REF], 2) creating new common languages with technical fields like IT, and support them in the implementation of innovative tools by creating user centered approaches to learning and skill development [START_REF] Bondarouk | HR Contribution to It Innovation Implementation: Results of Three Case Studies[END_REF], 3) ensuring the firm's functional flexibility through HRM job design practices, while also controlling the employee perception of High Commitment HRM, as it will positively impact Innovative Work Behavior. [START_REF] Dorenbosch | On-the-Job Innovation: The Impact of Job Design and Human Resource Management Through Production Ownership[END_REF], or 4) creating frameworks to manage the creation, flow and implementation of new knowledge at an organizational level [START_REF] Shipton | Managing People to Promote Innovation[END_REF]. The diversity of expectations and how HRM can act upon them is summarized in a recent systematic literature review on the intersection of HRM and innovation, [START_REF] Koster | Innovative HRM. A Review of the Literature[END_REF] which highlights three main lines of research on this issue: innovative HRM (instances the function which is able to innovate in its own mechanisms), the impact of innovation on HRM systems (more specifically the impact of technology on HR processes), and an innovation HRM (instances in which the HR contributes to the innovation of the firm). However, Koster also points out that these axes seem to have developed relatively separately, leaving room for work that would combine "several subthemes identified here to conceptualize organizations that are radical innovators in the field of HRM. Such a conceptualization would, for example, consist of organizations developing innovative HR policies and practices as a response to innovations in their external environment to reach high levels of organizational innovativeness. Such a full integration of the three research approaches reviewed in this article is not available in the literature yet." (ibid, p101).

The ongoing ambiguity of design activity in HRM

Yet, what is consistently lacking in the literature is a focus on how HRM actors are supposed to create, develop, introduce all these new processes, practices, and tools, as is expected of them. Indeed, while it seems clear that HRM systems must act to support innovative activities, particularly in intensive innovation contexts, existing works on situations of HR innovation, (where HR actors create and implement something new) seems to be mostly focused on the creativity of the individual actors. And yet, the term "design" is so often used by many if not all of the authors previously cited: whether it be "job design", "designing frameworks for the flow of knowledge", the biggest expectation seems to be that HRM actors are expected to resort to a structured process of creative action that can provide tailored solutions to the needs of innovative activities within a firm. This is where existing design theories, enriched by new analytical approaches to work, seem to offer promising opportunities to bridge existing limits and new paths of exploration. Beyond a few practical cases that have sought to apply innovation methods (such as the Design Thinking method) to design user-centered HR tools, or public-private cocreation experiments around innovative HR procedures [START_REF] Folcher | Construire et Conduire Une Intervention En Ergonomie, Questions Pour La Formation Professionnelle[END_REF][START_REF] Cappelli | HR Goes Agile[END_REF]), it appears that this type of approach, linking Design Theory and work analysis, has not yet been applied to the issue of HRM in the context of intensive innovation.

RESEARCH QUESTIONS

The existing research on HRM an innovation has brought us to wonder exactly how HRM actors design new HR practices and mechanisms, for the purpose of helping the firm maintain or pursue innovative activities that have a strategic value. The main questions we wish to address in this article are: are there historical cases of design by HRM actors? If so, where does individual instances of inventive activity or collective design endeavors? From that point on, it is also relevant to ask what kinds of objects (in HR practices, processes, tools, etc.) were being designed? Finally, were there different approaches to such design activities and can they be categorized?

METHODOLOGY AND MATERIALS

In order to provide answers to our research questions, we sought historical data that would permit a longitudinal yet explorative research to detect and identify key phenomena of change [START_REF] Pettigrew | Longitudinal Field Research on Change: Theory and Practice[END_REF]. The initial focus was on international HRM cases, but the scope was sharpened to focus on historical data regarding the evolution of French industrial firms that shed a light on the emergence and development of the HRM function (or Personnel function as it was known through most of its history). There are two sets of historical, secondhand data. The first one centers exclusively on the works of Jean Fombonne, a historian of the Personnel function in France, from 1830 to 1990. His seminal research work from "Personnel et DRH" (Personnel & HR Management) ( 2001), provided dozens of study cases and hundreds of testimonies and artifacts on the evolution of the HRM function within key industrial actors of the French economy. Furthermore, additional data was collected from his article from 1963 "Où en est la formation dans l'entreprise?" (Where are we with training within the firm?), published in the French journal of Sociology du Travail (Labor sociology). The second set of data is also composed of two research articles, that provide more study cases of innovative HRM practices in two French industrial firms that were extensively studied in Fombonne's work: the auto maker Renault [START_REF] Touvard | Chapitre 6. Renault crée une communauté d'innovation pour se décloisonner et penser le futur[END_REF], and the tire manufacturer Michelin [START_REF] Grab | Michelin développe l'innovation en écosystème[END_REF]. The data was analyzed by using the notions of design regimes developed in Design Theory [START_REF] Le Masson | Design theory: methods and organization for innovation[END_REF] and by creating a comparative framework inspired by prior endeavors to analyse the innovation regimes of new management tools in the service industry [START_REF] Canet | La fabrique des outils de gestion: quels régimes de conception?[END_REF].

PROPOSED MODEL Changing innovation contexts and design regimes

Having highlighted the recurring importance of a design activity necessary to illustrate the contributions of HRM systems to a firm's innovative activities, it now seems necessary to explore the type of literature that can help investigate the nature and processes of such design activities. This brings us to consider the body of work that has emerged from Design Theory, as it is presented by [START_REF] Le Masson | Design theory: methods and organization for innovation[END_REF]. Specifically, we focus on their conceptualization of design regimes that distinguish different processes of design through the history of industrial development. It is important to clarify that this type of work has been built by looking specifically at the activities of industrial design actors, that were primarily concerned with technical and/or technological exploration to develop new products, such as automobiles or trains. However, as it is rooted in management sciences, this approach does seem promising to apply to HRM design activities as they provide a framework to identify design actors, as well as the expansive reasoning and the performance criteria that can describe the collective process of a design activity.

Before we describe the different design regimes proposed, it is important to understand the use of the expression "design regimes" and its intrinsic link to innovative activities. The term "design" first comes to offer a more precise definition of "innovation", as the latter term encompasses a great variety of meaning and is used to describe many different objects, whether products or services, processes, etc. By using "design" the authors wished to focus on the activity at the heart of innovation, one that precisely implies the process of combining existing needs and available knowledge to achieve "the creation of knowledge, [and] the generation of alternatives and new worlds." (Le Masson, Weil & Hatchuel, 2017, p.10). As for the term "regime", its addition serves to add the idea of several varied, long-lasting, and stable phenomena of collective action, thereby creating a concept that can "describe the logic of collective action in design leading to innovation." (ibid, p.10).

Analytical framework

The notion of design regimes offered by Le Masson, Weil and Hatchuel is characterized by three core dimensions : 1) a particular design reasoning, which will describe how knowledge is mobilized and expanded during the design process; 2) a form of collective organization, which will describe how actors and resources participate in the design process; and 3) a performance logic (specific to a firm or ecosystem), which will define the expected value criteria of the object that should result from the design process [START_REF] Le Masson | La domestication de la conception par les entreprises industrielles : l'invention des bureaux d'études[END_REF]. It is by observing the diversity or change within these dimensions that the authors can identify four distinct types of regimes: 1) Wild Design; 2) Parametric rule-based design; 3) systematic rule-based design; and 4) Innovative design. As we will detail the characteristics of each regime, according to the three dimensions previously presented, in a historical manner, it is paramount to keep in mind that the chronical order of each regime is an indicator of when they emerged as new and distinct approaches of collective design (in the sense that they can be broadly observed through historical testimonies and artifact of industrial activities at a given time). Therefore, one should not assume that an earlier regime disappears at the following emerges: on the contrary, further research has shown that designer collectives regularly alternate between different regimes depending on their resources as a collective and the stability of an object's identity (here the term "object" refers to what is being designed, whether it is a product, a service, a tool, a process, etc.). It is by observing the diversity or change within these dimensions that the authors can identify four distinct types of regimes: 1) Wild Design; 2) Parametric rule-based design; 3) systematic rule-based design; and 4) Innovative design. As we will detail the characteristics of each regime, according to the three dimensions previously presented, in a historical manner, it is paramount to keep in mind that the chronical order of each regime is an indicator of when they emerged as new and distinct approaches of collective design (in the sense that they can be broadly observed through historical testimonies and artifact of industrial activities at a given time). Therefore, one should not assume that an earlier regime disappears at the following emerges: on the contrary, further research has shown that designer collectives regularly alternate between different regimes depending on their resources as a collective and the stability of an object's identity (here the term "object" refers to what is being designed, whether it is a product, a service, a tool, a process, etc.).

Wild" or Entrepreneurial Design

The first design regime can be observed since the earlier days of the Industrial Revolution in England: it was the collective design process of inventor-entrepreneurs, usually engineers with a strong business acumen who were the first industrialists to create fundamentally new objects, such as mechanical spinning machines, steam engines, machine tools, railways, steamships, etc. They fostered close communities of exchange, through learned societies such as the Lunar Society of Birmingham or the Smeatonian Society of Civil Engineers. As described by the authors, "the resources were limited to a few engineers, their initial knowledge was poor, and the learning process, primarily through the trial and error of successive generations, served as a design space and hence learning for those that followed.

In terms of expansion, there was still only a poor organisation for repeating or reproducing a particular design, and it was always just one product (and its associated process) that was designed: the designers attempted to discover the extent of its performance and gradually stabilize the object's identity." (Le Masson, Weil & Hatchuel, 2017, p.50). This type of design activity can easily be observed today, particularly in entrepreneurial startups, where founders and early team members must design everything from scratch, depend heavily on trial-and-error approaches, and often form communities of likeminded individuals for support.

Parametric Rule-Based Design: "Recipe-based" Design or Pathways for Industrial Catch-Up During the mid-19 th century, continental European countries, such as France and Germany, have taken notice of England's expanding industrial development and seek out ways to "catch up" as "one thing was certain: the process of building up a population of technical specialists and inventor-entrepreneur engineers similar to that which existed in England would take too long and seemed difficult to push forward." (ibid, p.50). The Swiss scientist and mechanical engineer Ferdinand Redtenbacher would develop an original approach known as parametric design. This approach is distinguished by 2 key steps that separate design actors into 2 distinct roles: the first actors would initially create "recipes for design" by determining all key design parameters of an object and scenarios that may require variety or adjustment of said object.

The second group of actors would use these recipes to design many kinds of machines by following the instructions and testing out methodically different variations in parameters. Through this approach, the design effort was displaced and more controlled: engineers could rapidly design many variations from one initial product, creating families of products. This made industrial firms start organizing teams of specialized designer engineers whose performance was measured by the rapid design of controlled variations on machines that could be quickly tested, and this limited the production of new knowledge and costly exploratory work to the select few who were part of the first group (those who design the recipes). Once again, such design regime is can still be observed today, in firms that come to standardize their product or service and will focus on providing variations of the same to increase market shares.

Systematic Rule-Based Design: The Invention of the Engineering Department By the beginning of the 20 th century and with the impulse of the 2 nd industrial revolution, the ruled-based design regime we previously exposed would come to evolve, to meet more demanding challenges. Indeed, "with the increase in production volumes, preoccupation with industrial efficiency became a priority. Customers (often in a B2B situation) became more competent and more demanding. Finally, the product dynamic made it essential to reconcile the new knowledge spaces: heat or electricity might be added to mechanics, for example." (ibid, p.56). It is in the United States and Germany that this new design regime took form and expanded outwards. In terms of organization, the previous logic of recipe designers and recipe users persisted but evolved as there was a major change in scale and intensity. Design actors increased in numbers and were structured into Engineering Departments and Design Bureaus (bureau d'études) and regrouped according to increasingly advanced specialties. As a result, they needed to develop several compatible design languages, that make it possible to discuss the same object at hand with different specialists and professionals within the firm. It is in Germany that a more universal 4-language model was developed, which relied on 4 dimensions to describe a design object: 1) the functional language, to express the customer's needs; 2) the conceptual language, which corresponded to engineering sciences, such as mechanics, thermodynamics, etc.; 3) the morphological language, to describe the assembly of the product, and 4) the language of detailed design, which sought to define even the most precise components of the product and their location in the manufacturing process. The expansive reasoning aimed high level improvements (optimization) and pursuing continuous incremental innovations on the product families. This led to performance criteria focused on optimizing the costs and resources of design activities and developing forecasting models or provisional tools. This design regime, defined by the advanced rationalization of product development is best embodied by the large R&D departments still present today and came to mark a new era: "the modern large enterprise was born" [START_REF] Segrestin | Refonder l'entreprise[END_REF], as cited in Le Masson, Weil & Hatchuel, 2017, p.58) Innovative Design As systematic rule-based design became the new industrial norm, "innovation in the 20 th century was not just a singular event, but was continuous, incremental, robust-powerful. It was intentional, organized, manageable and controllable." (ibid., p.125). However, as we have discussed at the beginning of this paper, the 21 st century saw the appearance of new technological computer and digital industries, and other crisis events that had the effect of destabilizing well implanted traditional industries. This has changed the strategic focus of industrial actors, as they strive to achieve an ambidextrous approach [START_REF] March | Exploration and Exploitation in Organizational Learning[END_REF] that will simultaneously help them maintain their market position and generate radical and disruptive ideas that can put them ahead of their competitors. In other words, "all modes of value formation are nowadays concerned by a logic of innovation and technology is only one of those modes. One of the major evolutions today is the transition from a problem of optimizing enterprise capacity (involving decision theory) to a problem of expanding enterprise capacity (involving design theory)" (Galarreta, 2018, p.63). Born from the theoretical work of C-K Theory [START_REF] Hatchuel | C-K Design Theory: An Advanced Formulation[END_REF][START_REF] Schoemaker | Identity in Flexible Organizations: Experiences in Dutch Organizations[END_REF], the innovative design regime seeks to offer a new design approach that is compatible with organizations structured around institutionalized systematic rule-based design activities. Indeed, a capital distinction with the previous regime is found in its novel approach to expansive reasoning: the priority of innovative design is to provide an alternative to the restrictive controlled imposed on knowledge creation and instead encourage the inclusion of new sources of knowledge, usually foreign or "exotic" to a particular community of designers. This approach aims to overcome cognitive "fixations" around a particular object, so that designers can "break" the limits of their integrated way of defining a product or service. An example of this could be to introduce automobile engineers to other forms of mobility that can help them expand the definition of what a car is and what it could be. In terms of performance criteria, there again, there is a fundamental switch to operate: "while systematic design gives value to minimizing expansions in order to attain a known objective, innovative design provides value to expansions" (Le Masson, Weil & Hatchuel, 2017, p.150). That is to say that the design process must mobilize knowledge (K) that will bring designers to propose new concepts (C) that open new avenues of exploration, avenues which would have remained closed or unknown without the injection of new knowledge. As an alternative to the CQT criteria (Cost-Quality-Time), the exploration through C-K Theory is determined with the V2OR model: Variety (of concepts discovered),

Value (of knowledge to break fixation effects), Originality (of concepts proposed) and Robustness (of knowledge mobilized). Finally, the organization of actors also takes a new form in this regime: there is 1) an incentive for design actors to engage in continuous learning processes and manage the dissemination of new knowledge, and 2) the community of design actors grows at an ecosystem level, to include new and original design partners and structure them into innovation communities. Such a design regime is best exemplified by the many cases of Open Innovation programs initiated by large firms, and their collaborations with startup incubators.

MATERIAL ANALYSIS

We used the framework of comparative design regimes, summarized in Table 1 as an analytical lens to read the changes and evolutions of HRM systems in the industrial history of France. Our aim is to seek instances of HRM design activity and see if they can be characterized according to the design regimes presented hereabove.

Starting with Fombonne's work, we intuitively chose to follow his own chronological segmentation of HRM history and identified multiple instances of design activity, embodied by collectives of HRM actors. When analyzed through the criteria of the different design regimes, we categorized them according to the 3 first ones (Wild, Recipe ruled-based and Systematic rule-based design). As mentioned, the author highlights a historical progression of the HRM function that first emerged out of necessity around the 1830s, but one which wasn't embodied by a dedicated department or profession before 1917. The second phase proposed by the author, that of a "personnel function with personnel departments", saw the appearance of multiple designations from 1950 onwards: personnel department, industrial relations office, social and human relations department, social affairs department. Then, in the 1980s, the term human resources management, which came from the United States, was translated, and widely adopted. However, this homogenization of the nomenclature did not prevent the appearance of many distinct HRM models.

Wild Design for a "Headless" HRM function: 1850s-1950s

According to Fombonne's historical accounts, for early French industrial firms in the latter part of the 19 th century and the beginning of the 20 th century, forms of organized management of employment relations were mostly burgeoning. These industrialists understood they needed to stabilize the availability of the workforce and sought to ensure the continued quality and rhythm of production. Initially, the core actors are members of the Board of Directors, who are going to directly oversee matters related to employment and payment. These entrepreneurs will gradually start to exchange managerial ideas and test out innovative tools, such as the livret ouvrier (Employment record books, p.55), the période d'essai (employment trial period, p.64), the restauration en entreprise (foodservices, p.82) or the règlements interieurs (employee by-laws, p.95). During that period, these mechanisms are quite new, and practices differ wildly. Later, some of these, like the Livret de travail will be made illegal, while others, such as the période d'essai will be codified into law. One of the first records of a Chef du Personnel (Head of Personnel, p.286) appears in 1906 at Schneider.

Wild design

The HRM function without a profession

Actors, resources, and organization

A few engineers or design actors

Poor initial knowledge

Learning by trial and error

Limited to the members of Board of Directors (Conseil d'Administration) who wish to retain control on all aspects of employment and relationship with the workforce.

Small communities of shared practices emerge, trying to align on certain recruitment processes

Expansive Reasoning

Gradually work out the dimensions and stabilize the identity of the objects Stabilize recruitment processes (integrating structured processes for entering a firm's workforce)

Increase security on the job (reducing the high number of work-related accidents)

Performance criteria

Construct a potential skill/ singular value Make the workforce more available (less likely to leave) and more dependable (ensuring they can fill a role)

HRM objects of design

Systematic design for a strategic HRM function: 1960s-1990s

With the increased internationalization of industrial activities and greater worldwide competition, French industrial firms seek future-oriented models to management human resources, on a global scale. During this phase, Fombonne details the emergence of HRM mechanisms that offer a more generic approach to organizing the workforce and provisional insights as to the possible evolutions of their talent pool.

Systematic design

The Innovative Design for an innovative HRM function: 2000s-Present Fombonne's historical analysis stopped at the end of the 1990s, but we propose to pursue his approach a little further, by including more recent historical events that shed a light of the latest evolutions in HRM practices and how its actors have adopted new design logics. We will therefore look at 2 major French industrial players that are featured prominently in Fombonne's work: Renault and Michelin. Both companies were case studies in a recent academic publication on the development of industrial ecosystems of innovation. In the case of Renault, the article details the CEO's desire to boost the firm's innovation and create a community of innovators to imagine the future of the auto-industry. To give an impulse to this strategy, the CHRO of Renault at the time, designs and organizes a Learning Expedition in North America, along with other CEOs of French firms. That expedition will mark the start of the first Renault innovation community, called the Cercle de Montréal (Circle of Montreal).

As for Michelin, the case details the creation and institutionalization of Open Innovation Challenges, by higher management and the HR departments, to regularly invite external innovators to come and enrich the Michelin products and processes.

Innovative design

The HRM function with HR Departments and CHROs

Actors, resources, and

Managing the learning process Extending the ecosystem of Innovation collectives managed by HR Departments that create ecosystems of design partners beyond the firm or its traditional industry limits.

RESULTS & CONCLUSION

We aim to obtain four new conceptual results: 1) a 4-part genealogy of the HRM function has been identified, confirming previous historical works while also offering a design-based language, better adapted to innovation management.

2) The evolution of HRM closely mirrors that of industrial innovation management over the last two centuries, which suggests that these two functions have, so far, shared a similar sensitivity to the environmental contingencies of different eras. 3) A new identification of historical stakeholders' roles in HRM innovation, one based on the nature of their involvement in the design process. 4) Certain HRM tools (such assessment forms) and processes (such as recruitment and training) have gone through several cycles of flow and stasis in the evolution of their identities, highlighting periods of varying experimentations and converging stabilization that coincide with major changes in the innovative context.

While our work will reinforce the well-established view that HRM as closely linked to innovation and transformative periods, we hope that it will provide a new framework to analyze and expand on the interactions between HRM and innovation, and therefore to better align Strategic Human Resource Management with the strategic nature of innovative activities. By looking at the design regime(s) of the firm's innovative activities, and comparing it to those of its HRM system, SHRM actors can determine whether there is a compatibility in innovative reasoning, expansion and performance. Compatibility shouldn't necessarily require an exact alignment in design regimes between the firm's core innovative activities and the way the HRM system supports it. Indeed, it is completely possible that some innovative processes that follow an innovative design pattern, such as automobile makers defining what the future of mobility can be, can be supported by updated HRM practices that are the result of a systematic rule-based design approach. On the other hand, this also opens up for a scenario where such an innovative ambition may require that SHRM actors simultaneously develop explorative HRM practices, so as to enable core design actors to, in turn, successfully explore the full diversity of conceptual proposals. This could take the form of new forms of financial incentives for employees, unprecedented career paths, original topics, and forms of training, etc. In terms of scientific research, we believe that this approach could provide the basis for a multi-level approach to innovation SHRM [START_REF] Shipton | HRM and Innovation: Looking across Levels[END_REF]. Furthermore, we see the possibility to move a step further in the ongoing debate between "Best fit" and "Best practices" [START_REF] Koster | Innovative Human Resource Management: Measurement, Determinants and Outcomes[END_REF] by incorporating them in a design driven approach, as sources of functional parameters and extended knowledge, respectively. Finally, this opens the possibility for further experimental work to test out the properties of other design theories in innovative HRM design activities, such as C-K Theory [START_REF] Le Masson | Design theory: methods and organization for innovation[END_REF], or Axiomatic Design [START_REF] Suh | Axiomatic Design Theory for Systems[END_REF].

Finally, we are certain that this will provide SHRM practitioners with the means to face increasingly unknowable situations (climate change and resource sustainability, blurring of personal and professional spaces through technological innovations and global crises, emergence of new and diverse expectations from employees and staff, etc.) and propose novel and tailored management answers.

Table 1 :

 1 Comparison of design regimes 2

		Wild design	Rule-based design Systematic design Innovative design
	Actors,	A few engineers or	Recipe (Generator	Design department	Managing the
	resources	design actors	model based on a		learning process
	and		conceptual model)	4 languages
	organization	Poor initial			Extending the
		knowledge	Distinction of	Division of labor	ecosystem of
			actors designing	and specialized	design actors and
		Learning by trial	the recipe and	skills	knowledge
		and error	those designing the		providers beyond
			product		the firm
	Expansive	Gradually work	Product family but	Product families	Expand on existing
	Reasoning	out the dimensions	the product is	(dominant	knowledge and
		and stabilize the	determined by the	design)	create new
		identity of the	generator model		definitions of
		objects		Limit innovation to	objects
				known
				performance
				Set out in ranges
				and families
	Performance	Construct a	Catch-up	Expansion	Measuring
	criteria	potential skill/		extended	expansions
		singular value	Maximize	by controlling	according to
			connectivity	learning processes	variety, originality,
			without producing		value, and
			additional	Still connective	robustness
			knowledge	(Robustness:
				minimize risk)
				Industrial system

  -Introduction of employment record books to establish employment history -Employment trial periods to assess new employees -Introduction of by-laws within factories to regulate workplace conduct -First cases of subsidized foodservices (p.309). The second part of the 20 th century sees the proliferation and institutionalization of Personal Management departments, in large part because French industrial firms wish to rebuild their competitive advantages after WWII and embrace a logic of catch up, by looking first towards the United States and later towards Japan. One example of this is the TWI method (Training Within Industry) which originated in the US army and provided a rigid but highly structured framework for training managers in what was seen as universal management attributes that all higher-level employees should possess. If its popularity is nearly immediate, the TWI gradually appears both rigid in form and limited in scope, its "one size only" does not fit all management situations. This will lead Personnel Departments to develop new HRM roles such as Training Managers, who are in charge of creating variations to the TWI method, focusing mostly on keeping the structure of these trainings and introducing a wide variety of new topics that could be better suited to different type of management professionals.

	Rule-based Design for an institutionalized HRM function: 1950s-1990s As Fombonne explains "it was in the 1960-1970 decade that the number of central personnel departments was multiplied, i.e. a personnel function attached to the general management, exercising a political role over the whole of the company and taking into in charge of new responsibilities: management of executives, training, human resources planning, part of The HRM function with a Head of Personnel Actors, resources, and organization Recipe (Generator model based on a conceptual model) Distinction of actors designing the recipe and those designing the product Gradually, most industrial firms create a role of Head of Personnel (Chef du Personnel), charged with overseeing matters of employee management and labor relations, and creating necessary mechanisms Secondary actors, such as training managers and facilitators create and deliver training programs and workshop sessions Expansive Reasoning Product family but the product is determined by the generator model Apply TWI training "recipes" to various topics of professional training Performance criteria Catch-up Maximize connectivity without producing additional knowledge Quickly adopt managerial innovations originating from the US and ensuring mass implementation throughout the firm or targeted populations of employees (such as managers) Maintain technological advances by disseminating regularly key technical knowledge within the firm. Identify opportunities for improved efficiency / lower cost HRM objects of design -The TWI model and its many variations -Institutionalization of continuing education processes in firms -T-group workshops and their many variations collective relations with staff representation." 3 Rule-based design -Suggestion offices and internal competitions

HRM function with HR Departments and CHROs

			assessment)
	Expansive	Product families (dominant	Organizational matrix structures, combining
	Reasoning	design)	hierarchical and functional forms of authority
		Limit innovation to known	Forecasted career paths and provisional workforce
		performance	plannings
		Set out in ranges and families	Standardized assessment systems for employees and
			work performances
	Performance	Expansion extended	Transition to more generic workforce "components"
	criteria	by controlling	by switching from traditional professions to an
		learning processes	organization of the workforce by skills possessed.
		Still connective	Incorporate uncertainties into HRM models and
		(Robustness:	decisions (such as provisional workforce planning)
		minimize risk)
			Capacity to intervene at both the macro (strategic)
		Industrial system	and micro (individual and operational) levels, on
			large scales, and often in international contexts
	HRM	-Strategic workforce planning tools and processes (provisional models, economic
	objects of	forecast reports)
	design
	Actors,	Design department	Structured HR departments with increasing levels of
	resources,		decentralized representation
	and	4 languages
	organization		Increasing multiplicity of technical and legal
		Division of labor and	languages
		specialized skills
			Division of HRM work in 5 core specialties
			(recruitment, training, careers, sanctions,

-Assessment tools and processes (assessment centers, performance evaluation charts)

« Personnel et DRH : l'affirmation de la fonction Personnel dans les entreprises (France, 1830-1990) -Translation from the authors.

Adapted from an earlier (partial) model from LeMasson, Weil & Hatchuel (2017, p.51)

« C'est dans la décennie 1960-1970 ont été multipliées les directions centrales de personnel, c'est à dire une fonction Personnel rattachée à la direction générale, exerçant un rôle politique sur l'ensemble de l'entreprise et prenant en charge des responsabilités nouvelles : gestion des cadres, formation, planification des ressources humaines, une partie des relations collectives avec la représentation du personnel.» (p.309)


Moussa Labbadi

Olivier Sename

Nonsingular Terminal Sliding-mode Lateral Control of an Autonomous Vehicle with Global Fixed-time Stability Guarantees

Keywords: Sliding mode control, Fixed-time stability, Autonomous vehicle, Lateral control, Automotive control

The present paper develops an output feedback fixed-time sliding mode control (FxTSMC) approach for lateral trajectory tracking of an autonomous vehicle subject to disturbances. Firstly, a nonsingular terminal sliding manifold is constructed to force the orientation error and lateral deviation to converge to desired values in finite time. Secondly, using variable exponent coefficients, a switching control law is proposed to provide fixed-time stability against external disturbances. By adjusting the control design parameters, the upper bound of the settling time for the closed-loop lateral system can be attained arbitrarily minimal without depending on the systems initial conditions. The effectiveness of FxTSMC is demonstrated by comparing the performance of closed-loop systems of the supplied controller and other existing controllers.

I. INTRODUCTION

Lateral control of autonomous vehicles mainly consists in following a predefined trajectory, given by a path planning algorithm. By adjusting the steering actuator, the lateral error is minimized, and the lane-keeping/lane-changing objectives are achieved. Many papers have investigated and focused on the lateral dynamics, see the survey papers [START_REF] Paden | A survey of motion planning and control techniques for self-driving urban vehicles[END_REF], [START_REF] Rokonuzzaman | Review and performance evaluation of path tracking controllers of autonomous vehicles[END_REF].

Different control approaches have been applied to the lateral control problem. In [START_REF] Atoui | LPV-Based Autonomous Vehicle Lateral Controllers: A Comparative Analysis[END_REF], a comparative study of Linear Parameter Varying controllers was presented. The authors of [START_REF] Yang | An expert fuzzy controller for vehicle lateral control[END_REF] developed a fuzzy controller for lateral dynamics. In addition, robust controllers have been used, such as H ∞ control [START_REF] Huang | Robust weighted gain-Scheduling H∞ vehicle lateral motion control with considerations of steering system backlash-type hysteresis[END_REF], and a feedback controller [START_REF] Jiang | Lateral Control of an Autonomous Vehicle[END_REF].

It is commonly known that sliding mode control (SMC) motion introduced by Utkin [START_REF] Utkin | Variable structure systems with sliding modes[END_REF], is usually robust with respect to disturbances and uncertainty. Therefore, the SMC approach may effectively suppress the lumped disturbances by adding a switching control law to the control input. The SMC approach ensures asymptotic convergence of state variables. Some works have been concerned with additional requirements as finite-time stability. Bhat and Bernstein first established finite-time stability in [START_REF] Bhat | Finite-time stability of continuous au-tonomous systems[END_REF] before applying it to finite-time stabilization, as seen in [START_REF] Hong | Finite-time stabilization and stabilizability of a class of control-lable systems[END_REF] among other works. Depending on the initial conditions, it makes sure that dynamical systems attain their equilibrium in a finite amount of time, known as settling time. Polyakov established the fixed-time stability in [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF]. Fixed-time stability guarantees that the settling time is independent of the beginning circumstances in addition to finite-time stability. For engineering 1 Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, 38000 Grenoble, France moussa.labbadi,olivier.sename@grenoble-inp.fr applications, fixed-time stabilization offers a predetermined convergence period toward the equilibrium. SMC has been used for autonomous vehicles in several works [START_REF] Sun | Path-tracking control for autonomous vehicles using double-hidden-layer output feedback neural network fast nonsingular terminal sliding mode[END_REF]- [START_REF] Wang | Robust adaptive sliding mode control for nonlinear four-wheel steering autonomous Vehicles path tracking systems[END_REF]. For example, the tracking performance of autonomous vehicle systems has been enhanced by the structural innovation and widespread application of SMC [START_REF] Hajjami | Neural Network Based Sliding Mode Lateral Control For Autonomous Vehicle[END_REF], [START_REF] Talj | Immersion and invariance control for lateral dynamics of autonomous vehicles, with experimental validation[END_REF]. The finite-time control using a nonsingular terminal sliding mode controller was studied in [START_REF] Sun | Path-tracking control for autonomous vehicles using double-hidden-layer output feedback neural network fast nonsingular terminal sliding mode[END_REF] for the lateral control of autonomous vehicles.

Based on the above works and inspired by finite-time (FnT) and fixed-time (FxT) controllers, a fixed-time controller combined with a nonsingular terminal sliding mode variable is proposed for the first time in this paper for the lateral dynamics of autonomous vehicles. Using a nonsingular terminal sliding mode variable, the lateral deviation and orientation error converging to zero in a finite time. The upper bound of disturbances affecting vehicle dynamics is coped with the fixed-time switching law. The latter used the sliding variable exponent coefficients. The proposed controller ensures for lateral dynamics time-independent settling time, nonsingularity, is easy to implement, and robust against disturbances. The main contributions of this work are:

• The finite-time convergence of the lateral and orientation errors, is ensured based on a nonsingular terminal sliding variable. • A switching control that has a sliding variable exponent coefficient is synthesised to ensure the fixed-time stability of the closed-loop lateral system. • Several nonlinear simulations under complex situations are given and a comparative study is presented with the proposed control method. The outline of the paper is as follows. The vehicle modeling and problem formulation are given in Section II. Preliminary theorems on finite/fixed time are presented in Section IV. Section II. Section IV is dedicated to designing a nonsingular terminal sliding-mode control of autonomous vehicles with global fixed-time stability guarantees. The nonlinear simulations with some scenarios are given in Section V. The last Section V concludes the paper.

II. VEHICLE MODELING AND PROBLEM FORMULATION

Assumption 1: The longitudinal speed v x is assumed be constant and strictly positive. Under the assumptions presented in [START_REF] Jiang | Lateral Control of an Autonomous Vehicle[END_REF], [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF], Assumption 1 and using the well-known Newton's Second Law for motion along the lateral axis, the equations for the yaw dynamics are produced by the moment balance about the vertical axis [6], [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF].

vy = -v x r + 1 m (F yf + F yr + F wy ) ṙ = 1 I z (l f F yf -l r F yr + l w F wy ) (1)
where F yf and F yr are respectively the front and rear wheels' lateral tyre forces. r is the yaw rate, v x represents the longitudinal speed. m is the total mass. I z stands for the vehicle's moment of inertia about the yaw axis, and l f and l r stand for the front and rear tires' respective distances from the mass center of the vehicle. The aerodynamic and tire/road interaction forces are the primary external forces acting on the vehicle while it is moving. While the tire's lateral slip forces F yi are approximated by linear stiffness with respect to the tire slip angles, the side wind force F wy is considered to be an external perturbation in this situation. Consequently, the state representation of the dynamic of the lateral vehicle is provided by

vy ṙ =           f 1 -2µ C f + Cr vxm f 2 2µ -C f l f + lrCr vxm -vx 2µ -l f C f + lrCr vxIz f 3 -2µ C f l 2 f + l 2 r Cr vxIz f 4           vy r +         g 1 2µ C f m µlf C f Iz g 2         δ + 1 lw Fwy (2)
All parameters presented in (2) are defined in Table I. These parameters correspond to an autonomous robot. The robot is from the SOBEN company of a small four wheels "ciTHy M" range (see Fig. 1). This autonomous robot is used for meal distribution and home delivery.

The vehicle positioning factors with regard to the traffic lanes must be understood in order to create a controller for a driving aid system. In order to minimize position and orientation errors in relation to the planned trajectory, it is required to maintain careful tracking of the reference trajectories provided by a planning module. This implies that the heading error ψ L and the lateral error y L should converge in finite-time to zero and that the vehicle axis should always be parallel to the planned trajectory tangent axis and the vehicle lateral deviation always be close to the

ψL =r -ρv x ẏL =v y + ψ L v x + rl s (3)
where ρ denotes the curvature of the reference trajectory (lane curvature). Note that the graphical definitions of the variables are illustrated in Fig. 2. The following formulation ls represents the control problem for lateral motion to find a robust controller such that the closed-loop system has the following characteristics given the system (2)-( 3) and the time history of the feasible road curvature ρ(t). P 1 ) The control effort δ(t) is bounded and satisfying δ(t) ≤ δ max and | δ(t)| ≤ v max . P 2 ) the tracking errors converge to zero in finite-time and the state variables v y (t) and r(t) converge to their references v yr (t) and r r (t) in short finite-time. P 3 ) the closed-loop system (2), (3) with the designed controller is fixed-time stable.

III. PRELIMINARIES ON FINITE/FIXED TIME STABILITY

Some definitions of finite/fixed-time stability are recalled. Consider the following system.

ż(t) =F(z(t)), z(t) ∈ R n (4a) z(0) =z0, F(0)) = 0.
(4b)

In the system (4a), F(z(t)) is a continuous function. Definition 1: Refer to [START_REF] Bhat | Finite-time stability of continuous au-tonomous systems[END_REF], [START_REF] Hong | Finite-time stabilization and stabilizability of a class of control-lable systems[END_REF] and consider the system (4a) and its candidate Lyapunov function (CLF) V (z(t)) with V (z(0)) = V (z 0 ) denotes the initial condition of CLF then the system (4a) is globally FnT stable if it is CLF stable for all z 0 ∈ R n there exists the settling time satisfying T (z 0 ) ≥ 0. T (z 0 ) depends on initial conditions Definition 2: [10] System (4a) is globally FxT stable if:

• (4a) is global FnT stable.

• T (z 0 ) ≤ T , which T is a constant for a given z 0 . Lemma 1: Consider the CLF as:

V (z) ≤ -β1V (z) α 1 -β2V (z) α 2 (5)
with β 1 , β 2 > 0 and 0 < α 1 < 1 < α 2 , then (4a) is global FxT stable and the settling time satisfies

T (z0) ≤ 1 β1(1 -α1) + 1 β2(1 -α2) (6)

IV. FIXED-TIME LATERAL CONTROL WITH NONSINGULAR TERMINAL SLIDING MODE FOR AUTONOMOUS VEHICLES

In order to design the lateral controller, the total system is viewed as an interconnected system. There is little doubt that the "Lateral Dynamics of the Vehicle" is the primary subsystem of the open-loop system, which also includes the "Lane Keeping Dynamics" and the "Steering System". The main results of this paper are addressed in this section. Consider the error surface to reduce lane tracking position and orientation errors as:

e = c 1 l p ψ L + c 2 y L , (7)
To balance the impact of the lateral deviation error and the heading error on the convergence of the sliding surface, c i > 0, i = 1, 2 are design parameters that need to be adjusted. These sliding surface gains have been suggested in SMC literature [START_REF] Utkin | Variable structure systems with sliding modes[END_REF] for the design of the surface gains and are simple to assure sliding surface convergence. Given that the system's relative degree to the error under consideration is 2, a linear sliding surface can be defined as:

s = ė + β 1 e, (8)
By setting the time derivative of s as

ṡ = ë + β 1 ė, (9)
and another hand, the dynamic of s is chosen as,

ṡ = - K {(c 1 l p + c 2 l s )g 2 + c 2 g 1 } sign(s), (10)
The double derivative of ( 7) is,

ë ={(c 1 l p + c 2 l s )g 2 + c 2 g 1 }δ + (c 1 l p + c 2 l s )(f 3 v y + f 4 r) + c 2 (f 1 v y + f 2 r) -c 1 l p ρv x + c 2 ψL v x , (11)
From ( 9), [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF], and [START_REF] Sun | Path-tracking control for autonomous vehicles using double-hidden-layer output feedback neural network fast nonsingular terminal sliding mode[END_REF], the control law can be given:

δ = - 1 {(c1lp + c2ls)g2 + c2g1} [(c1lp + c2ls)(f3vy + f4r) + c2(f1vy + f2r) -c1lp ρvx + c2 ψLvx + β1 ė + Ksign(s)] (12)
Proposition 1: 3), ( 7), ( 8) and ( 12) is globally asymptotically stable and the tracking errors (y L , ψ L ) are converging to zero.

If K > d ≥ |c 2 F wy + l w F wy (c 1 l p + c 2 l s )|, the closed-loop system (1), (
Proof: Consider the Lyapunov function V = 1 2 s 2 . We can prove that:

V =s(-Ksign(s) + c2Fwy + lwFwy(c1lp + c2ls))

≤ -(K -d)|s| ≤0. (13)
Note that ė = -β 1 e when the sliding surface is reached, ensuring the closed loop system's (1), ( 8), [START_REF] Utkin | Variable structure systems with sliding modes[END_REF], and ( 12) asymptotic stability in the direction of the origin. In the next part, a non-singular terminal sliding mode manifold is presented for the system to ensure the finite-time convergence of both tracking errors. Lemma 2 ( [START_REF] Feng | Chattering free full-order sliding-mode control[END_REF]): Consider the sliding variable

s = ë + κ 2 | ė| µ2 sign( ė) + κ 1 |e| µ1 sign(e), (14)
where µ j and κ j (j = 1, 2) are positive constants such that the polynomial p 2 + κ 2 p + κ 1 is Hurwitz and

µ 1 = µ 0 , µ 2 = µ1µ2 2µ2-µ1 . (15)
Once the sliding mode is established (i.e., s = 0), the system state converges to zero in FnT. Assumption 2: Define the disturbances

d(t) = c 2 F wy + l w F wy (c 1 l p + c 2 l s ) (16)
and assuming its derivative as ḋ(t) ≤ d 1 with d 1 > 0.

The equivalent control law can be obtained by setting s = 0 and d(t) = 0. Thus, one has

δ eq = - 1 {(c 1 l p + c 2 l s )g 2 + c 2 g 1 } [c 2 (f 1 v y + f 2 r) + (c 1 l p + c 2 l s )(f 3 v y + f 4 r) -c 1 l p ρv x + c 2 ψL v x + κ 2 | ė| µ2 sign( ė) + κ 1 |e| µ1 sign(e)] (17)
Remark 1: In [START_REF] Hong | Finite-time control for robot manipulators[END_REF] when s = 0 in [START_REF] Lee | Predictive control with sliding mode for autonomous driving vehicle lateral maneuvering[END_REF], it has been proved that the system state converges to zero in finite time. The same for the lateral control of (1), given the parameters µ j and κ j then both tracking errors (y L , ψ L ) converge to zero in a finite-time referred to as T 1 .

Employing the non-singular terminal sliding-dependent variable power coefficient in the reaching control law to address the disturbances problem and to ensure the fixedtime stability of the lateral system (2). Based on the work developed in [START_REF] Moulay | Robust Fixed-Time Stability: Application to Sliding-Mode Control[END_REF], the fixed-time controller is given by:

δs = - 1 {(c 1 l p + c 2 l s )g 2 + c 2 g 1 } [h|s| λs 2 1+ηs 2 sign(s)] (18)
with η, λ ∈ R * + such as a = λ 1+η > 1, and h > d 1 e

-λ 4 √ π . Then, the steering controller is resumed as:

δ =δ eq + δ s = - 1 {(c 1 l p + c 2 l s )g 2 + c 2 g 1 } [c 2 (f 1 v y + f 2 r) + (c 1 l p + c 2 l s )(f 3 v y + f 4 r) -c 1 l p ρv x + c 2 ψL v x + κ 2 | ė| µ2 sign( ė) + κ 1 |e| µ1 sign(e) h (|s| λs 2 1+ηs 2 sign(s))dτ ] (19)
Theorem 1: Consider the vehicle lateral model in ( 1)-(3) where d(t)is defined as [START_REF] Ma | A Path Following Scheme using Sliding Mode Prediction Control for Autonomous Vehicle with Uncertainty Estimation[END_REF] and satisfies the Assumption 2, then the closed-loop lateral extended system (1), ( 3), ( 7), ( 8), [START_REF] Lee | Predictive control with sliding mode for autonomous driving vehicle lateral maneuvering[END_REF], and ( 19) is globally finite-time stable and the settling time satisfies

T g ≤ T 1 + 1 2 α (h -d 1 )(α -1) + 2 1 2 (he -λ 2e -d 1 ) (20)
However, the variables y L and ψ L converge to zero in an FnT T g . Proof: From the system (2) and ( 3), the nonsingular terminal sliding mode variable can be rewritten as:

s ={(c 1 l p + c 2 l s )g 2 + c 2 g 1 }δ + (c 1 l p + c 2 l s )(f 3 v y + f 4 r) + c 2 (f 1 v y + f 2 r) -c 1 l p ρv x + c 2 ψL v x + κ 2 | ė| µ2 sign( ė) + κ 1 |e| µ1 sign(e) (21)
Using ( 17) into ( 21) results in:

s = -h (|s| λs 2 1+ηs 2 sign(s))dτ + d(t) (22)
Consider now the subsequent quadratic Lyapunov function and its time derivative as:

V = 0.5s 2 , V = s ṡ. ( 23
)
with ṡ = -h|s| (25)

Therefore, the proposed fixed-time stability in [START_REF] Moulay | Robust Fixed-Time Stability: Application to Sliding-Mode Control[END_REF] depends on the disturbance d(t). However, the stability analysis reported in (24) depends on the derivative of disturbances ḋ(t). Similar to the work developed in [START_REF] Moulay | Robust Fixed-Time Stability: Application to Sliding-Mode Control[END_REF], we have two cases:

• When V (s) ≥ 1, then one has λs 2 1+ηs 2 +1 ≤ λ 1+η +1 > 2. Define α = λ 1+η + 1. One get V ≤ -(h -d 1 )|s| λ 1+η +1 ≤ -(h -d 1 )2 α+1 2 V α+1 2 (26) As h -d 1 > 0 and α+1 2 > 1.
Based on Lemma 2 and [START_REF] Moulay | Robust Fixed-Time Stability: Application to Sliding-Mode Control[END_REF], the solution of (26) reaches in fixed-time T 2 = 1 2 α (h-d1)(α-1) .

• Consider the second case when V (s) ≤ 1, from the result of theorem 1 in [START_REF] Moulay | Robust Fixed-Time Stability: Application to Sliding-Mode Control[END_REF], we have V ≤ -(he

-λ 2e -d 1 )2 1 2 V 1 2 . ( 27
)
As h -d 1 > 0 and based on Lemma 2, the solution of (27) reaches in fixed-time

T 3 ≤ 2 1 2
(he

-λ 2e -d1)
.

Finally, system (2) reaches the origin in a fixed time T s ≤ T 2 + T 3 and the global system including the tracking errors (3) and variable sliding [START_REF] Lee | Predictive control with sliding mode for autonomous driving vehicle lateral maneuvering[END_REF] reaches in finite-time T g ≤ T 1 + T 2 + T 3 .

V. SIMULATION RESULTS AND ANALYSIS

A nonlinear vehicle dynamics model is used in the simulation to evaluate the performances of the previously presented methodologies. This model enables the simulation of 7DOF: yaw angle, longitudinal and lateral displacements, and the four-wheel angular velocities to see then with the tire longitudinal slip. To show the effectiveness of the developed FxTSMC with nonsingular terminal sliding manifold, two case studies are presented: uniform circular motion and a realities trajectory from real data. To demonstrate the interest of the proposed FxTSMC, the classical SMC and backstepping controllers are used for the comparison [START_REF] Norouzi | Lateral control of an autonomous vehicle using integrated backstepping and sliding mode controller[END_REF]. The resistive wind force is represented by the external disturbance in all of the aforementioned circumstances by the equation

F wy = c y v 2
x . The control parameters are: µ 1 = 9 16 , µ 2 = 9 23 , κ 1 = 192, κ 2 = 72, h = 2, and η = 0.1. It should be noted that in both scenarios, the car is driven at a constant forward speed of v x = 8m/s. Additionally, Table I provides the parameter values for the 7DOF vehicle in each simulation.

A. Case 1: Uniform Circular Motion

The curvature of trajectory in this case is assumed constant as ρ(t) = 0.02. In other words, the vehicle following a consistent circular motion. Be aware that in the case study, we make the assumption that the reference trajectory abruptly transforms at t = 0 into a circle with a radius of 50m. The performance of the FxTSMC, backstepping, and SMC techniques as represented in Fig. 3. It is clear that the proposed FxTSMC ( 19) is capable of controlling the system because all four crucial variables related to the vehicle's lateral dynamics converge to reference values. Backstepping and SMC techniques are not capable enough to accurately track the desired values of the system in this situation. Additionally, while the path's curvature is consistent, the vehicle equipped with either controller can follow the reference path precisely. However, it is evident by comparing the simulation results for the variables ψ L , y L , r, and v y that the closed-loop system's response time with the provided FxTSMC controller is far faster than that of the other controllers. In comparison to the closed-loop system with backstepping and SMC approaches, the lateral deviation is almost negligible with the FxTSMC within 0.5 seconds. Additionally, by using the FxTSMC, the maximum lateral deviation is lowered from 0.2m (backstepping) and -0.1m (SMC) to 0.02m. The steering input control provided by the .

SMC have a chattering problem and have high amplitudes as presented in Fig. 4. The xy trajectory is plotted in Fig. 5, which the three controllers track the desired trajectory.

B. Case 2: Curvature for a real trajectory

In this case, the reference of the curvature is presented in Fig. 6. Figure 7 is depicted the four variables y L , ψ L , r, and v y . The results of this scenario confirm the above three cases, in which the backstepping approach provided a high deviation in the error lateral motion and the other three variables are similar with more some performances on the FxTSMC. It is clear that the proposed FxTSMC provided a good settling time for a closed-loop system for all scenarios than the backstepping and SMC approaches. For example, the settling-time in case 1 satisfying T g < 0.4 seconds, which demonstrates the results of Theorem 1. In other words, the lateral error converges to the desired value in a finite-time and has a faster response compared to other controllers. Figures 9 and8 show respectively the input control and the xy trajectory of three controllers in the case 2.

We note that the closed-loop system with the fixed-time settles in less than 0.3s regardless of the reference by comparing the settling times for the three scenarios. In other words, regardless of the variance in the variable ρ, the lateral deviation and heading error tend to zero, which coincide requires the vehicle to operate and these results are similar to some LPV approaches recently published as [START_REF] Atoui | LPV-Based Autonomous Vehicle Lateral Controllers: A Comparative Analysis[END_REF], [START_REF] Kapsalis | A reduced LPV polytopic look-ahead steering controller for autonomous vehicles[END_REF].

VI. CONCLUSIONS

A solution to the fixed-time stability of the closed-loop lateral system of an autonomous vehicle has been proposed. The lateral controller solution is based on both a fixedtime switching approach that provided fixed-time stability against disturbances/external factors and a nonsingular terminal sliding mode manifold that ensured the finite-time convergence of the lateral deviation and the orientation error.

Fig. 1 .

 1 Fig. 1. Autonomous Robot ciTHy M (https://www.twinswheel.fr/).

  vy, vx lateral and longitudinal velocities δ, ψ vehicle steering and yaw angles y L , ψ L lateral and heading error ρ, µ curvature and road friction m, Iz mass and z inertia (160kg, 40kg.m 2 ) C f , Cr tires stiffness (6e3, 5e5)N/rad l f , lr wheelbase (0.8, 0.7)m lw, ls preview distance (0.4, 0.5)mnearest trajectory point. The dynamics of both variables used in this paper are[START_REF] Kapsalis | A reduced LPV polytopic look-ahead steering controller for autonomous vehicles[END_REF]

Fig. 2 .

 2 Fig. 2. Graphical definitions of variables y L and ψ L in lane keeping cases;

λs 2 1+ηs 2 2 1+ηs 2 Remark 2 : 2 1+ηs 2

 2222222 sign(s) + ḋ(t). One obtains V = -sh|s| λs sign(s) + s ḋ(t) The fixed-time controller proposed by Moulay et al. [20] can be presented as V = -2sh|s| λs sign(s) + 2sd(t).

Fig. 3 .

 3 Fig. 3. Case 1: Illustration of the lateral deviation y L , heading error ψ L , yaw rate r and lateral speed vy: (solid blue) SMC, (solid green) backstepping, and the proposed controller (solid red).

Fig. 4 .

 4 Fig. 4. Case 1: Illustration of the steering control: (solid blue) SMC, (dashed green) backstepping, and the proposed controller (dashed red).

Fig. 5 .

 5 Fig. 5. Case 1: Illustration of the trajectory: (solid blue) SMC, (dashed green) backstepping, and the proposed controller (dashed red)

Fig. 6 .Fig. 7 .

 67 Fig. 6. Case 2: Road map curvature.

Fig. 8 .

 8 Fig. 8. Case 2: Illustration of the steering control: (solid blue) SMC, (dashed green) backstepping, and the proposed controller (dashed red).

Fig. 9 .

 9 Fig. 9. Case 2: Illustration of the trajectory: (solid blue) SMC, (dashed green) backstepping, and the proposed controller (dashed red).

TABLE I SYSTEM

 I ABBREVIATIONS, NOTATION, AND PARAMETERS.

ACKNOWLEDGMENT

The authors gratefully acknowledges the financial support from the ANR (French National Research Agency) through the project by FRANCE RELANCE Grenoble INP-SOBEN.


Eric Kvaalen

A FASTER BROYDEN METHOD

Keywords: Broyden method, systems of equations, roots, quasi-Newton methods

A variation of the Broyden update is found to require less operations and to work as well as the usual Broyden update.

Introduction.

In C. G. Broyden's 1965 paper "A Class of Methods for Solving Nonlinear Simultaneous Equations," [START_REF] Broyden | A class of methods for solving nonlinear simultaneous equations[END_REF] two such methods were given. Method One has become the accepted "Broyden method," whereas Method Two was not discussed further because Broyden found that "success�ve step vectors ... were identical and further progress became impossible. " The aim of the present note is to indicate that this conclusion was apparently invalid. Broyden's second method has been tried on several types of problems (including those given by Broyden) and has encountered no such difficulty. In fact, the second method has at least one advantage over the usual Broyden method in that it requires less operations to do the update. A shortcut is also suggested which can reduce the computation to about half of the usual method. The number of iterations needed to converge may be different from Method One, but in comparisons there has usually been no difference. Broyden's second method has been used successfully in the past for solving systems of ordinary differential equations by Brown et al. [START_REF] Brown | Experiments with quasi-Newton methods in solving stiff ODE systems[END_REF].

2. Broyden's two methods.

Our problem is to find a vector x which satisfies

(2.1) f(x) = 0
where f is a vector-valued function. In both Broyden methods, the ith approxi mation of the solution, X ;, is replaced by a better one, namely

(2.2)
where f; = f(x;), H; is the negative inverse of an approximation B; of the Jacobian matrix of the function evaluated near the solution or near x;. Then if t; is equal to one, and the approximations are good, the procedure will converge rapidly to the solution. After each iteration, the matrix H; is updated to H; + 1 to make it a better approximation of the negative inverse of the Jacobian. Method One and Method Two differ in how this update is done.

In both methods, H; + 1 satisfies two conditions. First, it is given by the difference of H; and a rank-one matrix, that is, one that can be written as the outer product of two vectors:

(2.3) (in which u and v actually depend on i). Second, H; + 1 should express the relationship between the change in x and the change in f. That is, or From this it is seen that v can be almost arbitrary (it cannot be orthogonal to Af) and u will then depend on v. In Method One, v is chosen to be Hr Ax while for Method Two it is chosen to be Af.

(2. 4a) (2.4b) H;+1(f;+1 -f; ) = X ; -X ;+ i.
From these formulae one can see that the number of multiplications and additions involved in the calculation of the update for Method One with n variables is about 3n2, whereas in Method Two it is only about 2n2• At this point, a comment should be made regarding the implementation of either of these methods. The approximately n2 multiplications and additions normally needed for finding the next value of x by Equation 2.2 can actually be avoided. First note that the term H;Af needed in the calculation of u can be calculated as H;f;+ 1 -H;f;, the latter part of which has already been found. If we save H;f;+ i. we can find H;+ 1f;+ 1 (2.6) Equation 2.6 involves only about 2n multiplications and additions. In Method One without Equation 2.6, the number of multiplications and addi tions is of order 4n2, whereas in Method Two with Equation 2.6, it is of order 2n2 per iteration. The use of these two improvements thus halves this basic amount of computation. However, if n is small or if much time is needed in the evaluation off, then this improvement is of minor importance.

Test cases.

Broyden gives the results of ten sample problems, but explicitly states only six of them. Each of these six has been retried with Method Two and found to converge in the same number of iterations as with Method One, except Case 10 which does not converge with either method due to a poor initial guess. The method proposed here was faster than the usual method in each case, ranging from 6 % faster for a problem of two variables, to 70% faster for a problem of twenty variables. In both methods the initial Jacobian was estimated by finite differences.

As a further test, the problems given by More and Cosnard [START_REF] More | Numerical solution of nonlinear equations[END_REF] were tried. In every case in which Method One was able to solve the problem without adjusting the step size or restarting, Method Two was also able to do so in the same number of iterations. For those problems that could not be solved so easily, a programme was used which would cut the step size in half whenever the original step caused the norm of the function to increase, and which also recalculated the Jacobian when "in trouble. " This algorithm succeeded in solving more of the problems, using both methods, without much difference in number of iterations. Only in one problem ("Brown's almost linear problem"), with starting point very far from the solution, did Method Two fail (due to floating point overflow or singular Jacobian), while Method One succeeded after many iterations. But with the original starting point, Method Two succeeded, and with less iterations.

The invariance of Broyden's second method with linear transformation.

There is a further property of Broyden's second method, having to do with the scaling of the variables, which may or may not be considered an advantage over the first method. In Method One, if one transforms the problem by performing a linear transformation on the independent variables, then the iterations will not follow the same path as before. However, with Method Two, the path will be the same (but transformed). It should be pointed out that Method One has the same property with respect to linear transformation of the function.

Conclusions.

Broyden's second method, coupled with an improved algorithm, seems to have advantages over his first method, with no apparent disadvantages. In short, Broyden's second method should be given a second chance.

  This gives, with (2.3), u(vT Af) =Ax+ H;Af or (2.5) u =(Ax + H;Af)/(vT Af).

Acknowledgement.

I would like to thank C. G. Broyden for his interest and for conversations and correspondence regarding this note. Thanks also to Mordochai Shacham for his suggestions.