Charles Girard
email: charles.girard@univ-lyon3.fr

Making democratic contestation possible: public deliberation and mass media regulation

Keywords: contestatory democracy, deliberative democracy, contestation, public sphere, mass media, regulation, marketplace of ideas

Making democratic contestation possible: public deliberation and mass media regulation

While recent democratic theory has insisted upon the contestability of public policies as a key source of democratic legitimacy, the role mass media should play in fostering public contestation has yet to be specified. The role of deliberative theory in particular should be to try to identify which standards mass communication needs to meet in order to provide a general forum for democratic contestation. This article focuses on the theoretical foundations of such a project.

It puts forward three claims. First, the contestability of public policy can only be a source of democratic legitimacy if citizens have the opportunity to participate in public contestation through -among other forums -a general public forum that is endowed with a deliberative dimension. Second, media pluralism alone is insufficient in providing such a public forum: both external standards (related to the independence of mass media) and internal standards (related to the norms guiding media practices) need to be satisfied. Third, and as a consequence of the preceding two, the democratic contestability of public policies presupposes effective mass media regulation that goes beyond the promotion of media pluralism. Democratic contestation requires more than a "marketplace of ideas".

evolution of democratic regimes, namely, the increasing degree to which political discourses and decisions may be contested, not only by parties, unions or lobbyinggroups, but also by citizens, associations, and social movements. In formal and informal arenas, in local meetings or online forums, citizens have growing opportunities to participate in the public evaluation of public policies, and they sometimes manage to compel public authorities to revise them.

While deliberative democrats have especially insisted on the way opportunities for contestation provide "grist for deliberative engagement" (Dryzek 2006, vii), the role the mass media should play in the structuring of a contestatory public sphere has yet to be convincingly specified. Since Habermas' seminal work on the public sphere [START_REF] Habermas | The Structural transformation of the Public Sphere[END_REF], political theorists have tried to identify the conditions under which public communication could at the same time favor the grounding of public opinion on citizens' discursive interactions within civil society, and ensure that public opinion efficiently contest public policies when needed (Habermas 1992;Calhoun 1992;Gripsrud and al. 2011). While normative media studies offer various accounts of what role mass communication should play in the public sphere [START_REF] Dahlgren | Media and Political Engagement: Citizens, Communication, and Democracy[END_REF][START_REF] Blumler | The Internet and Democratic Citizenship. Theory, Practice, Policy[END_REF], normative political theory has mostly stayed at a distance, focusing on face-to-face or online deliberations rather than on the enabling role of mass media in the context of deliberative democracy [START_REF] Chambers | Deliberation and Mass Democracy[END_REF]. As a consequence, productive encounters between the two fields have been relatively limited (for exceptions, see Chambers and Costain 2000;Maia 2012).

The recent "systemic" turn in deliberative democracy [START_REF] Dryzek | Foundations and Frontiers of Deliberative Governance[END_REF][START_REF] Parkinson | Deliberative Systems: Deliberative Democracy at the Large Scale[END_REF] has nonetheless underscored the decisive role of mass media as a transmitter that connect multiple local arenas and sub-publics (scattered throughout the democratic system), and provide a platform within which public issues can be discussed and publicly criticized at the large scale. Analyses grounded in deliberative theory tend to focus especially on the role mass media can play either as broadcasters of smallgroup deliberations [START_REF] Parkinson | Rickety Bridges: Using the Media in Deliberative Democracy[END_REF] or of televised debates [START_REF] Coleman | Debate on Television: The Spectacle of Deliberation[END_REF]. But mass media also constitute an overarching public stage, on which a small set of discourses may gain maximal public visibility and thus contribute to shaping citizens' political opinions, and potentially influence political decision-makers. Normative political theory thus needs to define which standards mass media should meet if they are to provide a forum for the deliberative contestation of public policies. This article focuses on the theoretical foundations of such a project. While concerned with the importance of media regulation for deliberative democracy, it does not deal specifically with the difficulties associated with the rise of new media and the challenges the new media landscape creates for democracy [START_REF] Keane | Democracy and Media Decadence[END_REF].

I put forward three main claims. First, the contestability of public policy can only be a source of democratic legitimacy if citizens have the opportunity to participate in public contestation through -among other forums -a general public forum that is endowed with a deliberative dimension. Second, media pluralism alone is insufficient in providing such a public forum: both external standards (those related to the independence and inclusiveness of mass media) and internal standards, (those related to the norms guiding media practices) need to be satisfied. Third, and as a consequence, the democratic contestability of public policies presupposes effective mass media regulation that goes beyond the promotion of media pluralism.

To make this case, I draw not only on recent contributions to deliberative theory, but also on other works linking the idea of democratic contestation to public deliberation. In the first section, I offer a critical take on "contestatory democracy," as developed in Philip Pettit's neo-republican account. While this model explains why genuine contestability requires deliberation, as well as inclusion and responsiveness, it excessively narrows the scope of citizens' participation. Contestation can only be made democratic if the task of identifying and sorting out competing contestatory claims is not entirely delegated to experts. There should be, in particular, a general forum in which criticisms of public policies as advanced by citizens, associations, and social movements can be publicly expressed and made accessible to all. In the second section, I consider how mass media could help create such a forum and foster the conditions for democratic contestability. In order to refute the assumption that media pluralism alone suffices to endow the general public forum with a deliberative dimension, I criticize attempts to make the "marketplace of ideas" a model for mass deliberation, focusing in particular on Benjamin Page's pluralist approach. In the concluding section, I make a deliberative case for mass media regulation, and identify regulatory resources -in journalism ethics, social criticism and media law -with which deliberative theory needs to engage to understand how democratic contestation can be made possible.

Making democracy contestatory, making contestation democratic

Why contestatory democracy?

Contestatory democracy makes public contestability the keystone of self-government.

What makes contestatory democracy appealing, according to Pettit, are the shortcomings of the majoritarian and pluralist conceptions of democracy with which it breaks. These concern mainly their inability to ensure that public policies track citizens' interests and views, and their inability to offer a convincing model of citizens' control over government.

Pluralist conceptions [START_REF] Truman | The Governmental Process[END_REF][START_REF] Dahl | A Preface to Democratic Theory[END_REF][START_REF] Stigler | The Theory of Economic Regulation[END_REF] suggest that free political competition between interest-groups can, under an appropriate institutional framework, prevent lasting forms of domination. According to this conception, the pursuit by each group of its own self-interest will spontaneously result in equilibrium, with competing interests balanced as best as possible. Interest-group pluralism renounces the idea that democratic government requires a collective intention guided by collective reason, because it assumes an "invisible hand" at work, thus conjuring some "sort of magic" (Pettit 1997, 203).

Pluralists conceptions also offer a weak interpretation of what is required for enabling citizens' control over government: the idea is that "if people are generally represented in such interest-groups, then the influence of those groups over government, proportional as it is likely to be to their numbers, can be expected to be utilitarian in its overall effect" (Pettit 2012, 250). But there is no reason to believe that the influence of the various interest-groups on the government will be based on the size of their constituencies rather than on their financial and organizational resources. Neither the tracking of citizens' interests and ideas by public policies, nor the actual influence of citizens can be secured via the free competition of uncoordinated forces. By contrast, majoritarian, or "populist," conceptions of democracy acknowledge the importance of collective action. Drawing on [START_REF] Rousseau | Rousseau:'The Social Contract' and Other Later Political Writings[END_REF], they understand political decision-making as a procedure through which the will of the people is expressed. Such a view is also problematic, however, since nothing guarantees that the "will" supposedly expressed will be rational and serve the genuine interests and ideas of the citizenry, as opposed to the particular interest of a tyrannical majority. When they fail to take into account the minorities' interests, majoritarian procedures can easily lead to the "ultimate form of arbitrariness" (Pettit 1997, 8). Moreover, while populist conceptions hail citizens' participation as the best way to ensure the people's actual control of the government, they do not offer a plausible view of citizens' motivations. To ask each citizen to adopt the general viewpoint of the people rather than her own particular viewpoint is to set an impossible task. This problem, according to Pettit, plagues any theory that sees citizens as intentionally imposing a common direction on government. "The intention-direction model fails" because it requires citizens (and politicians) to "form judgments about what the public interest requires (...) in full sensitivity to the range of relevant concerns, however altruistic", and to act on the basis of those judgments "without letting private interest of other pressures warp the exercise" in any way (Pettit 2012, 247). One does not need to suppose every man a knave, to find this view to be too demanding.

Contestatory democracy, by contrast, does not foreground spontaneous equilibrium or virtuous will: it foregrounds reason. For citizens to control government, and for their ideas and interests to be tracked by public policies, what is needed is neither pluralism alone, nor consent, but contestability: it must be possible for citizens to publicly criticize policies and to impose their revisions when needed. While the will of the people and the individual consent of each citizen cannot be at the origin of all (or most) policies in large and complex societies, all policies can be open to some form of contestation. What makes democracy different, then, is that it allows citizens to ask for amendments of the rules to which they are submitted. An individual is autonomous, not when she is the author of all her desires and belief, but when she is able to revise any given belief or desire if she has good reasons to do so. Similarly, the people are "selfruling or democratic" when "they are able to contest decisions at will and, if the contestation establishes a mismatch with their relevant interests or opinions, able to force an amendment" (Pettit 1997, 186). But can public policies be made contestable in such a way that citizens can both genuinely influence them and truly be protected against political domination?

Conditions for contestability

Three minimal conditions for contestability can be distinguished: there must be a basis for contestation, a channel for the expression of dissenting voices, and a forum where they can be met with a response. Democracy should be deliberative, inclusive and responsive.

The need for a basis offers an original justification for public deliberation. For contestation to be possible there must be something to be evaluated and criticized. This includes not only the policies that have been decided, but also the reasons supporting them and motivating their selection. If deliberation, in which participants exchange reasoned opinions on the merits of competing policies, is preferable to bargaining, in which participants exchange threats and promises, it is only insofar as deliberation produces reasons, which can still be contested after the decision has been made. In bargaining, only the fairness of the bargaining rules is open to criticism, since no justifications are required. Deliberation, on the contrary, brings out justifications that are open to contestation by anyone, since "you do not have to have any particular weight or power, at least not in principle, in order to be able to mount a reasonable challenge to a reasoned decision" (Pettit 1997, 188).

Citizens also need a channel through which they can make their voice heard. It must be possible for them to make their complaints and criticisms public. Democracy, as a consequence, needs to be fully inclusive. Each social group should be represented wherever decisions are made or wherever they are criticized. A social group can only be said to have a voice for contestation insofar as its members are capable of protesting against the legislature, administration or judiciary if "they can make themselves heard in decision-making quarters" (191).

Finally, democratic institutions should be responsive to contestation. Citizens' claims must not only be heard, but also assessed and, when found to be compatible with the public interest, satisfied. They must provoke an adequate response in a political forum. This means that forums that make it possible for individuals or groups to claim that some policies unduly sacrifice their interests, to appeal against them, and to sometimes win on appeal, should be available.

Now, what kinds of forums are best suited to enable and support contestability? An obvious candidate is the general public sphere, especially in such instances where social movements manage to attract public attention on particular forms of domination that call for policy revisions. But Pettit refuses to consider the rise of "popular movement, widespread controversy and debate, and progressive, legislative adjustment" as a typical image of the contestation process. When considering the implementation of contestability, to think of the public success of the women's movement, the green movement or the gay rights movements "is not inappropriate [...] but it is important to recognize that there should also be other, less heroic procedures available for giving a hearing to contestation" (195). Relying on "heroic" debates in the general public sphere as a systematic mechanism for citizens' contestations is admittedly impossible, but Pettit's concerns do not stop with feasibility: his reluctance to consider the general public sphere as an essential forum for public contestation stems from what he perceives to be the irrationality of mass-media dominated "popular debate". Suggesting alternatively to create cross-party committees, standing appeals boards or expert commissions, he claims that complaints should in many cases "be depoliticized and should be heard away from the tumult of popular discussion and away, even, from the theatre of parliamentary debate" (196). Issues related to crime, for instance, should not be left to public, politicized forums, since mass-media mediated popular debates systematically sanction excesses of leniency and never excesses of harshness in the treatment of criminals. This "unbalanced pattern of triggering controls is exacerbated by the fact that the politicians who are called upon to respond to any outrage are expected to manifest concern in a manner that can communicate itself in the newspaper headline or in the television sound-bite" (197). In this view, contestatory forums can only ensure a rational assessment of citizens' complains if they are isolated from public opinion. But can they then remain democratic?

Is contestatory democracy democratic enough?

By making the idea of citizens' control plausible in complex societies, contestatory democracy offers a view of self-government that is more feasible than the populist model and less irresponsible than the pluralist one. Furthermore, it echoes the growing tendency in recent democratic theory to take into account the fragmentation and multiplication of the decision-making arenas (Habermas 1992[START_REF] Dryzek | Foundations and Frontiers of Deliberative Governance[END_REF]. However, its insistence on protecting the rationality of the contestatory processes by "depoliticizing" them appears to undermine its democratic ambition.

Pettit's sombre diagnosis regarding the irrationality of mass-media mediated debates would be neither out of place, nor excessive, in a work of media criticism. It is less expected in a normative theory of democracy, which attempts to sketch a view of democratic government that might guide the reform of existing regimes. Although Pettit thinks that democracies should provide public funding for quality mass media (Pettit 2012, 234), he does not consider the possibility of reforming mass media into a more rational forum, as well as more welcoming to the expression, discussion and evaluation of citizens' complaints. He is correlatively reluctant to give any substantive role to citizens in the evaluation of public complaints. This reluctance echoes his scepticism regarding the judgment citizens can exert when they vote, as he insists that his model "does not presuppose that when people go to the polls, they assume a deliberative profile and express their considered, detached judgment on what is for the best overall" (Pettit 1997, 191-192). Similarly, when participating in processes of public contestation, citizens are not expected, in his view, to a "assume a deliberative profile". This makes it difficult to consider the "contestatory citizenry" he evokes (Pettit 2012, 225) as self-ruling. For public policies to be appropriately contested, it is necessary that citizens' complaints be rationally assessed in the light of the public interest. In the same way an individual who revises her beliefs regularly, but in an entirely arbitrary way, will not be said to be autonomous, a people who contest its government's decisions regularly, but in an arbitrary way, will not be said to be selfruling. But it is also required that citizens have the possibility in some cases to judge by themselves -in a context favourable to collective and individual deliberation -whether the complaints publicly expressed should indeed compel public authorities to revise previous decisions. The evaluation of these complains in the light of the public interest should not be left exclusively to experts. In the same way an individual who regularly considers changing some of her beliefs or desires, but only does so when a tutor gives her the permission or the order to do so, would hardly be seen as autonomous, a people who asks for revisions of its laws, but systematically relies on experts to sort out and judge these demands, can hardly be seen as self-ruling Pettit believes that "contestatory democracy requires that the demos, and the legislative representatives of the demos, generally tie their hands and gag their mouths" (197), by delegating decisions regarding citizens' contestations to groups of independent and depoliticized experts, but how can it then retain its democratic character? Inclusion appears to have been sacrificed to rationality: contestatory democracy so conceived considerably narrows the scope of citizens' participation.

The alternative would be to endow citizens with the ability not only to form a judgment on their particular interests and to appeal particular policies that concern them, but also to form a rational judgment on the appropriateness of other policies and complaints in light of the public interest. But how can they do so if they do not have access to a general forum where they are able to deliberate on these topics? An essential condition to enable citizens to adopt such a perspective on the public expression of political complaints is the existence of mass media that do not impede, but favour largescale deliberative exchanges.

To consider the deliberative potential of the mass-media dominated public sphere is not to suggest going back to the discredited view according to which the people could gather in a single arena to express its will, nor is it to ignore the many structural sources of irrationality and power asymmetries embedded in media systems.

The social complexity and fragmentation of the contemporary political process should not prevent us from investigating the potential of a reformed mass media system to offer a forum for contestatory democracy, which would allow citizens to participate not only in the formulation of political claims but also in the evaluation and criticism of those claims.

Why pluralism is not enough: the inadequacy of the "marketplace of ideas" model

The deliberative dimension of the public sphere

A deliberative view of democracy that makes contestability a core principle but rejects Pettit's narrow view of citizens' participation cannot avoid asking what standards mass media should comply with in order to sustain a general public sphere that could also be a forum for the contestation of public policies. Yet, normative political theory falls short in providing a satisfactory answer to this question. This is mainly due to the frequent assumption that protecting mediated communication from corrupting influences is enough. While the "marketplace of ideas" model is the most obvious and radical expression of this assumption, and as such deserves special attention, it is worth noting that some version of this assumption can be identified in many theories of deliberative democracy, including the one advocated by Habermas.

Having partially renounced his initial and rather pessimistic account of contemporary mass media's democratic potential [START_REF] Habermas | Further Reflections on the Public Sphere[END_REF]), Habermas now insists on the importance of mass communication via the mass media. If the public sphere is to operate as "an intermediary system of communication between formally organized and informal face-to-face deliberations in arenas at both the top and the bottom of the political system" (Habermas 2006, 415), mass communication needs to connect the main institutions of the political systems (parliaments, courts, administrative agencies, government) to networks of communication that disseminate the opinions originating from politicians and political parties, lobbyists and pressure groups, as well as citizens, activists and actors of civil society. These opinions, selected and shaped by the mass media, reach broad and partly overlapping audiences. In Habermas' view, the flows of communication produced not only by citizens, journalists and politicians, but also by advocates, lobbyists, experts, moral entrepreneurs and intellectuals, make it possible for deliberation to operate "across the system as a whole" (416).

Notwithstanding the asymmetrical structure of mass communication, Habermas claims that the general public sphere could generate considered public opinions provided that two conditions are met. Firstly, the independence of the media system, which implies its clear demarcation both from the state and the market, is necessary in order to prevent asymmetries in media power from corrupting mass communication.

Secondly, a feedback loop must exist between mass communication and civil society, to allow for citizens' participation in public discourse, as they need "to give voice to society's problems and [to] respond to the issues articulated in elite discourse" (421).

These two "external" standards are hardly sufficient, however. It is admittedly essential that social, economic and political groups not be allowed to turn the mass media into private instruments of propaganda. It is equally essential that whole subpublics not be prevented from publicly reacting to media discourses and from making their reaction heard, as is so often the case. But what are the "internal" standards that the mass media should also satisfy? When asserting that the media system should not only be independent but also self-regulating, Habermas suggest it should operate "in accordance with its own normative code" (419), but he does not specify what the democratic requirements of such a code would be. While it is plausible that under conditions of independence and inclusion, mass communication could shape influential opinions, it is much more difficult to understand how media practices and discourse could spontaneously produce considered opinions. The existence of a free press and independent media, the absence of corrupting influence and the engagement of citizens cannot be expected to ensure the sort of deliberative rationality that a contestatory forum requires. As we saw in the first section, such a forum needs to be both inclusive and deliberative: it needs to favour informed collective reasoning on public policies.

While certainly needed, pluralism alone is not enough to create such setting.

The Marketplace of Ideas

The claim whereby the implementation of external standards securing pluralism is all the media regulation needed in a democracy has found its clearest expression in the marketplace of ideas model. What was simply a metaphor when Oliver Wendell Holmes wrote, in his minority opinion in Abrams vs. United States, that "the best test of truth is the power of the thought to get itself accepted in the competition of the market", has since been systematized into a model (Schmuhl and Picard 2005) that remains authoritative in many areas of media law, media research and public policy. The idea that true (or at least enlightened) opinions will prevail in public debate, under conditions of diversity and competition, still exerts a considerable influence on conceptualizations of both public debate and media regulations (see for instance Blocher 2008, Harris, Rowbotham and[START_REF] Harris | Truth, law and hate in the virtual marketplace of ideas: perspectives on the regulation of Internet content[END_REF][START_REF] Klocke | Foreclosing Deliberation[END_REF]. It has also inspired, within normative political theory, a definition of normative standards for mass deliberation. While deliberative democracy usually rejects the interest-group pluralism view of democracy, it sometimes adopts the marketplace pluralism view of public debate. A paradigmatic case is Benjamin Page's model of mediated deliberation [START_REF] Page | Who Deliberates? Mass Media in Modern Democracy[END_REF], which still informs many analyses of media debates [START_REF] Barabas | How Deliberation Affects Policy Opinions[END_REF][START_REF] Church | YouTube Politics: YouChoose and Leadership Rhetoric During the 2008 Election[END_REF]. In Page's view, public deliberation is needed to ensure that the public's policy preferences, upon which democratic decisions should be based, "are informed, enlightened and authentic" (Page 1996, 1). The main threat to the democratic process is not the irrationality of the public, which is a myth (Page and Shapiro, 1992), but the lack of a proper deliberative context in which the general public can participate in the evaluation and contestation of public policies. In complex societies, public deliberation can only be drastically decentralized or mediated. Mediated deliberation presupposes a division of labor in which "a small set of representatives or surrogate deliberators, perhaps (…) professional policy experts and communicators" is in charge of studying policy and addressing the public (Page 1996, 4). It also presupposes the engagement of the citizenry: large audiences of ordinary citizens should be involved in political mass communication. But how can such a division of labor ensure the formation of considered opinions? Page assumes that "if extensive political information is available somewhere in the system, not everyone has to pay attention to it all the time; a lot of information, and reasonable conclusion from it, will trickle out through opinion leaders and cue givers to ordinary citizens, who can deliberate about it in their own small, faceto-face groups of family, friends, and co-workers" (7).

This assumption explicitly draws on the marketplace of ideas model, according to which competition between a sufficiently diverse set of views will guarantee that truth defeat falsehood in public debate. Under this assumption, diversity and free competition are enough to guarantee that public communication produce enlightened judgments. Page indeed considers that "the average citizen has a good chance of arriving at sound opinions, so long as good information is available somewhere, and so long as there is vigorous competition among different ideas and interpretations, even if the media are full of bias or contaminated by untruths" (8). This allows him to identify three sets of normative standards for the evaluation of the deliberative quality of mass communication. They relate to the diversity and representativeness of the views expressed in the mass media; the quality and expertise of the discourses produced by media elites; and the competitiveness of public debate, which should not be limited by some organizations' excessive influences or propaganda techniques.

When applied to the autumn 1990 debate in the New York Times concerning whether or not the United State should go to war with Iraq, this set of standards reveals the defectiveness of the alleged deliberation constructed by the journal. The range of viewpoints represented on the Times' op-ed page was in fact narrow and made the view of the Times' own editorial the center of gravity of the discussion. As a consequence, views favorable to the war were over-represented, social organizations and experts hostile to any military action were ignored, available information about the likely consequences of a military conflict were absent. In brief, this alleged deliberation "did not meet standards of content diversity, presentation of all major points of view, democratic representativeness, or the communication of relevant facts and expertise" (Page 1996, 34). This analysis offers a convincing illustration of the fruitfulness of an approach that does not simply assume that the mass-media dominated public sphere is structurally irrational, but attempts to clarify the conditions for rationality and turn them into standards for the critical analysis of actual media practices. It also showcases the shortcomings of the marketplace of ideas model.

According to Page, the lesson of this case study is that "under certain conditions, certain media outlets may not serve as neutral, passive transmitters of societal ideas and interpretations, but may take an active part as articulators of policy views" (37). When his analysis reveals that the construction of the debate by the Times is biased, he does not denounce the particular bias so much as the construction itself. In fact, because he assumes that mass media could possibly act exclusively as "neutral, passive transmitters", he equates a biased deliberation with a constructed deliberation. Under certain discussions, he argues, "editors at the Times and elsewhere may consciously or unconsciously construct a public debate" (20). To support a genuine deliberative forum, it is assumed that mass media should transmit without determining in any way the content which is transmitted. While the unavoidable selection of opinions that mass communication implies makes it clearly impossible to imagine a spontaneous, unconstructed debate via the mass media, this fact is simply obscured by the reference to the marketplace of ideas.

The problem is not only that Page sets low standards for the marketplace of ideas -despite his criticisms of the Times, he ends up concluding that in the current U.S. media system "the marketplace of ideas actually works reasonably well, most of the time" (124). To simply point out the imperfection of the marketplace of ideas -and merely suggest that its failures should be fixed by corrective policies [START_REF] Coase | The Economics of the First Amendment: The Market for Goods and the Market for Ideas[END_REF][START_REF] Gary | The American Vision of a Free Press[END_REF]) -would be to miss the broader problem. There is simply no reason to believe that "true and useful ideas, once they are enunciated somewhere, have a way of spreading, willy-nilly, everywhere" (Page 1996, 124). Not only does this assumption ignore the many epistemic biases to which free discussion is vulnerable (Sunstein 2006), but it is also blind to the fact that, in order to circulate from one area of the media system to another, an opinion has to go through a series of selection filters, not only neutral transmitters. This blindness makes it impossible to develop a clear view of the conditions for citizens' participation in the general public sphere.

Who contests? Media elites and citizens' uprisings

Another of Page's case studies focuses on the channels through which mass communication can generate popular contestation of public policies. When it was revealed, on January 14, 1993, that Zoe Baird, President Clinton's nominee for U.S. Attorney General, had employed two Peruvians living illegally in the US as her chauffeur and nanny, the media elite and the public reacted in strikingly different manners. While the study of opinion polls, letters to newspapers editors and calls to members of Congress reveal a reaction of deep public outrage, an analysis of the coverage of the story in the main national newspapers and TV networks indicates that media elites minimized the matter. Surprisingly, however, the public eventually forced the media elites to change discourse (Page 1996, 88). Radio talk-show call-ins had started before the confirmation had begun and before any substantial criticisms of Baird had been expressed in the mainstream media. By January 19, when the confirmation hearings began, public outrage had been vehemently expressed, compelling both members of Congress and mainstream media to adopt a much more critical stance in the coverage of the event, eventually forcing Baird to withdraw her name from consideration for the position on January 22. In Page's view, "a relatively autonomous popular uprising -based on the bare facts of the case as reported on television, and reinforced by hearing fellow citizens' anger on talk radio -overcame the complacency of Washington officials and media elites, changed public discourse, and overturned Baird's sure thing confirmation" (89-90).

In this case, the three conditions for contestability might seem to have been satisfied: a) publicity regarding official reasons and relevant facts enabled a public contestation of a political decision based on publicly expressed reasons (such as the integrity of the nominee); b) citizens' claims found media channels to make themselves heard despite the lack of reaction from the media elites; c) Congress was eventually responsive to these claims by submitting Baird to an unexpected, hostile treatment. While he acknowledges that the "Nannygate" was not "an entirely happy story of democratic triumph" (99), as Baird and the illegal immigrants suffered a violent public humiliation partly fuelled by sexism and racism, Page concludes that ordinary citizens were able to impose "not patently unreasonable legal and ethical standards" (100) to a major government appointment despite the failures of the mainstream mass media to organize an appropriated deliberation. To this extent, it was "a demonstration that, under certain circumstances, unrepresentative surrogate deliberators can be overcome by direct, populistic deliberation" (100). This understanding of mass media-supported contestability is, however, problematic. The opposition, in the mass media context, between delegated deliberation, which should be inclusive but sometimes fails to be, and populistic deliberation, which can arise from time to time in order to compensate for the shortcomings of media elites, ultimately rests on two mistaken premises.

First of all, it identifies the particular citizens' voices that manage to make themselves heard through the selection filter of the mass media with the voice of a homogeneous public, and thus ignores the issue of their representativeness, as well as the ways through which they manage to make their opinion prevalent in the general public sphere. What was, in Page's eyes, a popular uprising has been described by other commentators as a partisan action fomented by far-right activists associated with the Republican Party. In James Fishkin's view, for instance, it was a campaign orchestrated by conservative talk radio shows. "An affecting sound bite led to thousands of listeners feeling sound-bitten. They responded with enraged calls and faxes to members of Congress." (Fishkin 1997, 973) Focusing on a superficial issue, the public contestation entirely sidetracked, according to Fishkin, the merits and implications of the nomination choice: it was neither rational nor deliberative. Whatever might have been the case in this specific instance, it is clear that Page does not offer any criteria to distinguish between inclusive, reasoned contestation and manufactured, partisan flak, no more than he offers criteria to distinguish between genuine deliberation and non-deliberative communication. Once again, the crude dichotomy between the spontaneous and the constructed is ineffective, because both citizens' engagement in public contestation and the deliberative shaping of mass communication require some form of organization and construction. To think otherwise is to rely again on some sort of 'magic'.

Secondly, the delegated/populistic opposition superimposes the distinction between delegation and direct participation on the distinction between professionals (the media elite) and lay citizens (the public). However, to believe that media elites ordinarily act as "representatives" or "delegates" of the public is largely unfounded.

While lay citizens might sometimes recognize some of their ideas or interests defended in mass communication, they usually do not feel that media speakers are, in any political sense, acting as their representatives. Moreover, there is no reason to consider that media representation can only be provided by media professionals. Some division of labor is certainly required if mass communication is to enjoy a deliberative dimension. Nevertheless, we should not assume that the need for social groups to accumulate media resources and skills automatically translate into the emergence of a specific cast endowed with representative authority. The frontiers separating the respective functions of journalists, media experts and spokespersons for social groups should not be blurred if one wants to avoid relegating citizens' participation to the exceptional -and truly "heroic", as Pettit would say, -uprisings. While only a few voices can gain maximal visibility on the mass media stage, there is no reason to renounce the inclusion of lay citizens and social movements' spokespersons alongside the media elites, in order to participate in the evaluation of public policies.

External standards ensuring adequate media pluralism are simply not enough to create the conditions for public contestation in the general public sphere. The lack of cultural, economic, racial or gender representativeness frequently observed in mass communication, and the frequent irrationality of public reactions it provokes, are clear enough indications of the fact that economic competition between a diversity of media outlets does not make for sufficient inclusiveness or rationality.

Towards mass deliberation: resources for mass media regulation

Mass deliberation requires much more than a marketplace. It requires internal regulation guided by deliberative standards, which relate, among other things, to systematic reason-giving, representativeness of published opinions, and fair access to media speech. The prospect of deliberative media regulation -going beyond the promotion of media pluralism -in the name of contestatory democracy, is likely to raise fierce objections.

First of all, one could argue that the rise of the digital media and the increasingly easy access to media speech for lay citizens has abolished the selection process, and with it the need for regulation. In the twenty years since the radio talk shows triumphed in "Nannygate", media systems have been drastically restructured. Indeed, political communication no longer happens in a media environment exclusively dominated by professional journalistic gatekeepers, as in the old media regime [START_REF] White | The Gate Keeper[END_REF]. But that is not to say that the selection process itself has disappeared. Rather, it has become more fragmented, diffuse and competitive: a multitude of gatekeepers with divergent motivations and opinions have replaced the limited, unified selection system of the previous era [START_REF] Williams | After Broadcast News. Media Regimes, Democracy and the New Information Environment[END_REF]. While a growing diversity of habits, goals and ideologies preside over the transmitting and framing of media discourses, only a small minority of these discourses reaches the mass media and gain maximum public visibility. Media power remains, first and foremost, a selection power, related to the "sheer force of mass distribution of information" (Schudson 2003, 28), even when it is exerted in a fragmented, uncoordinated way by a multiplicity of media agents. The fragmentation and lack of coordination only make regulation more urgent, if more difficult.

Furthermore, any project of regulation by the state might appear to threaten media independence by creating opportunities to impose excessive restrictions on the freedom of the press. The state might be as corrupting an influence as the market. regulation does not need to come exclusively from the state. In a pluralist media system -whether it corresponds to "polarized pluralist", "democratic corporatist" or "liberal" model [START_REF] Hallin | Comparing Media Systems[END_REF]) -, a pluralist regulation can be conceived. Such pluralist regulation must combine self-regulation, informal social regulation and legal regulation. In particular, a plurality of media agents can act together, in the absence of centralized coordination but in partial agreement on the normative standards associated with contestatory democracy, to contribute to making the general public sphere an appropriate forum for public contestation. It does not require all media agents to be exclusively motivated by the ideal of contestatory democracy. It does require, however, that the normative resources already existing in journalism ethics, media criticism and media law be exploited and developed. This is all the more plausible given that journalistic, social and legal conceptions of media regulation have, for a long time, if often ineffectively, been driven by the idea that the first social responsibility of the mass media is to create a forum for democratic deliberation [START_REF] Siebert | Four Theories of the Press[END_REF].

The self-regulation of the media system starts with journalistic ethics. While lay citizens play a growing role in the mass media selection process, the influence of professional journalists remains paramount, especially when it comes to filtering discourses that circulate in areas of medium visibility so as to make them reach maximum visibility. Ethical rules embedded in journalistic habits, codes of deontology, international or national professional guidelines, and alternative standards put forward by social movements such as public journalism [START_REF] Dzur | Public Journalism and Deliberative Democracy[END_REF], although not binding and oftentimes ignored, play an actual role in media practice and constitute a powerful, if insufficient, source of regulation. The practice of demanding reasons from political institutions and holding them accordingly accountable -something that is required to unearth a basis for contestation -is at the core of journalism, conceived not only in its role of providing information, but also in its role of enhancing deliberation [START_REF] Ettema | Journalism as Reason-Giving: Deliberative Democracy, Institutional Accountability, and the News Media's Mission[END_REF].

Critical evaluation of media practices is not only produced by journalists themselves, but constantly fed by (lay or expert) media criticism. While sometimes reluctant to endorse a positive theory of democratic deliberation, media critics tend to focus on the lack of inclusiveness, representativeness and rationality in the mass media's construction of mass communication. For more than a century, radical media criticism has denounced the stifling of minority voices and the biased construction of discursive confrontations (Mc Chesney and Scott 2004). When Pierre Bourdieu condemns the "de facto monopoly" of a media elite "on the large-scale informational instruments of production and diffusion of information (Bourdieu, 1996, 46), he points to the bias in the selection process. The problem, in his view, is the control media elites exert on the access of ordinary citizens to the public sphere: discourses "have to face this trial of journalistic selection in order to catch the public eye. The effect is censorship, which journalists practice without even being aware of it." (47) While such criticism remains incomplete without a proper conception of what an appropriate massmedia structured public sphere could and should look like, it contributes in submitting media practices to normative standards.

Self-regulation and informal social regulation can neither thrive nor contribute to endowing mass communication with a deliberative dimension without an appropriate legal framework. Legal regulation, when it is not stringently limited by the reference to the marketplace of ideas, is already aiming at promoting an (admittedly vague) idea of public debate, thus justifying, for instance, public subsidies for media outlets satisfying specific standards. Existing restrictions on media ownership, or proposals to institutionalize principles such as net neutrality, offer a clear idea of the already existing trends towards ambitious regulation and of how quickly they can be made meaningless by empty reforms or anti-regulation policies (Baker 2007, Lunt andLivingstone 2012).

Since speech in the areas of maximum visibility is a limited good, to bestow upon each citizen an equal formal right to freedom of expression is not enough to ensure fair access to speech in the mass media: legal rules are needed to make sure that asymmetries in media resources and media power do not transform an equal formal right into deeply unequal opportunities to participate in, and thereby influence, the public evaluation of public policies.

The contestability of public policies is an essential feature of democratic regimes, which deliberative theory should promote. It requires deliberative, inclusive and responsive institutions. But it can only be made democratic if the evaluation of citizens' claims is not entirely delegated to experts. A general public forum, supported by the mass media, should make it possible for all to form a deliberate judgment at least on some of the most important issues. This can only be made possible if this forum is not reduced to a "marketplace of ideas": effective media regulation, going beyond the promotion of media pluralism, need to contribute in endowing it with a deliberative dimension. Current media practices might make the idea of a mass media-structured forum implausible, if not downright naïve. Political theory should not confuse, however, structural necessities and contingent pathologies. Conceptual resources are not lacking to elaborate a deliberative theory of the mass media.

Acknowledgments

I thank Selen A. Ercan, Florence Hulak, Matthew Udkovich and an anonymous reviewer for their precious comments on a first draft of this paper.


Marianne ´jover

M Clark

Daniela Corbetta

DEVELOPMENT OF HEAD-TURN PREFERENCE FROM 2 TO 12 WEEKS OF AGE: THE INFLUENCE OF ENVIRONMENTAL CONTEXTS

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

DEVELOPMENT OF HEAD-TURN PREFERENCE FROM 2 TO 12 WEEKS OF AGE: THE INFLUENCE OF ENVIRONMENTAL CONTEXTS

Jover, M. 1 , Clark, M. 2 , Corbetta, D. 2 1 Aix Marseille Univ, PSYCLE, Aix-en-Provence, France 2 Department of Psychology, the University of Tennessee, Knoxville, TN, USA

CONTEXT

The development of head control in early infancy has been studied for quite a long time (e.g. [START_REF] Turkewitz | Head turning in the human neonate: Spontaneous patterns[END_REF][START_REF] Rönnqvist | Head position preference in the human newborn: A new look[END_REF][START_REF] Lima-Alvarez | Early development of head movements between birth and 4 months of age: a longitudinal study[END_REF]. Two overarching conclusions from these studies are that prior to 2 months of age, infants' head control is weak, especially when in the supine position, and that the majority of infants prefer turning their head to the right side. In most of these studies, an experimenter momentarily held the infants head at midline before releasing it, or presented a visual stimulus to induce head turn. Very little is known, however, about the development of infants' spontaneous head movements during this early period.

METHOD Spontaneous head movements were explored in 3 infants (GM, LN and MA) followed weekly from the age of 3 to 12 weeks, until they acquired head control. Infants were observed in 5 conditions: baseline, toys-in-view (preferred head turn side), musical toys outof-view (non-preferred head turn side), mobile overhead, and mother reading to infant. The toys-in-view condition was not considered for this poster since the head would more likely remain turned in that condition. Each condition lasted 5 min while infants laid in supine on a white padded surface surrounded by two vertical white panels to block distractions from the surroundings. Two synched cameras, placed overhead and to the side, recorded the whole session.

Head movements were coded with respect to the frontal plane at a frequency of 15 Hz with a custom software which provided 2D kinematic data of selected points on the video (VideoAnalyser, Mare, 2010, figure 1). Data were filtered. The infants' nose was the body point of interest selected to describe head turns (figure 2). The rate of change, direction, and duration of head turns were extracted and compared across conditions within each session. Non-parametric Friedman and Durbin-Conover tests were used to compare the head movements between the conditions.

RESULTS

Infants acquired head control at 11 (CG), 13 (LN) and 14 (MA) weeks. Descriptively, the mean amplitude of spontaneous head displacement was already large at 3 weeks and did not vary before head control was acquired (figure 3). However, the percentage of time the infant was motionless tended to decrease during that time (figure 4).

DISCUSSION

These preliminary results show that infant spontaneous head turns were present well before they acquired head control, however we do not see any developmental trend. Infants were more active in the Baseline and Mobile conditions and more still in the mother condition, possibly due to a motor inhibition mechanism triggered when attention (to the music and mother voice) is engaged (Robertson & Johnston, 2009). This indicates that young babies can move their head differentially depending on the context. Infants did not move their head the same depending of the context (figure 5). The difference between the condition was significant, showing that infants were longer motionless in the musical toy (toyNP) and mother conditions as compared to the other conditions. Moreover, their head movements were slower in those conditions. All pairwise comparison showed that each condition differed from each other. However, the amplitude of the infant's nose displacement did not differ significantly according to the condition.

Figure 1 :

 1 Figure 1: Video Analyser interface.The small circular marker on the nose represents the 2D position (in pixels) that was coded frame by frame for each infant.

Figure 2 :

 2 Figure 2: Graphic of the infant's nose displacement during 5mn used to describe head turns.

Figure 3 :

 3 Figure 3: Amplitude of the infant's nose displacement on the x and y axis according to the time to head control

Figure 5 :

 5 Figure 5: Percentage of the time spent motionless (green), mean velocity (red) and amplitude of the infant's nose displacement according to context.


Dan Ma
email: madan@mail.neu.cn.edu

Islam Boussaada
email: islam.boussaada@centralesupelec.fr

Catherine Bonnet
email: catherine.bonnet@centralesupelec.fr

Silviu-Iulian Niculescu
email: silviu.niculescu@centralesupelec.fr

Jie Chen
email: jichen@cityu.edu.hk

Multiplicity-Induced-Dominancy extended to neutral delay equations: Towards a systematic PID tuning based on Rightmost root assignment

Keywords: Time-delay systems, Neutral systems, Asymptotic stability, α-stability, Frequency domain analysis, PID tuning, Pole placement, Rightmost root, Delay margin

Recently, the conditions on a multiple spectral value to be dominant for retarded time-delay system with a single delay were deeply explored. Such a property was called Multiplicity-Induced-Dominancy. It was then exploited in the design of delayed stabilizing controllers. As a matter of fact, the approach is merely a delayed-output-feedback where the candidates' delays and gains result from the manifold defining the maximal multiplicity of a real spectral value. This can also be seen as a pole-placement method, which unlike methods based on finite spectrum assignment does not render the closed loop system finite dimensional but consists in controlling its rightmost spectral value. This work aims at extending such a design approach to time-delay systems of neutral type occurring in the classical problem of PID stabilizing design for delayed plants. More precisely, the controller's gains (ki, kp, k d ) are tuned using the intentional multiplicity's algebraic constraints allowing to the stabilization of unstable delayed plants. The specificity, of such a design is related to the analytical assignment of the closed-loop solution's decay rate.

I. INTRODUCTION

Linear systems with commensurate delays (all delays are multiple of a nominal delay) are described in the Laplace domain by transfer functions involving quasi-polynomials and then possibly admit an infinite number of poles. Studying the stability properties of retarded systems (they admit a finite number of poles in any right half-plane) is much easier than studying those of neutral systems which may have an infinite number of poles, in chains asymptotic to vertical axes possibly located in the open right half-plane or clustering the imaginary axis from left or right. Both situations prevent to get exponential stability for these systems. A subclass of neutral systems of interest is the one with all asymptotic axes in the open left half-plane guaranteing that there is a finite number of poles in an extended right half-plane. For such systems, the concept of α-stability) will play an important role.

In the Laplace domain, a number of effective methods have been proposed, see for instance [START_REF] Bellman | Differential-difference equations[END_REF], [START_REF] Cooke | On zeroes of some transcendental equations[END_REF], [START_REF] Walton | Direct method for tds stability analysis[END_REF], [START_REF] Stépán | Retarded Dynamical Systems: Stability and Characteristic Functions[END_REF], [START_REF] Hale | Introduction to functional differential equations[END_REF], [START_REF] Michiels | Stability and stabilization of timedelay systems[END_REF], [START_REF] Olgac | An exact method for the stability analysis of time delayed linear time-invariant (lti) systems[END_REF], [START_REF] Sipahi | Stability and stabilization of systems with time delay[END_REF].

Even with the significant advances that have been reported on the topic of Delay systems, the question of determining conditions on the equation parameters that guarantee asymptotic stability of solutions of linear time-invariant time-delay systems remains an open question.

Once stability conditions are established; further questions related to performance occur. What is about the estimation of the corresponding rightmost roots of the system characteristic equation? Such a rightmost root corresponds to the so-called α-stability problem, itself is related to the solution's decay rate. Also a fundamental measure of robust stabilization against uncertain time delays is the so-called delay margin, which addresses a central issue in the study of feedback stabilization of time-delay systems: What is the largest range of delay so that there exists a single controller that can stabilize the delay plant within that entire range? This question is also longstanding and remains open except in particular cases, see for instance [START_REF] Ma | Delay margin of low-order systems achievable by pid controllers[END_REF]. PID controllers have been extensively used to control and regulate industrial processes which are typically modeled by reduced-order dynamics. In [START_REF] Ma | Delay margin of low-order systems achievable by pid controllers[END_REF], the delay margin achievable using PID controllers for reduced order linear time-invariant (LTI) systems subject to variable, unknown time delays is investigated. An explicit expressions of the exact delay margin is carried out and its upper bounds achievable by a PID controller for low-order delay systems with unknown constant and possibly time-varying delays. The effect of non minimum phase zeros is also investigated and the fundamental limits of delay within which a PID controller may robustly stabilize a delay process is emphasized.

In recent works, the characterization of multiple spectral values for time-delay systems of retarded type were established using a Birkhoff/Vandermonde-based approach; see for instance [START_REF] Boussaada | Tracking the algebraic multiplicity of crossing imaginary roots for generic quasipolynomials: A Vandermonde-based approach[END_REF], [START_REF]Characterizing the codimension of zero singularities for timedelay systems[END_REF], [START_REF]Computing the codimension of the singularity at the origin for delay systems: The missing link with Birkhoff incidence matrices[END_REF], [START_REF] Boussaada | Multiplicity and stable varieties of time-delay systems: A missing link[END_REF]. In particular, in [START_REF]Characterizing the codimension of zero singularities for timedelay systems[END_REF], it is shown that the admissible multiplicity of the zero spectral value is bounded by the generic Polya and Szegö bound denoted P S B , which is merely the degree of the corresponding quasipolynomial 1 , see for instance [START_REF] Pólya | Problems and Theorems in Analysis[END_REF]. In [START_REF] Boussaada | Tracking the algebraic multiplicity of crossing imaginary roots for generic quasipolynomials: A Vandermonde-based approach[END_REF], it is shown that a given crossing imaginary root with a non vanishing frequency never reaches P S B and a sharper bound for its admissible multiplicities is established.

Moreover, in [START_REF] Boussaada | Multiplicity and stable varieties of time-delay systems: A missing link[END_REF], the manifold corresponding to a multiple root for scalar time-delay equations defines a stable manifold for the steady state. An example of a scalar retarded equation with two delays is studied in [START_REF] Boussaada | Tracking the algebraic multiplicity of crossing imaginary roots for generic quasipolynomials: A Vandermonde-based approach[END_REF] where it is shown that the multiplicity of real spectral values may reach the P S B . In addition, the corresponding system has some further interesting properties: (i) it is asymptotically stable, (ii) its spectral abscissa (rightmost root) corresponds to this maximal allowable multiple root located on the imaginary axis. Such observations enhance the outlook of further exhibiting the existing links between the maximal allowable multiplicity of some negative spectral value reaching the quasipolynomial degree and the stability of the trivial solution of the corresponding dynamical system. This interesting property induced by multiplicity appears also in optimization problems since such a multiple spectral value is indeed the rightmost root, see also [START_REF] Vanbiervliet | A nonsmooth optimisation approach for the stabilisation of time-delay systems[END_REF]. Also notice that the property was already observed in [START_REF] Ramirez | Design of proportional-integral-retarded (pir) controllers for second-order lti systems[END_REF], where a tuning strategy is proposed for the design of a delayed Proportional-Integral controller by placing a triple real dominant root for the closed-loop system. However, the dominancy is only checked using a Mikhailov curve and QPmR toolbox, see for instance [START_REF] Vyhlidal | Mapping based algorithm for large-scale computation of quasi-polynomial zeros[END_REF].

It is worth noting that the rightmost root for quasipolynomial function corresponding to stable retarded time-delay systems (also in the neutral case under some assumptions) is actually the exponential decay rate of its time-domain solution, see for instance [START_REF] Mori | On an estimate of the decay rate for stable linear delay systems[END_REF] for an estimate of the decay rate for stable linear delay systems. To the best of our knowledge, the first time an analytical proof of the dominancy of a spectral value for the scalar equation with a single delay was presented in [START_REF] Hayes | Roots of the transcendental equation associated with a certain difference-differential equation[END_REF]. The dominancy property is further explored and analytically shown in scalar delay equations in [START_REF] Boussaada | Multiplicity and stable varieties of time-delay systems: A missing link[END_REF], then in second-order systems controlled by a delayed proportional is proposed in [START_REF] Boussaada | Further remarks on the effect of multiple spectral values on the dynamics of time-delay systems. application to the control of a mechanical system[END_REF], [START_REF] Boussaada | On the coalescence of spectral values and its effect on the stability of timedelay systems: Application to active vibration control[END_REF] where its applicability in damping active vibrations for a piezo-actuated beam is proved. An extension to the delayed proportional-derivative controller case is studied in [START_REF] Boussaada | On the dominancy of multiple spectral values for time-delay systems with applications[END_REF] where the dominancy property is parametrically characterized. We emphasize that the idea of using roots assignment for controller-design for time-delay system is not new. As a matter of fact, an analytical/numerical stabilization method for retarded timedelay systems related to the classical pole-placement method for ordinary differential equations is proposed in [START_REF] Michiels | Continuous pole placement for delay equations[END_REF], see also [START_REF] Zitek | Dimensional analysis approach to dominant three-pole placement in delayed pid control loops[END_REF] for further insights on pole-placement methods for retarded time-delays systems with proportional-integralderivative controller-design.

A stimulating idea from [START_REF] Niculescu | Stabilizing a chain of integrators using multiple delays[END_REF], which consists in exploiting the properties of Vandermonde matrices to control one chain of integrators by delay blocks, enticed the authors in [START_REF]Computing the codimension of the singularity at the origin for delay systems: The missing link with Birkhoff incidence matrices[END_REF] and 1 The quasipolynomial degree is exactly the number of the involved polynomials plus their degree minus one [START_REF] Boussaada | Computing the codimension of the singularity at the origin for time-delay systems in the regular case: A Vandermonde-based approach[END_REF] to extend the investigation of such structured matrices. A complete characterization of the root at the origin can be found in [START_REF]Computing the codimension of the singularity at the origin for delay systems: The missing link with Birkhoff incidence matrices[END_REF], [START_REF] Boussaada | Computing the codimension of the singularity at the origin for time-delay systems in the regular case: A Vandermonde-based approach[END_REF]. It is shown that the multiplicity of such a root for a generic system is bounded by the degree of the associated quasipolynomial, i.e, the sum of the number of its polynomials plus their respective degrees minus one. This bound was already depicted in [START_REF] Pólya | Problems and Theorems in Analysis[END_REF]. Furthermore, in the sparse case, a sharper bound is established in [START_REF]Computing the codimension of the singularity at the origin for delay systems: The missing link with Birkhoff incidence matrices[END_REF] exhibiting the link between the maximal multiplicity and the number of the non vanishing parameters rather than the degree of the quasipolynomial.

The present work is a natural continuation of [START_REF]Computing the codimension of the singularity at the origin for delay systems: The missing link with Birkhoff incidence matrices[END_REF], [START_REF] Boussaada | Computing the codimension of the singularity at the origin for time-delay systems in the regular case: A Vandermonde-based approach[END_REF], [START_REF]Characterizing the codimension of zero singularities for timedelay systems[END_REF], it aims at proposing a systematic PID controller tuning to achieve the asymptotic stability of a reduced order linear time-invariant dead-time plant by extending and exploiting the Multiplicity-Induced-Dominancy property for quasipolynomial functions corresponding to neutral systems with single delay.

The remaining paper is organized as follows. Some prerequisites in complex analysis and the problem formulation are presented in section 2 . The main results are presented in Section 3 where a PID tuning based on the MID property is provided. Section 4 illustrate the obtained result. A conclusion ends the paper.

II. PREREQUISITES AND PROBLEM FORMULATION

The principle of the argument is a consequence of Cauchy theorem and connects the winding number of a curve with the number of zeros and poles of a given complex variable function inside that curve. More precisely, it asserts that the integral of the ratio of the single valued complex variable functions f (z)/f (z) on a single passage along a closed path in the positive sense (in counter clockwise direction) gives exactly 2iπ(N -P ) where N (respectively P ) is the sum of multiplicities of zeros (respectively of poles) of the function f enclosed in such a contour. A natural and direct application of the principle of the argument is the stability analysis of dynamical systems. As a matter of fact, in frequency domain, showing the asymptotic stability of the trivial solution of a given dynamical system amounts to prove that the zeros of the corresponding characteristic equation are located in the open left-half complex plane. The principle of argument is applied to determine the stability of closed loop system by choosing a closed path which encircles the entire right half complex plane in counter clockwise direction.

The next proposition follows directly from [START_REF] Krall | On the real parts of zeros of exponential polynomials[END_REF] and [28, Prop. 2.1] which gives an explicit localisation of the spectrum chain's asymptote for quasipolynomial functions corresponding to the following neutral systems.

G(s) = r(s) Q 0 (s) + Q τ (s)e -sτ
(1) So, under the condition |α| < 1 one is able to prove that at most a finite number of poles of G are located in the righthalf plane. By scaling, this property extends to any parallel to the imaginary axis located at the right of the spectrum chain's asymptote. Remark. The above result naturally apply to any transfer function

such that deg(r) ≤ deg(Q 0 ) = deg(Q τ ). Proposition 1. Let α = lim |s|→∞ Q τ (s)/Q 0 (s) 1
G(s) = r(s)e -s T Q 0 (s) + Q τ (s)e -sτ for any T > 0. (2)

A. Problem formulation

Consider the feedback control system depicted in Figure .1 where P (s) is a reduced order plant which is subject to an unknown delay. Thus, the corresponding transfer function is given by P τ (s) = P 0 (s) e -τ s

where P 0 (s) is a delay-free plant. Our aim in this paper is to provide an easy way in tuning the standard PID gains (k i , k p , k d ) achieving the stabilization of the closed-loop system. More precisely, the approach consists in intentionally impelling an appropriate multiplicity for a given root of the quasipolynomial function such that the induced algebraic constraints on the quasipolynomial's coefficients leads to robustly stabilize the considered unstable delayed plant. This approach showed its efficiency in studying stability and stabilization of retarded time-delay systems. However, in the present paper context, the obtained characteristic equation in closed-loop system corresponds to a delay system of neutral type, a class for which the stability analysis becomes more involved. As a matter of fact, the structure of the controller we consider is given by:

K P ID (s) = k p + k d s + k i s , (4)

III. MAIN RESULTS

A. Robust PID stabilizing design for first-order delayed plants

First-order plants are typical benchmarks usually considered for controlling industrial processes. In this aim, let consider P 0 (s) = 1 s-p where p is a positive (instable) pole.

It is shown in [START_REF] Silva | New results on the synthesis of pid controllers[END_REF] that the delay margin τ P ID = 2 p , see also [START_REF] Ma | Delay margin of low-order systems achievable by pid controllers[END_REF]. The resulting closed-loop plant is given by:

M (s) = k p s + k i + s 2 k d e -sτ s 2 -sp + e -sτ k p s + e -sτ k i + e -sτ s 2 k d . (5)
Since we are dealing with stability aim, le us focus on the corresponding characteristic equation:

∆(s) = Q 0 (s) + Q τ (s)e -sτ where Q 0 (s) = s 2 -sp and Q τ (s) = k d s 2 + k p s + k i . (6)
The following theorem provides a bound on the multiplicity of the corresponding zeros as well as a systematic manier to tune the parameters k p , k i , k d such that the closed-loop system becomes stable.

Theorem III.1.

i) For arbitrary real parameters k p , k i , k d and arbitrary positive delay τ , the multiplicity of a given root of the quasipolynomial function ( 6) is bounded by 4.

ii) The quasipolynomial (6) admits a multiple real spectral value at

s ± = -τ p -6 ± τ 2 p 2 + 12 2τ (7)
with algebraic multiplicity 4 if, and only if, Remark. • It is worth-noting that the degree of the quasipolynomial function ( 6) is equal to the generic Polya-Szegö bound denoted P S B = 5. Such a bound cannot be reached due to the sparsity of the polynomial Q 0 . • From a robustness point of view, the solution s -is not of interest since in such a case the open loop system

               k d = (4 + 2 τ s ± -τ p) e τ s± 2 , k p = - 8 τ + τ 2 s ± p -18 -12 τ s ± e τ s± τ , k i = (τ s ± + 3) τ 2 p 2 +(-12τ s ± -60) τ p+108+84 τ s ± e τ s±
L 0 (s) = P 0 (s) K P ID (s) (9)
do not satisfy the requirement for rolling off at high frequencies since |L 0 (∞)| > 1, see for instance [START_REF] Ma | Delay margin of low-order systems achievable by pid controllers[END_REF]. • If ( 8) is satisfied, then the quasipolynomial function [START_REF] Michiels | Stability and stabilization of timedelay systems[END_REF] can be written as a Fredholm equation of the first type,

∆(s) = 1 0 K(t, s)φ(t)dt, (10)
where its smooth kernel function is given by:

K(t, s) = e -tτ (s-s+) , (11)
and its solution to the inhomogeneous equation writes:

φ(t) = s + (s + -p) 2 t 2 + (2 s + -p) t τ + 1 τ 2 . ( 12
)
In a (p, τ ) parametrization such a solution is:

φ(t) = 12 -3 τ 2 p 2 + 12 t 2 + 2 τ 2 p 2 + 12 -12 t + 2 2τ 2 . (13)
Other Fredholm equations of the first type appeared in [START_REF] Boussaada | Multiplicity and stable varieties of time-delay systems: A missing link[END_REF], [START_REF] Boussaada | Further remarks on the effect of multiple spectral values on the dynamics of time-delay systems. application to the control of a mechanical system[END_REF], [START_REF] Boussaada | On the dominancy of multiple spectral values for time-delay systems with applications[END_REF] in investigating the dominant root of quasipolynomial functions corresponding to retarded delay systems. The specific polynomial structure of the solution to the inhomogeneous equation and the exponential structure of the kernel seem to be related to such a dominancy property. Such a question will be explored in a future work.

• In terms of the delay margin, the dominant root writes:

s + = -(τ + 3 τ P ID ) + √ τ 2 + 3 τ P ID 2 τ P ID τ . ( 14
)
Proof: [Sketch of the proof] The degree of the quasipolynomial function defined in ( 6) is equal to 5. So, using the Polya and Szegö result from [START_REF] Pólya | Problems and Theorems in Analysis[END_REF] one asserts that 5 is the generic bound of the multiplicity of any root of ( 6). First, the vanishing of the quasipolynomial ∆ given in ( 6) yields the elimination of the exponential term as a rational function in s:

e -τ s = -s 2 + ps k d s 2 + k p s + k i (15)
Next, to investigate potential roots with algebraic multiplicity 4, one substitutes of the obtained equality [START_REF] Vanbiervliet | A nonsmooth optimisation approach for the stabilisation of time-delay systems[END_REF] in the ideal I 4 generated by the first three derivatives of ∆ that is

I 4 =< ∂ s ∆, ∂ 2 s ∆, ∂ 3 s ∆ >.
This allows to investigate the following variety of three algebraic equations in 6 unknowns k i , k p , k d , τ, p, s :

0 =τ k d s 4 + (k p -k d p) τ s 3 + ((k i -k p p) τ + k d p + k p ) s 2 + (2 k i -τ pk i ) s -pk i , 0 = -τ 2 k d s 4 + (-k p + k d p) τ 2 + 4 τ k d s 3 + (k p p -k i ) τ 2 + (-4 k d p + 2 k p ) τ s 2 + τ 2 pk i + 2 k p + 2 k d p -2 τ k p p s + 2 k i , 0 =τ 3 k d s 4 + (k p -k d p) τ 3 -6 τ 2 k d s 3 + (k i -k p p) τ 3 + (6 k d p -3 k p ) τ 2 + 6 τ k d s 2 + -pk i τ 3 + 3 τ 2 k p p -6 τ k d p s (16)
Using standard elimination techniques, one obtains the following set of admissible solutions:

1)

k i = k p = k d = 0. 2) k i = s = 0. 3) k p = -2 ((τ s±+8)τ p-12 τ s±-18)k d τ (τ (2s±-p)+4)
,

k i = ((τs±+3)τ 2 p 2 +(-12 τ s±-60)τ p+84 τ s±+108)k d (2 τ s±+4-τ p)τ 2
where the explicit expression of s ± is given by [START_REF] Olgac | An exact method for the stability analysis of time delayed linear time-invariant (lti) systems[END_REF]. Observe that the first solution corresponds to the openloop system while the second solution is inconsistent with respect to the transcendental-term elimination [START_REF] Vanbiervliet | A nonsmooth optimisation approach for the stabilisation of time-delay systems[END_REF]. So that, these two solutions are discarded. Furthermore, substituting conditions of the third solution in [START_REF] Vanbiervliet | A nonsmooth optimisation approach for the stabilisation of time-delay systems[END_REF] yields the explicit values of the gain k d allowing to tune the parameters as provided in [START_REF] Sipahi | Stability and stabilization of systems with time delay[END_REF].

Note that, when forcing multiplicity 5 complex gains and delay are obtained. Since we deal only with quasipolynomials with real coefficients, such a multiplicity is discarded.

Next to show that s + is the dominant root of the quasipolynomial ∆ (with coefficients satisfying (8)), one has to apply the principle of the argument. However, it is necessary first to prove that only a finite number of roots of ∆ may occur in the right-half complex plane C + s+ = {s ∈ C, s.t. (s) > s + }. According to Proposition 1, one has to check that τ s + > log(|α|), see also [START_REF] Partington | h∞ and bibo stabilization of delay systems of neutral type[END_REF]Prop. 2.1]. In our case, α is nothing but k d . So that,

log(k d ) = log (-1 + 1/2 τ 2 p 2 + 12)e -1/2 τ p-3+1/2 √ τ 2 p 2 +12 = τ s + -log(2) + log(-2 + τ 2 p 2 + 12) Since, τ < τ P ID = 2
p , then τ 2 p 2 + 12 < 4. So that, log(k d ) < τ s + which proves that only a finite number of roots of ∆ may occur in the right-half complex plane

C + s+ = {s ∈ C, s.t. (s) > s + }.
Equivalently, by using the following scaling s → z + s + and the new parametrization δ = τ 2 p 2 + 12, one has to show the dominancy of zero spectral value for the following quasipolynomial function:

∆(z) =z 2 + δ -6 τ z + 12 -3 δ τ 2 + δ -2 2 z 2 + 2 δ -6 τ z + 3 δ -12 τ 2 e -zτ (17)
Obviously, z = 0 is a root of ( 17) with multiplicity 4. To apply the principle argument on the standard Bromwich contour, which allows to counting the roots of the quasipolynomial [START_REF] Vyhlidal | Mapping based algorithm for large-scale computation of quasi-polynomial zeros[END_REF] on the right half-plane, one needs first to introduce a deflation eliminating the roots on the imaginary axis. To do so, let us first investigate nonzero imaginary roots for [START_REF] Vyhlidal | Mapping based algorithm for large-scale computation of quasi-polynomial zeros[END_REF]. Assume that there exists ω > 0 such that z = iω is a root of [START_REF] Vyhlidal | Mapping based algorithm for large-scale computation of quasi-polynomial zeros[END_REF]. Let define R(ω) = ( ∆(iω)) and S(ω) = ( ∆(iω)), which gives :

                       R(ω) = 2 -δ 2 ω 2 + 3 δ -12 τ 2 cos (ω τ ) + (2 δ -6) ω sin (ω τ ) τ -ω 2 + 12 -3δ τ 2 , S(ω) = δ -2 2 ω 2 + 12 -3 δ τ 2 sin (ω τ ) + ω (2 δ -6) cos (ω τ ) τ + δ -6 τ ω. (18)
This means that for any z = iω a root of (17) one has: R(ω) = 0, S(ω) = 0 Some algebraic manipulations allows to eliminate the trigonometric functions. Next, using the standard trigonometric identity cos 2 (ω τ ) + sin 2 (ω τ ) = 1 which is equivalent where Ω = τ 2 ω 2 . Since τ < τ P ID = 2/p, then , one shows that no positive solution exists for both cases ( ± ). Now, one is able to apply the principle of the argument to investigate the dominancy of s + as a root of ∆, given by ( 6) such that its coefficients satisfy [START_REF] Sipahi | Stability and stabilization of systems with time delay[END_REF], which is equivalent to investigate the dominancy of zero as a root of ∆ given by [START_REF] Vyhlidal | Mapping based algorithm for large-scale computation of quasi-polynomial zeros[END_REF]. Clearly, apart from zero, ∆ and ∆(s) = ∆(s) s 4 have the same roots. But it seems to be more computationally convenient to investigate roots of the deflated analytic function ∆ rather than ∆. Let define the integration contour γ = ∪ 2 k=1 C k which is taken as a counterclockwise closed curve, then an integral over γ is defined as the sum of the integrals over the directed smooth curves that make γ up, as depicted in Figures 2.

Elementary calculations give parametrization of γ:

   z 1 (θ) = Re iθ , for - P i 2 ≤ θ ≤ P i 2 , z 2 (t) = it, for R ≤ t ≤ -R, (20)
where z k (t) designates the parametrization of z along C k for k ∈ {1, 2}. Since ∆ is analytic and dont vanish on the imaginary axis (including zero) then the argument principle asserts:

1 2iπ γ ∂ s ∆(s) ∆(s) ds = Z, (21)
where Z designates the number of the quasipolynomial roots enclosed by γ. Furthermore, the left-hand side of [START_REF] Boussaada | On the coalescence of spectral values and its effect on the stability of timedelay systems: Application to active vibration control[END_REF] gives:

γ ∂ s ∆(s) ∆(s) ds = lim R→∞ 2 k=1 I k (t)
where Since we are interested in the limit when R goes to infinity, one uses the uniform convergence theorem and then explores the dominant terms in R respectively the numerator and the denominator of the integrants of I 1 and I 2 . The last tedious step, consists in evaluating the limit of the resulting integrals I 1 and I 2 which gives Z = 0.

I 1 (θ) = π 2 -π 2 ż1 (θ) ∂ s ∆(z 1 (θ)) ∆(z 1 (θ)) dθ, I 2 (t) = - R -R ż2 (t) ∂ s ∆(z 2 (t)) ∆(z 2 (t)) dt. (22)

IV. ILLUSTRATIVE EXAMPLE

Consider the plant depicted in Figure 1 where P (s) is a delayed first order instable plant and K(s) is a PID controller. So that the performance properties of the closedloop plant are characterized via the transfert function [START_REF] Hale | Introduction to functional differential equations[END_REF] and its stability is characterized by [START_REF] Michiels | Stability and stabilization of timedelay systems[END_REF]. Assume the K(s) admits an instable pole at p = 1 and a delay τ = 1. One tunes the controller gains as prescribed in (8) and such that s + is a root of (6), which gives:

k d = 1/2 -2 + √ 13
This guarantees that s + = -5/2 + 1/2 √ 13 is a root of the quasipolynomial function [START_REF] Michiels | Stability and stabilization of timedelay systems[END_REF] with multiplicity 4. Obviously, s + is negative and its dominancy is asserted by Theorem III.1. Figure 3 exhibits the roots z k corresponding to ∆ which are commun roots with ∆ apart from zero. So that, the roots of ∆ are obtained by the scaling s k = z k + s + apart from the quadruple root at s + . Furthermore, since ∆ is analytic inside and on a closed contour γ, and is not zero on γ, then the number of its within γ is equal to the number of times the image curve of γ under the mapping ω = ∆(z) encircles the origin in the ω-plane. The point ω = ∆(z) moves around as depicted in FigVar, where θ increases from -π/2 to π/2, and y increases from -R to R. Since the curve dont encircles the origin, the argument of ∆ dont increase, see for instance [START_REF] Bellman | Differential-difference equations[END_REF].

V. CONCLUSION

In this paper we investigated the effect of multiple spectral values on the stability of time-delay systems. An extension of the MID property to linear functional differential equations of neutral type is provided. We showed the potential applicability of such a property in a control design problem. The proposed approach yields a systematic tuning for PID controllers satisfying the requirement for rolling off at high frequencies.

)Fig. 1 . 2 )

 12 Fig. 1. A feedback control system

  If (8) is satisfied and |k d | < 1 then s = s + is the rightmost root of quasipolynomial function (6).

Fig. 2 .

 2 Fig. 2. The used contour for applying the argument principle to investigate the dominancy of the multiple root.

Fig. 3 .

 3 Fig. 3.The distribution of the spectrum corresponding to deflated characteristic function ∆ (6) for p = τ = 1 and the controller gains given in[START_REF] Michiels | Continuous pole placement for delay equations[END_REF].

e - 5 /

 5

Fig. 4 .

 4 Fig. 4. The argument variation: the behavior the point ω defined by the mapping ω = ∆(z).

ACKNOWLEDGMENTS

IB & SIN are partially financially supported by a public grant overseen by the French National Research Agency (ANR) as part of the "Investissement d'Avenir" program, through the "iCODE Institute project" funded by the IDEX Paris-Saclay and a grant from Hubert Curien (PHC) BALA-TON, project number 40502NM.