Emma Bell

Soft power and corporate imperialism: maintaining British influence

Keywords: corporate imperialism, international development, soft power, UK foreign policy

It is often suggested that Britain has lost its great power status since the fall of Empire. Yet, whilst its military and economic power has undoubtedly been weakened, it continues to exert power and influence and to promote national interests via the exercise of both hard and soft power. Far from representing a novel strategy, this simultaneous deployment of both forms of power may be considered as a continuation of the dual imperial strategy of gunboat diplomacy and winning hearts and minds at home and abroad. Yet, the postcolonial era does represent some novelty: as was clear under the Conservative-led coalition and is now evident under a Conservative majority government, soft power is no longer exercised principally via cultural diplomacy, through, for example, the British Council, but increasingly via large companies which promote British economic and political interests through corporate imperialism.

British imperialism is alive and well. Far from being overshadowed by American imperialism, it has actually been reinforced by the Anglo-American 'special relationship'.

Although defence spending has been reduced, Britain remains an important military power: it is one of just nine states to possess nuclear arms and it retains 'a highly developed and experienced intelligence capability'. [START_REF] Morris | How Great is Britain? Power, responsibility and Britain's future global role[END_REF] It is still capable of exercising 'hard power' and is currently deployed in over eighty countries around the globe. But it also exercises what Joseph Nye famously referred to as 'soft power'. [START_REF] Nye | Soft Power: the means to success in world politics[END_REF] Whilst hard power relies on military force to retain power and influence, soft power relies above all on the propagation of influence via cultural and political institutions which serve to render the country more attractive in the eyes of the world. There are moments in history when hard power prevails, such as when the dominant position of a country is threatened -an example is that of the United States which increasingly fell back on hard-power strategies following its loss of absolute economic dominance from the early 1990s onwards. [START_REF] Go | Global fields and imperial forms: field theory and the British and American empires[END_REF] Similarly, during the last days of empire, Britain often attempted to maintain its authority via the use of military force. Today, following the fall of empire, Britain only tends to exercise hard power with the support of the Americans.

The two forms of power are not mutually exclusive. Even in Iraq, military force was accompanied by propaganda campaigns aimed at winning the hearts and minds of the Iraqi people, as was often the case in Britain's colonial territories. For Nye, the deployment of both strategies is essential if one hopes to exercise power effectively -what he describes as 'smart power'. [START_REF] Nye | Public diplomacy and soft power[END_REF] This article will demonstrate how one form of power effectively supports the other.

There is nothing new about the simultaneous use of hard and soft power. Yet the means via which power is exercised has changed considerably. It goes without saying that the methods of modern warfare are far removed from those of the past but this article focuses on new strategies of soft power. This continues to be exercised via cultural institutions such as the British Council but is now more likely to be promoted through large companies that are involved in what the British government refers to as 'commercial diplomacy'. As former Conservative foreign secretary William Hague put it, 'We have put commercial diplomacy at the heart of Britain's foreign policy, not only in order to secure our economic recovery and future prosperity but also to ensure our influence in the world is not diminished, promoting both our security and our ability to project our values.' [START_REF] Hague | speech to the Inward Investment Business Summit[END_REF] Soft power is thus not limited to a cultural imperialism that entails making British culture attractive to foreign nations: it now includes the promotion of corporate imperialism -imperialism promoted via British-based companies -which strengthens their economic power whilst simultaneously promoting the interests of the British state. Soft power thus allows British imperialism to spread in a new form.

In order to understand the role of corporations in contemporary British imperialism, it is first necessary to briefly highlight the symbiotic relationship that has always existed between the British state and private companies. Next, it is shown how hard power can serve to reinforce the soft power of companies and of the state itself. But corporations also contribute to the dissemination of soft power independently of the exercise of hard power, notably by providing consultancy services and becoming involved in humanitarian aid projects. Finally, I argue that it is above all soft power, as exercised via corporations, that serves to legitimise the power and authority of the British state today. The role of international development aid provided by the UK's Department for International Development (DfID) is crucial, ensuring much continuity between Conservative-led governments and their New Labour predecessors in the simultaneous pursuit of what are termed ethical as well as realist foreign policies.

A symbiotic relationship between political and economic power

There has always been considerable interdependence between economic and political power. Indeed, English commercial dominance in India predated military and political dominance over the territory, having been established by the East India Company when it was granted a commercial monopoly by the British Crown in 1600. It was the quest for economic dominance and the desire to eliminate competition from other colonial powers that led the Company to establish its own army and administrative department long before the British state took control of the region. [START_REF] Bowen | The Business of Empire: the East India Company and imperial Britain, 1756-1833[END_REF] In other words, it was in large part the existence of a commercial empire which facilitated the creation of a political empire.

Moreover, the British succeeded in establishing informal control over various territories outside the empire proper thanks to commercial links which helped to forge close relations between British and foreign business elites. One notable example is Argentina, [START_REF] Knight | Latin America[END_REF] which is often considered as part of the so-called 'informal' empire on account of the weight of commercial interests that Britain established in the country. [START_REF] Darwin | The Empire Project: the rise and fall of the British world system 1830-1970[END_REF] Up until 1914, Argentina represented a source of investment for British companies and many banks were owned by the British. More generally, the country was 'at least during the years 1880-1914 . . . more important than Egypt or China and, perhaps, even than India as a source of foodstuffs and raw materials, a market and a place for the investment of capital'. 10 The local elites, los estancieros, adopted the gold standard in order to attract investment from London, which meant that they had to accept the rules of the City. 11 Economic dependence thus entailed a degree of political dependence.

For economic historians Cain and Hopkins, the links between economic and political power were essential to the development of British colonial policy since the political and economic elites formed a unified class that shared a close correspondence of interests. They described them as 'gentleman capitalists', unified by the same world view. [START_REF] Cain | British Imperialism[END_REF] According to Cain and Hopkins, since at least 1688, 'the barriers between business and government were no more than mobile Chinese walls'. [START_REF] Cain | [END_REF] Their theory has been criticised, notably for having exaggerated the common mind-set of these elites. White, for example, argues that whilst elites from the City had privileged access to ministers, that did not necessarily mean that they managed to influence colonial policy in a significant way. [START_REF] White | The business and politics of decolonisation: the British experience in the twentieth century[END_REF] In addition, he suggests that there was often a great divergence of opinions between the commercial elites in London and those in the colonies.

Cain and Hopkins' theory actually seems to be more relevant to the recent postcolonial era. The revolving door which allows ministers and other civil servants to move fluidly back and forth between the public and private sectors [START_REF]Appointments taken up by former ministers[END_REF] is without precedent, at least in the period since the second world war. Moreover, the lobbying industry gives large companies privileged access to Parliament, [START_REF]Commons Public Administration Select Committee, Lobbying: access and influence in Whitehall[END_REF] whilst government policies serve to promote the interests of the private sector. [START_REF]ANON. 18 ANON. 19 Nye[END_REF] As I have explained in detail elsewhere, 18 there is a genuine coalition of interests between the public and private sectors in Britain today. This may be said to closely resemble the class of 'gentlemen capitalists' described by Cain and Hopkins.

The power exercised by these elites might be considered as a form of soft power, even if it is often accompanied by hard power strategies. Nye's definition of soft power is generally limited to public diplomacy which seeks to influence the media and public opinion abroad and to forge closer relations with foreign countries through cultural and educational exchanges. 19 Whilst these strategies have always played an important role in the promotion of national interests, companies have become ever more important as a means of extending British influence throughout the world. This trend should be placed in the context of significant budget cuts which have affected the more traditional vehicles for the dissemination of soft power, notably the British Council and the BBC. The BBC World Service had its budget cut by 16 per cent over a fouryear period from 2010-2014 following the last government's spending review in 2010, leading to fears that its capacity to spread soft power throughout the world will be significantly diminished. [START_REF] Bbc | The Future of News[END_REF] The British Council was also negatively affected by the 2010 spending review, subject to a funding cut of 26 per cent over a five-year period up to 2015. [START_REF]FCO performance and finances 2011-12[END_REF] Added to this are the defence spending cuts highlighted above which may make the exercise of hard power somewhat more difficult. This hostile context, combined with the rapprochement between the private and the public sectors, helps us to understand why the British government is increasingly turning to corporations as the principal means of exercising soft power. Whilst spending has been cut elsewhere, the budget for overseas aid increased to reach the UN target of 0.7 per cent of gross national income in 2014. The new Conservative-led government has recently passed the International Development Act which enshrines this minimum aid spending target in law. As will be demonstrated below, much of this spending has actually gone to private companies that are to contribute to the spread of British soft power abroad. This now seems to be the central prong of foreign policy. Yet, this does not mean that hard power has been abandoned. Indeed, hard power often facilitates the exercise of soft power, particularly in the form of corporate imperialism.

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Hard power in the service of soft power

The occupation of Iraq serves as an example of the deployment of hard power in the service of soft power. Whilst the occupation might initially have appeared to be a failure of soft power to the extent that it damaged the reputation of Americans and their British allies, [START_REF] Nye | Unrevised transcript of evidence taken before The Select Committee on Soft Power and the UK's Influence[END_REF] it nonetheless allowed them to establish important economic interests in the region and thus to spread their influence. Shortly after the invasion of Iraq in March 2003, the coalition government established by the Americans launched extensive reforms of the Iraqi economy in order to open it up to international markets. For the US Department of State, a planned economy represented yet another example of Sadaam Hussein's tyranny, creating an unfavourable business environment. [START_REF] Whyte | The crimes of neoliberal rule in occupied Iraq[END_REF] Consequently, emergency legislation was passed, notably a law allowing the privatisation of public services and companies, opening them up to foreign investment. [START_REF] Whyte | The crimes of neoliberal rule[END_REF] Numerous British (but more US) companies benefited from this law. DfID, the British Department of State responsible for providing humanitarian aid to Iraq, played a central role, granting contracts worth millions of pounds sterling to British companies. Many of these were consulting agencies, notably Adam Smith International, affiliated to the Adam Smith Institute, the influential think-tank whose reports largely helped to justify privatisation programmes in Britain at the end of the 1980s. [START_REF]Corporate Carve-up: the role of UK Corporations in Iraq[END_REF] This agency advised the Iraqi government on the privatisation of public industries and utilities. [START_REF] Smith | Devising a strategy for state enterprise reform in Iraq[END_REF] Private military and security companies also benefited from contracts awarded by DfID, which provided them with £278 million of its total reconstruction budget. 27 The extent of this subcontracting of security services to foreign companies is without precedent in British history.

The establishment of numerous British companies in Iraq allowed the British government to exercise soft power whilst projecting a positive national image which highlighted its role in the reconstruction and the securitisation of a country ravaged by years of famine and conflict. It represented an attempt to demonstrate that foreign policy was indeed guided primarily by ethical concerns, allowing the then New Labour government to inscribe its policy in Iraq as the continuity of its humanitarian interventions in Kosovo and Sierra Leone. Indeed, the companies benefiting from DfID contracts advertise their commitment to the positive values to which the United Kingdom professes to adhere. Adam Smith International, for example, proudly declares that it is working 'to fight against poverty and for economic growth for the citizens of countries facing the need for good governance, economic and social change'. 28 The democratic reconstruction of such countries is thus seen to go hand in hand with the neoliberal reconstruction of its economy.

Whilst the British experience in Iraq allowed the UK to follow its own post-imperial project, boosting the power of British capital abroad, ironically the country could not have benefited from the occupation had it not been part of the 'coalition of the willing' alongside the Americans. Yet, there are many fields in which the UK is not so dependent on its favoured allies, notably in the sale of arms and security expertise, areas in which it does not need to rely on the superior strength of American hard power.

The defence and security industries as tools of soft power

The main British company engaged in the sale of arms to foreign countries is BAE Systems which brings in profits totalling approximately 32 billion dollars per year. 29 But the company is not only important for the British economy. Its sales play an important diplomatic role, allowing the UK to extend its soft power to the countries with which it does business. This viewpoint was clearly expressed by Nicholas Beadle, a researcher associated with the Royal United Services Institute, a think tank specialising in research on security and defence. In his witness statement to the parliamentary Select Committee on Soft Power and the Influence of the United Kingdom, he asserted that the sale of arms to countries which would otherwise be incapable of defending themselves is a particularly useful way to forge good long-term diplomatic relations and exercise political influence. Summing up this statement, the chair of the select committee declared, 'what is being done on the military side, including defence sales, is all part of supporting national interests more effectively than in the past, and it is very positive'. [START_REF]House of Lords Select Committee on Soft Power and the UK's Influence[END_REF] The consulting industry in transnational security also has significant value for the UK in the exercise of soft power. These companies, notably the British-based Control Risks, propose a large range of high-range security services and operate on a global scale. [START_REF] Reilly | L'évolution de l'offre des professionnels du risque mondialisé[END_REF]

Control

Risks does not only sell its consulting services but also seeks to influence the governments which buy its services. Although it claims to be independent, it works closely with the agency Security in Complex Environments, which seeks to influence the British government to 'contribute to the debate, shape policies and influence international fora'. [START_REF]SCEG 'Security in Complex Environments Group[END_REF] For these companies to exert an effective influence on foreign governments, they need to be more than the purveyors of security and defence solutions but also to embody positive values and be seen to promote democracy and humanitarianism. As Nye suggests, soft power often involves the transmission of the desire to emulate the other. [START_REF] Nye | Public diplomacy and soft power[END_REF]

Soft power and the legitimation of imperialism

The British government and private companies share the same aim with regard to foreign policy: 'to promote British business interests, to work for open economies, to combat protectionism, and to work to remove barriers to business'. [START_REF] Mitchell | Oral evidence provided to the UK Parliamentary Committee on Soft Power and the UK's Influence[END_REF] Yet, these aims must be presented as being in the best interests of the countries concerned. Imperialist aims are thus dressed in the language of democracy and humanitarianism. The concern with self-interest represents a perpetuation of the realist tradition in British foreign policy as espoused by Lord Palmerston, then Conservative foreign secretary, in 1848, when he declared that Britain has 'no eternal allies, and we have no perpetual enemies. Our interests are eternal and perpetual, and those interests it is our duty to follow.' [START_REF] Cited In | Interpreting ethical foreign policy: traditions and dilemmas for policymakers[END_REF] Yet this realist approach, commonly associated with the Conservative Party, has been combined with a more ethical approach to foreign policy, most often identified with New Labour. In practice, New Labour never abandoned a realist approach, just as the Conservatives also identify with a broader ethical tradition. 36 Indeed, it could be argued that this dual approach to foreign policy characterised the era of empire, just as it characterises the contemporary post-colonial era, even if there is more overt emphasis placed on the humanitarian/ethical dimension today. The Conservative Party has largely embraced the New Labour rhetorical emphasis on values whilst continuing to work towards the furtherance of British self-interest in foreign policy. [START_REF] Beech | Humanitarian intervention and foreign policy in the Conservative-led coalition[END_REF] Presenting economic protectionism as a barrier to democracy, as in the case of Iraq discussed above, is a good example of such a dual approach to foreign policy. The 'democratisation' of the country justified the imposition of anti-democratic neoliberal reforms which in practice took power away from local people by opening local markets up to exploitation by foreign capital. As Ayers has explained with reference to imperial strategies more generally, 'self-determination does not mean autonomy. Rather it means the "freedom" to embrace the rules, norms and principles of the emerging (neo)liberal global order.' [START_REF] Ayers | Imperial liberties: democratisation and governance in the "new" imperial order[END_REF] Democratic ideals are thus used to promote neoliberalism and the interests of private capital which depend upon it.

The same might be said for humanitarian projects which are underpinned by neoliberal values promoting commercial interests whilst simultaneously spreading British soft power abroad. DfID plays a vital role here: in the words of Adam Smith International, it 'is arguably one of the most influential voices in the world of international development, second only, perhaps, to the World Bank and IMF'. [START_REF] Smith | Devising a strategy for state enterprise reform in Iraq'. 40 OECD, 'Aid to developing countries rebounds in 2013[END_REF] In terms of aid funding, it is the largest contributor of aid in real terms, after the United States, and ranks fifth overall if we take its contribution as a percentage of its gross national income. 40 Much of this money is given directly to the Private Infrastructure Development Group (PIDG), a multilateral organisation founded in 2002 by DfID and three other international partners. It brings together different national aid agencies which provide its funding which it then uses to finance private companies which carry out infrastructure projects in poor countries. DfID is the single biggest contributor to the PIDG, contributing $662m out of a total of almost $959m of the total amount received for investment in private companies, affiliated programmes, project development and administration between 2002 and 2013. [START_REF] Pidg | PIDG Annual Report[END_REF] It is notable that the level of investment has increased significantly in recent years, rising from an annual expenditure of over $23m in 2010 to $233m in 2013. 42 DfID aid funding also goes directly to private companies. One example is that of the British multinational Price Waterhouse Coopers which has been granted £355 million between 2011 and 2017 to manage the 'Girls' Education Challenge', a project to encourage private investment in the provision of private education for girls in poor countries. [START_REF] Curtis | Profiting from Poverty, Again: DfID's support for privatising education and health[END_REF] Facilitating private sector investment in poor countries is now regarded as central to DfID's aid initiative. As its strategic framework made clear in 2014:

The private sector is the engine of growth. Successful businesses drive growth, create jobs and pay the taxes that finance services and investment. Foreign investment, and particularly exports, can accelerate domestic development. Governments have a role in encouraging responsible investment, and helping markets work better for poor people. [START_REF]DfID, Economic development for shared prosperity and poverty reduction: a strategic framework[END_REF] Yet, as the Independent Commission for Aid Impact (ICAI), the body charged with scrutinising UK aid spending, highlights, there is limited evidence to demonstrate that donor support for private investment brings tangible benefits to the poor. [START_REF] Curtis | ICAI, Business in Development[END_REF] It notes that it is unclear 'whether DfID's contribution achieves impact beyond what would have been achieved anyway'. 46 In the case of privatised education, many families will be unable to afford even the lowest of fees and private schools may only benefit children who would have previously been enrolled in public schools in any case. 47 Furthermore, there is some evidence that DfID funding of private investment may do more harm than good. The National Audit Office (NAO) criticised DfID for lack of oversight over how the large sums of public money donated to the PIDG are spent. [START_REF] Nao | Oversight of the Private Infrastructure Development Group[END_REF] Consequently, there have been allegations that money has inadvertently encouraged the very kind of corruption that is said to be a hindrance to economic development in poor countries. In January 2015, for example, it was alleged that £19m of money donated to PIDG had fallen into the hands of a convicted money launderer in Nigeria. [START_REF] Syal | Hundreds of millions given to foreign aid fund that lacked scrutiny, say MPs[END_REF] The British government's commitment to private sector investment as a means of combating poverty and promoting the development of poor countries suggests, in the absence of hard evidence concerning its efficiency, that policy is driven more by self-interest than by genuine humanitarian concerns. Yet, as noted above, one does not necessarily preclude the other, realist and value-based approaches to foreign policy having always been adopted simultaneously. The British government may possibly be motivated by humanitarian concerns but these appear to be ideologically driven and may conflict with 'realist' motives. Indeed, it would seem that the receipt of aid is conditional on recipient countries accepting neoliberal ideology and policies, notably economic restructuring to open markets up to investment for British companies. Neoliberal values are not only spread by government but have also permeated humanitarian aid agencies, whether they are independent of government or not.

These agencies increasingly adopt a business culture characterised by a concern with presentation and the setting or clear and quantifiable targets. [START_REF] Barnett | Humanitarianism transformed[END_REF] This leads to the promotion of a form of 'soft' neoliberalism whereby ideological objectives are dissimulated behind humanitarian rhetoric. [START_REF] Roberts | The state, empire and imperialism[END_REF]

Commercial diplomacy and great power status

Using humanitarian aid to boost British business and investment abroad goes hand-in-hand with the government's broader aim of using commercial diplomacy as the primary means to enable Britain to achieve great power status. As William Hague recently noted, reflecting on his time as foreign secretary under the coalition government, We've … opened twenty new embassies and consulates and are closing very few. As a result, the link between economic and foreign policy has been restored, with a far more commercial emphasis on trade which I think if you believe, which is obviously true, that the economy is at the heart of everything else, should be the case … I think our creation of new British chambers of commerce in about twenty countries is an excellent idea, more on the German model of how trade is promoted overseas. If those twenty are shown to be successful, that is the right sort of direction to go into for the long term … Britain is expanding again, in terms of commercial effort, ministerial visits, members of embassies and consulates and we've done all that at this time of austerity. [START_REF] Hague | An interview with William Hague[END_REF] Discussion of foreign policy in the most recent Conservative manifesto was limited to how it can be used 'to boost our prosperity' by 'push [ing] for freer global trade, concluding major trade deals with the US, India and Japan and reinvigorating the World Trade Organisation'. [START_REF]Conservative Party, The Conservative Party Manifesto 2015: strong leadership, a clear economic plan[END_REF] Rather than retreating from the world stage, it would seem that the Conservative government is simply seeking to secure power and influence by focusing on economic influence as a tool of soft power rather than on military influence. Yet its commitment to hard power has not been abandoned either. Under Prime Minister Cameron, the UK has been a willing partner in the military coalition to fight Islamic State and, at the time of writing, the new Home Secretary Michael Fallon, is discussing the possibility of seeking parliamentary support once again to launch air strikes against Syria. In this scenario, intervention may be justified on ethical grounds, just as military intervention in Afghanistan or Iraq was presented under New Labour, but it is also undoubtedly a means for Britain to project its power on the world stage.

It has certainly not abandoned its old imperial pretensions, falling back on its economic and military power to maintain influence and authority throughout the globe. However, Britain does not act alone but in partnership with old allies, notably the United States, which shares the goal of spreading neoliberal ideology throughout the globe.

Consequently, US companies have also benefited from British foreign policy: one example is the consultancy firm GRM International which has received DfID funding to promote market solutions to healthcare problems in poor countries. 54 Whilst America provides the military might, it would seem that the UK is becoming ever more adept at bringing its expertise in the spread of soft power to the Anglo-American relationship. This mutually beneficial partnership thus supports both British and American imperialism.

Conclusion

This discussion has identified a number of new inflections in British foreign policy, notably the Conservative Party's more overt commitment to an 'ethical' foreign policy focused on international development. Yet it has shown that the Party in government has also pursued a classic conservative realist approach, ensuring that such a policy works in its own best interests. There is much continuity with New Labour which had already experimented with corporate imperialism in Iraq whilst focusing on the ethical dimension of military intervention. However, there is also significant continuity with the imperialist project of the colonial era: foreign nations rendered economically dependent on Britain find themselves under her political influence, forced to accept fundamental ideological transformation; there is today a new class of 'gentleman capitalists', economic and political elites who share the same neoliberal vision of the world; hard power and soft power strategies are often combined to ensure the maximisation of national self-interest. Britain remains a significant player on the world stage yet, unlike in the past, she often needs to act in partnership with like-minded allies such as the United States. Nevertheless, whilst the latter may possess superior military might, it would seem that Britain's soft power credentials are more important. It is Britain that is currently doing most to support corporate imperialism, notably through significant spending on international development and the funding of organisations such as the PIDG, and it is Britain that comes out top on the international soft power ranking, just ahead of the US. [START_REF] Mcclory | The New Persuaders III: an international ranking of soft power[END_REF] British companies, particularly in the consulting and security sectors, are today the principal means of exporting soft power around the world. Engaging with humanitarian projects and the promotion of democracy, they are essential actors in a new project of British imperialism. It would thus seem that the 'informal' British empire is alive and well.


Yann Denieul

Joël Bordeneuve

Daniel Alazard

Clément Toussaint

Gilles Taquin

Ph D Student

Professor Av Edouard Belin

Aiaa Member

Multicontrol Surface Optimization for Blended Wing-Body Under Handling Quality Constraints

Control architecture sizing is a main challenge of blended-wing-body design. This aircraft configuration typically features redundant elevons located at the trailing edge of the wing, acting simultaneously on pitch and roll axes. Consequently, a proper sizing requires one to consider coupled longitudinal and lateral criteria. Moreover, significant hinge moments due to large control surface areas, combined with high deflection rates in order to safely control the longitudinal instability, may result in excessive power consumption and actuator mass penalty. Therefore, it is highly desirable at the preliminary design level to minimize control surface areas, while ensuring adequate closed-loop handling qualities, with limited deflections and deflection rates. The problem of integrated design of control surface sizes and flight control laws for an unstable blended-wing-body aircraft is addressed here. The latest tools for H ∞ nonsmooth optimization of structured controllers are used to optimize in a single step the gains for both longitudinal and lateral control laws, as well as a control allocation module, while minimizing the control surface span. The following constraints are ensured: maximal deflection angles and rates for 1) pilot longitudinal pull up, 2) pilot bank angle order, and 3) longitudinal turbulence. Using this coupled approach, significant gains in terms of the outer elevon's span as compared to the initial layout are demonstrated, whereas closed-loop handling quality constraints are guaranteed. Nomenclature n controls = number of control surfaces y 2 inb = inboard y position of elevon 2 y 5 out = outboard y position of elevon 5 y init 2 inb = initial inboard y position of elevon 2 y init

I. Introduction

A MONG other disruptive aircraft configurations, the blended- wing-body (BWB) has been considered for years as a promising candidate for the future of civil aviation [START_REF] Liebeck | Design of the Blended Wing Body Subsonic Transport[END_REF]. The rationale for this game-changing configuration is as follows: instead of considering separate geometrical components for each basic function of an aircraft (namely, lift, transport, control, and propulsion), the BWB gathers the three functions of lift, transport, and control into a single lifting surface. As a consequence of merging these functions, an overall improved efficiency is expected, involving significant gains in terms of fuel consumption. Most gains come from a decreased wetted area as compared to a classical tube-and-wing design; but, also, structural mass improvements are expected. As a result, studies from the literature have regularly forecast gains ranging from 15 to 20% in terms of fuel consumption as compared to a conventional configuration [START_REF] Liebeck | Design of the Blended Wing Body Subsonic Transport[END_REF][START_REF] Martínez-Val | Medium Size Flying Wings[END_REF][START_REF] Qin | Aerodynamic Considerations of Blended Wing Body Aircraft[END_REF][START_REF] Bolsunovsky | Flying Wing-Problems and Decisions[END_REF]. This paper focuses on an Airbus long-range BWB configuration.

Major challenges are yet to be solved before a potential entry into service, including control-related issues [START_REF] Roman | Aerodynamic Design Challenges of the Blended-Wing-Body Subsonic Transport[END_REF]. These issues first originate from the nature of the control devices used for this configuration: the BWB (also known in the literature as hybrid wingbody [START_REF] Garmendia | Alternative Trim Analysis Formulations for Vehicles with Redundant Multi-Axis Control Surfaces[END_REF]) is controlled with multicontrol surfaces (also named elevons), usually spanning the whole trailing edge and acting as pitch and roll devices. Among the challenges associated with this technology, new handling quality criteria are required in order to take into account the combined authority of control surfaces on longitudinal and lateral axes. This was already addressed in previous work [START_REF] Saucez | Handling Qualities of the Airbus Flying Wing Resolution[END_REF][START_REF] Saucez | Optimization of Engine Failure on a Flying Wing Configuration[END_REF]. Then, these control surfaces typically lack longitudinal pitch authority because of their small longitudinal lever arm as compared to that of a conventional elevator.

Flight control system (FCS) sizing is also affected by these unusual effectors: failure cases of actuators or control surfaces induce a loss of authority on both pitch and roll axes. The failure case analysis then tends to be more complex than for classical aircraft configurations, where functions are segregated for each axis. However, we will make the assumption that a failure case analysis sizes the number of redundant elevons and the number of actuators per elevon [START_REF] Roskam | Airplane Design[END_REF] rather than the elevon's total area, which is the main concern of this paper. Failure scenario analyses for elevon layout sizing, which were already addressed by Garmendia et al. [START_REF] Garmendia | Method for Evaluating Electrically Actuated Hybrid Wing Body Control Surface Layouts[END_REF] and in the Active Control for Flexible 2020 Aircraft 2020 studies [START_REF] Kozek | Modeling and Control for a Blended Wing Body Aircraft[END_REF], are therefore outside the scope of our study.

Then, concerning control surface area sizing, two phenomena have a combined detrimental effect on both the actuator mass and power consumption:

1) On the one hand, trailing-edge elevons induce high aerodynamic hinge moments (HMs) due to their large area [START_REF] Roskam | Airplane Design Part VI: Preliminary Calculation of Aerodynamic, Thrust and Power Characteristics[END_REF]. It has been known since the work of Roman et al. [START_REF] Roman | Aerodynamic Design Challenges of the Blended-Wing-Body Subsonic Transport[END_REF] that hinge moments are related to the control surface area through a "squarecube" law: control surface areas increase as the square of the scale λ 2 , whereas hinge moments increase with the cube of this scale λ 3 . Large BWB control surfaces lead to high hinge moment requirements. Moreover, the control surface's inertia is proportional to λ 5∕2 .

From this perspective, during the preliminary design phase, attention should be focused on downsizing the control surface areas.

2) On the other hand, high deflection rates result from the longitudinal stabilization of an unstable configuration [START_REF] Roskam | Airplane Flight Dynamics and Automatic Flight Controls[END_REF]. Indeed, the Airbus BWB features a negative static margin, especially at low speed (see Sec. II.B). Otherwise said, our aircraft features an unstable short-period mode. For this reason, it requires a permanent stability augmentation system (SAS) in order to guarantee adequate safety and handling qualities. However, it was shown in a previous study [START_REF] Denieul | Interactions of Aircraft Design and Control: Actuators Sizing and Optimization for an Unstable Blended Wing-Body[END_REF] that, the more unstable an aircraft, the faster its control surfaces need to move in order to maintain the equilibrium under disturbance. This effect is further increased on the BWB because elevons lack a longitudinal lever arm with respect to the center of gravity (CG). As a consequence, their longitudinal efficiency is reduced as compared to an elevator of the same area on a conventional airplane. Thus, more demanding deflections and deflection rates are needed to create the same pitching moment. As a result, deflection rates of 60 deg ∕s [START_REF] Garmendia | Multidisciplinary Approach to Assessing Actuation Power of a Hybrid Wing Body[END_REF] and even 100 deg ∕s [START_REF] Wildschek | Flight Dynamics and Control Related Challenges for Design of a Commercial Blended Wing Body Aircraft[END_REF] are often considered when presizing the FCS for a BWB, as compared to nominal values of about 30 deg ∕s for conventional tail-aft design actuators. During the preliminary design phase, control surface pitch efficiency should then be sought to be maximized (for instance, by increasing the control surface area as much as possible), which conflicts with the previously mentioned requirement on hinge moment limitation.

Both large hinge moments and high deflection rates have a direct impact on the FCS sizing and secondary power consumption. Indeed, as stated by Garmendia et al. [START_REF] Garmendia | Assessment of Electrically Actuated Redundant Control Surface Layouts for a Hybrid Wing Body Concept[END_REF], secondary power for the FCS P FCS may be evaluated in a preliminary way by Eq. ( 1):

P FCS X n controls i1 HM max i ⋅ _ θ max i (1)
where HM max i and _ θ max i are the maximum hinge moment and maximum deflection rate of the ith control surface, respectively; and n controls is the number of control surfaces.

At the preliminary design phase, when the actuator's sizing is not yet frozen, the deflection rate is a direct consequence of the control law design. Also, the traditional way of sizing conventional control surfaces considers simplified open-loop handling qualities criteria, such as the roll rate target for the ailerons or the pitch rate target for the elevator. Such an approach is no longer valid for BWB control surface sizing due to the natural pitch instability: control surface areas may be largely sized by stabilization requirements, so sizing requires considering control laws at the early design phase. Control law design in turn depends on the effectiveness of control surfaces. This coupled problem is known in the control community as plant-controller optimization or integrated design and control. The classical way of handling this problem involves an iterative approach: effectors are sized based on engineering rules, and then a control law is designed. If the requirements are not met, then the sizing is changed based on the existing control law, and so on. However, it has been proved [START_REF] Fathy | On the Coupling Between the Plant and Controller Optimization Problems[END_REF] that, in addition to being time-consuming, this approach may miss the optimum because of the tightly coupled nature of the problem. Consequently, several approaches seek to solve these combined problems in a single step. "Plant-controller optimization" has been studied in a variety of fields, such as chemistry [START_REF] Ricardez-Sandoval | Integration of Design and Control for Chemical Processes: A Review of the Literature and Some Recent Results[END_REF][START_REF] Ricardez-Sandoval | Simultaneous Design and Control of Chemical Processes with Application to the Tennessee Eastman Process[END_REF], autonomous underwater vehicles [START_REF] Silvestre | Plant-Controller Optimization with Applications to Integrated Surface Sizing and Feedback Controller Design for Autonomous Underwater Vehicles (AUVs)[END_REF], and astronautics [START_REF] Denieul | Interactions of Aircraft Design and Control: Actuators Sizing and Optimization for an Unstable Blended Wing-Body[END_REF][START_REF] Alazard | Avionics/ Control Co-Design for Large Flexible Space Structures[END_REF][START_REF] Denieul | Integrated Design and Control of a Flying Wing Using Nonsmooth Optimization Techniques[END_REF].

In the field of aeronautics, two complementary approaches were studied. The first method considered integrating a stability and control module into a multidisciplinary optimization (MDO) process [START_REF] Welstead | Conceptual Design Optimization of an Augmented Stability Aircraft Incorporating Dynamic Response and Actuator Constraints[END_REF][START_REF] Perez | A Multidisciplinary Optimization Framework for Control-Configuration Integration in Aircraft Conceptual Design[END_REF]. Longitudinal and lateral control laws were synthesized using a linear quadratic technique, and they were incorporated into an optimization process, together with an aerodynamics, propulsion, weight, and performance module. Optimization was run with and without taking into account the stability augmentation system for a conventional tail-aft design. The results showed improved performance for the relaxed static stability due to smaller elevator sizing. Other MDO studies also considered stability constraints for BWB planform optimization [START_REF] Mader | Stability-Constrained Aerodynamic Shape Optimization of Flying Wings[END_REF][START_REF] Lyu | Aerodynamic Design Optimization Studies of a Blended-Wing-Body Aircraft[END_REF][START_REF] Haghighat | Aeroservoelastic Design Optimization of a Flexible Wing[END_REF].

A second, more control-oriented approach takes advantage of optimization tools developed for controller design in order to simultaneously optimize a controller and some meaningful physical parameters. Niewhoener and Kaminer [START_REF] Niewoehner | Integrated Aircraft-Controller Design using Linear Matrix Inequalities[END_REF] optimized, in a single loop, a longitudinal controller and elevator control surface using a linear matrix inequality (LMI) framework. The handling quality constraints were expressed in terms of the LMI. The resulting controller was full order. This approach was later developed in other studies of integrated longitudinal controller/control surface sizing [START_REF] Liao | An LMI-Based Optimization Approach for Integrated Plant/Output-Feedback Controller Design[END_REF][START_REF] Liao | Mixed H2∕H∞ Sub-Optimization Approach for Integrated Aircraft/Controller Design[END_REF] and design of hypersonic vehicles [START_REF] Sridharan | Integrated Design and Control of Hypersonic Vehicles[END_REF]. More recently, nonsmooth optimization methods enabling structured linear varying parameter (LPV) controllers were applied to the longitudinal integrated design and control problem [START_REF] Lhachemi | A Structured-Based Optimization Approach for Integrated Plant and Self-Scheduled Flight Control System Design[END_REF].

We propose to extend this approach to longitudinal/lateral integrated design and control of a BWB by optimizing together a three-axis control law and a control surface total span, using nonsmooth optimization techniques for fixed structure controllers. More precisely, we propose to optimize in a single step the control surface span, the control allocation module, and the flight control laws in order to guarantee longitudinal and lateral handling quality constraints with a minimum control surface size. The main contributions of this paper as compared to the state-of-the-art literature are threefold: 1) applying the methodology of integrated design and control to a BWB, 2) extending the plant-controller optimization problem to a coupled longitudinal/lateral case, and 3) using optimization tools for fixed-structured controllers in order to solve this problem.

This paper is organized as follows: in Sec. II, the flight dynamics models are presented. Then, Sec. III introduces the strategy for parameterizing the elevon's total span and obtaining a parametrized state-space representation suitable for optimization. The integrated design and control problem of computing structured longitudinal/ lateral control laws gains together with optimal elevons size is presented in Sec. IV, and the results are discussed in Sec. V.

II. Aircraft Flight Dynamics and Control

In this section, the models used in Secs. III and IV for design, control, and simulation are described. The configuration studied in this paper is a long-range BWB for which the planform results from optimization studies on high-speed performance with constraints on the low-speed pitching moment [START_REF] Meheut | Aerodynamic Shape Optimizations of a Blended Wing Body Configuration for Several Wing Planforms[END_REF]. The focus of this work is the sizing of control surfaces; thus, the planform is considered constant. The planform and initial control surface layout are visible on Fig. 1a This work focuses on the elevon's sizing based on closed-loop longitudinal and lateral handling qualities criteria. Consequently, three-axis nonlinear flight dynamics equations are used to simulate the aircraft motion. Using Newton's second law,

m _ V g mg F aero F thrust (2) J _ Ω G M G aero M G thrust ( 3
)
where m and J are the aircraft mass and inertia, respectively; V g is the ground speed; _ V g is its derivative with respect to the R 0 x 0 ; y 0 ; z 0 Earth reference frame; and Ω G is the aircraft rotation vector around its center of gravity. F aero and M G aero denote the aerodynamic forces and moments, respectively; whereas F thrust and M G thrust are the forces and moments resulting from engine thrust, respectively. For an extensive development of these equations, the reader may refer to [START_REF] Boiffier | The Dynamics of Flight: The Equations[END_REF].

To perform the control law synthesis and linear analysis, these equations are linearized around equilibrium flight points. These initial equilibria are computed for the following conditions: zero flight-path angle, sideslip, and bank angle. The choice of different flight points in terms of mass, Mach, and altitude is discussed in Sec. II.E. The following state-space representation was used:

δ _ V δ _ α _ q δ _ θ _ β _ r _ p _ ϕ 3
x V x α x q -g x β 0 0 0 -z V -z α 1 -z q 0 0 0 0 0 0 m α m q 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 y β y r -
1 0 • • • 0 0 . . . . . . . . . 1 0 0 • • • 0 1 V e g z V V e g z α V e g z q 0 • • • 0 3 7 7 7
where X δV δα q δθ β r p ϕ T is the state vector composed of δV V -V e , which is the relative airspeed with respect to the equilibrium speed; δα α -α e and δθ θ -θ e are the relative angle of attack and pitch attitude with respect to the equilibrium, respectively; sideslip is β; and p, q, and r are the rotation rates of the aircraft with respect to the Earth reference frame in roll, pitch, and yaw, respectively. Comprehensive expressions of the matrix coefficients are given in the Appendix. The controls are Δδx δx -δx e , which is the relative symmetric thrust with respect to the equilibrium thrust; and Δδm i δm i -δm e , i 1; : : : ; 10, is the relative deflection of the ith elevon control surface with respect to the equilibrium position. Elevons for the BWB are acting not only for maneuvers but also as trim devices. Equilibria for this study were performed using all elevons with equal deflection. Other studies showed the benefits of segregating trim and maneuver control surfaces [START_REF] Garmendia | Alternative Trim Analysis Formulations for Vehicles with Redundant Multi-Axis Control Surfaces[END_REF], but this trade is out of the scope of our work. However, for maneuvers, each of the 10 elevons is actuated independently, through a control allocation strategy presented in Sec. IV.B. The elevon's layout, shown in Fig. 1a, is ordered in the control vector as follows:

δm i;i1; : : : ;10 LDQ1 : : : LDQ5; RDQ1 : : : RDQ5

The control vector also contained the rudder deflection δn. Although two rudders are visible on the configuration of Fig. 1a, it was chosen for the sake of clarity to group them as a single control with twice the efficiency of one rudder; the aim of our study is, indeed, not to size vertical surfaces, but only elevons. The output vector Y gathers the aircraft states and vertical load factor δNz Nz -1, with the vertical load factor for the initial equilibrium being equal to one. Finally, a turbulence effect is included as a vertical velocity w z expressed in the Earth reference frame, under the assumption that it acts as an increment of angle of attack. The model used for turbulence is described in Sec. II.C.

B. Mode Analysis

The modes of the eight-state aircraft dynamics presented in Sec. II. A are depicted in Fig. 2. For the sake of clarity, poles are shown only for one mass and altitude, with the Mach number varying between 0.45 to 0.85. More generally, all of this work is performed for the maximum aft CG allowed for each mass in order to be at the most penalizing configuration from a longitudinal stability perspective.

The main point to retain from Fig. 2 is the strong instability of the short-period oscillation, with a frequency of approximately 1.2 rad ⋅ s -1 . Even though it is not comparable to the most unstable fighter aircraft ever designed, such as the X-29 (see [START_REF] Rogers | X-29 H-Infinity Controller Synthesis[END_REF], with instability levels approaching 6 rad ⋅ s -1 ), this instability level is an order of magnitude beyond current civil aircraft: these are currently designed to be at least neutrally stable in an open loop. Identifying and minimizing the impact of this unstable mode on control surface sizing are the main concerns of this paper.

In Fig. 2, a poorly damped Dutch roll (DR) mode is also visible. For some flight points, it even turns slightly unstable. A previous study [START_REF] Saucez | Handling Qualities of the Airbus Flying Wing Resolution[END_REF] sized the vertical surfaces of this configuration in order to guarantee a minimum instability level of ξ -5% for the Dutch roll on the whole flight envelope.

C. Turbulence Model

A Dryden continuous turbulence model is used for simulating vertical continuous turbulence w z . A band-limited white noise e z is passed through a forming filter approximating the Dryden velocity spectra. The transfer function has the following expression from [START_REF] Standard | Flying Qualities of Piloted Aircraft[END_REF]:

H w z s w z e z s σ z 2L z πV r 1 3 p L z ∕Vs 1 L z ∕Vs 2 (6)
where V is the aircraft airspeed, and L z is the vertical turbulence scale length that was set to 500 m. This value is typical from [START_REF] Gage | Creating a Unified Graphical Wind Turbulence Model from Multiple Specifications[END_REF] for simulating turbulence at altitudes above 2000 ft. Note that σ z is the turbulence intensity; in this work, the value σ z 5 m ⋅ s -1 is chosen in order to simulate severe turbulence intensity. The underlying rationale for this choice is that we should size control surfaces in order to guarantee a stable closed-loop behavior, even for extreme turbulence events. For an unstable aircraft such as the BWB studied here, if the deflection stops or rate limits of the actuators are reached during stabilization, then the aircraft dynamics may revert back to an unstable behavior. This may lead to the aircraft loss. Thus, extreme turbulence was chosen for control surface sizing.

D. Actuator Model

A second-order actuator model accounting for their bandwidth and damping is used:

y act u act ω 2 0 s 2 2ξω 0 s ω 2 0 ( 7
)
A single bandwidth and damping of, respectively, ω 0 8.8 rad ⋅ s -1 (1.4 Hz) and ξ 0.8 was used. Previous studies considered allocating different bandwidths for all control surfaces [START_REF] Denieul | Interactions of Aircraft Design and Control: Actuators Sizing and Optimization for an Unstable Blended Wing-Body[END_REF][START_REF] Denieul | Integrated Design and Control of a Flying Wing Using Nonsmooth Optimization Techniques[END_REF]; however, this is outside the scope of the present paper. A 100 ms delay accounting for sensors, computers, and data processing is included in the control law synthesis and simulation. Such a value is typical for an actual control chain delay; challenging the current technology would, of course, improve the control law performance but is not considered here. During synthesis on the linearized model, a second-order Padé approximation is used; for simulation, the actual delay is taken into account. Actuators and delay will be embedded in the control law structure in Sec. IV.C.

E. Description of the Flight Envelope

To properly size the aircraft from a stability and control perspective, our integrated design and control process was run on the entire flight envelope in terms of mass, Mach, altitude, and for aft CG. More precisely, four masses were examined: m 200T; 250T; 300T; 350T. Mach numbers were discretized from 0.25 to 0.85 with steps of 0.1; for each Mach number, a sweep was performed on the airspeed V air with steps of 20 kt. This is equivalent to varying the altitude for a given Mach. Admissible altitudes were given by a velocity maximum operating limitation for the low-altitude limit and a maximum of the buffet lift coefficient and operational ceiling for the high-altitude limit. The resulting flight points are depicted in Fig. 3.

III. Parametric Representation of Control Surface Span Variation

In this section, a process for obtaining a continuous approximation of control surface efficiencies for varying the outer elevon's span is presented. This part aims at obtaining a continuously parametrized state-space representation so that the continuous optimizer presented in Sec. IV may use the continuous variable representing the elevon's size as an optimization variable.

A. Geometric Parametrization of Control Surfaces Span

In "classical" plant-controller optimization problems applied to aeronautics, attention is drawn to minimizing the elevator size under longitudinal constraints. For the BWB lacking (by definition) an elevator, this process is not applicable here. Therefore, the outer elevon's total span is chosen as a plant figure of merit to be minimized. More precisely, a variable η representing the ratio of the outer elevon's total span as compared to the initial control surface's span is introduced. In Fig. 1a, the initial elevon's layout, corresponding to a parameter value of η 1, is presented. This initial layout features five control surfaces on each side of the wing spanning the whole trailing edge, except a gap between elevons 1 and 2 for engine pylon integration: elevons are numbered from inboard to outboard. The elevon's relative chord is limited in the x-wise position by cabin integration for elevon 1, and by the rear spar for elevons 2 to 5. So, it was decided to keep the elevon's relative chord constant, as well as the elevon's number; as mentioned in the Introduction (Sec. I), the elevon's number is mostly a failure case problem that is out of the scope of our study. Also, the initial elevons are split equally on the outer wing; that is, elevons 2 to 5 have the same span.

From this initial layout, control surfaces were derived using the following assumptions:

1) Note that η y 2 inb ∕y init 2 inb , y 2 inb and y init 2 inb are the current and the initial inboard y position of elevon 2, respectively.

2) Note that y 5 out y init 5 out , y 5 out and y init 5 out are the current and the initial inboard y position of elevon 5, respectively.

3) The elevon's relative chords are kept constant and equal to 22%. 4) Elevons 2 to 5 are split equally. 5) Elevon 1 is kept constant.

It is then clear that 0 ≤ η ≤ 1, η 1, corresponding to an initial elevon's layout, and η 0, corresponding to the lack of any control surface on the outer wing. Examples of layouts for η 0.8 and η 0.4 are presented on Figs. 1b and1c, respectively.

B. Computation of Aerodynamic Models for Discretized Control Surface Span

The computation of an aerodynamic model for different parameter values of η is described in this section. For the sake of clarity, the initial layout (namely, the configuration for η 1) is called the "reference" aircraft; whereas derived layouts (namely, configurations for η < 1) are called "project" aircraft. The reference flight dynamics model described in Sec. II.A is used for all configurations described in Sec. III.A as a planform, and airfoils are kept constant for all configurations; therefore, only control surface aerodynamic efficiencies need to be evaluated for the project aircraft. Once again, the goal of Sec. III is to obtain state-space representations continuously parametrized by elevon span parameter η. The process to obtain such a continuous approximation is presented in Fig. 4. A first step is to compute calibrated aerodynamic models for discrete values of η, namely, for values between 0.1 and 1, with steps of 0.1. For that purpose, the Athena Vortex Lattice (AVL) software [START_REF] Drela | AVL Aerodynamic Analysis, Trim Calculation, Dynamic Stability Analysis, Aircraft Configuration Development[END_REF] was used together with calibration factors coming from the supposedly known aerodynamic coefficients of the initial BWB design. This allows one to take into account the Mach effects and nonlinearities due to a high angle of attack or high deflection, which AVL cannot account for. Continuous approximation of discretized models is treated in Sec. III.C, whereas computation of calibrated aerodynamic models is explained hereafter.

More precisely, the aircraft total lift coefficient C L embedded in the F aero term of Eq. ( 3) comprises an ith control surface deflection dependency C L i , which can be written as follows:

C L i k NL δm i α; δm i C L δm i δm i ( 8
)
where k NL δm i α; δm accounts for loss-of-control surface efficiency as a function of the angle of attack α and control surface deflection δm i , and C L δm i is the lift gradient of the ith control surface. Note that k NL δm i α; δm is known for reference aircraft, and it is kept for project aircraft. The reference lift gradient C ref

L δm i
is supposed known from previous studies [START_REF] Meheut | Aerodynamic Shape Optimizations of a Blended Wing Body Configuration for Several Wing Planforms[END_REF], and it is compared with the lift gradient computed by AVL for the reference configuration C AVL L δm i (see Fig. 5a). From these data, a calibration factor accounting for AVL's lack of accuracy is computed:

ΔC AVL L i C ref L δm i C AVL L δm i (9)
Then, an AVL computation is run on project geometry in order to compute the lift gradient of the project aircraft C AVL proj L δm i . This gradient is finally calibrated using the previously computed calibration factor ΔC AVL L i of Eq. ( 9) (see Fig. 5b), giving the following:

C proj L i k NL δm i ΔC AVL L i C AVL proj L δm i δm i (10)
A similar process is used for computing the control surface's pitching moment efficiencies of project configurations. More precisely, the aircraft pitching moment coefficient C m embedded in M G aero of Eq. ( 3) comprises an ith control surface deflection dependency C m i , which can be written as follows:

C m i k NL δm i α; δm i C L δm i δm i X CG -X F i (11)
where X CG denotes the x-wise CG position, and X F i denotes the ith control surface aerodynamic center x-wise position. However, X F i is not a direct output of AVL and has to be computed as follows:

Fig. 4 Process for obtaining LFR approximation of state-space representations as a function of the η parameter.

X AVL F i X CG - C AVL m δm i C AVL L δm i (12)
This leads to computing the aerodynamic center calibration factor for the reference aircraft, knowing the actual aerodynamic center X ref

F i
for the ith elevon:

ΔX AVL F i X ref F i X AVL F i (13)
Finally, an AVL computation is run on project aircraft in order to compute the pitching moment gradient of the project aircraft's ith elevon C AVL proj m δm i , and the resulting aerodynamic center is computed and calibrated as follows:

X AVL proj F i ΔX AVL F i ⋅ X CG - C AVL proj m δm i C AVL proj L δm i ! (14)
A similar process is used for computing the control surface's rolling moment efficiencies of project configurations. More precisely, the aircraft rolling moment coefficient C l embedded in M G aero of Eq. ( 3) comprises an ith control surface deflection dependency C l i , which can be written as follows:

C l i k NL δm i α; δm i C l δm i δm i ( 15
)
The calibration factor for the rolling moment coefficient is then computed as follows:

ΔC AVL l i C ref l δm i C AVL l δm i (16)
And, finally, the rolling moment coefficient of the project aircraft is calibrated using the calibration factor of Eq. ( 16):

C proj l i k NL δm i ΔC AVL l i C AVL proj l δm i δm i (17)
This method, summarized in Fig. 4, combines the advantages of fast data generation through a light computational fluid dynamics computation, and with far better accuracy than AVL direct output through accurate knowledge on a reference configuration. Here, it is applied only to control surface aerodynamic coefficients, but the process would be similar for computing any aerodynamic coefficient of a project aircraft with a baseline knowledge on a similar reference aircraft.

C. Polynomial Approximation and LFR Representation of Control Surface Aerodynamic Efficiencies

Once aerodynamic coefficients are computed and calibrated for discretized values of η (namely, η 0.1; 0.2; : : : ; 1; see Sec. III.B), the final step concerning the modeling problem consists of obtaining an approximation of state-space representations for the continuously varying parameter η. For that purpose, it was chosen to work with the linear fractional representation (LFR) framework because this representation is suited to the optimizer coming from the control community presented in Sec. IV. Moreover, efficient algorithms for approximating a set of numerical data as an LFR were developed by ONERA-The French Aerospace Lab [START_REF] Roos | Polynomial and Rational Approximation with the APRICOT Library of the SMAC Toolbox[END_REF]. A LFR is a model where all fixed dynamics are gathered in a single linear time-invariant plant M, whereas uncertainties or varying parameters are contained in a block-diagonal matrix Δ (see Fig. 6). Polynomial and rational expressions are, for instance, easily convertible into a LFR. For a comprehensive LFR theory, please refer to [START_REF] Zhou | Robust and Optimal Control[END_REF].

More precisely, the problem is that of finding a LFR approximating as closely as possible the state-space representations computed for different values of η. Uncertainties are not considered in this study, so the Δ block is only composed of the η parameter repeated several times. Moreover, as mentioned previously, in this work, the BWB planform is kept constant; therefore, state matrix A from the statespace representation is assumed to be independent from η. As a consequence, only control matrix B containing the elevon's efficiencies is approximated by the LFR. However, the described process would be similar if we would include some planform variables having an impact on the A matrix. It was decided to restrict the search for LFR approximations to polynomial approximations in order to keep the LFR order (that is, the amount of times η is repeated inside the Δ block) as small as possible. From a physical perspective, this can be justified by the fact that control surface efficiencies should vary smoothly with respect to their span. The least-squares routine lsapprox from the APRICOT library in MATLAB [START_REF] Roos | Polynomial and Rational Approximation with the APRICOT Library of the SMAC Toolbox[END_REF] was used. The problem consists of finding a polynomial P of degree n p minimizing the following criterion:

C X N k 1 B η k -Pη 2 ( 18
)
with B η k being the control matrix for sampled values of η, η k;k1: N 0.1; : : : ; 1.

The degree of the polynomial P is set to n p 5, and the resulting LFR size is 20; this number represents how many times η is repeated inside the Δ block. The maximum root mean square is 9.36 ⋅ 10 -3 , and the maximum local absolute error is 2.01 ⋅ 10 -2 . Note that lsapprox instead of the orthogonal least-squares olsapprox routine was used because it achieves higher accuracy. The LFR order is, however, is higher but remains acceptable; that is, the computation cost for simulation using Simulink [START_REF]Robust Control Toolbox™[END_REF] is only increased by a few seconds. The resulting approximated aerodynamic gradients are plotted in Fig. 7.

IV. Integrated Design and Control

In this section, the integrated design and control problem of simultaneously minimizing the elevon's span parameter η while satisfying the handling qualities and maneuverability constraints is developed.

A. Structure of Control Laws

As stated in Sec. II.B, the longitudinal instability on this BWB requires a SAS to make it flyable. Moreover lateral control laws are also mandatory to enhance lateral handling qualities. Considering both these longitudinal and lateral/directional control laws is, moreover, necessary for a proper sizing of control surfaces. A main contribution of this paper is, indeed, to provide a methodology for simultaneous longitudinal, lateral, and directional control law's synthesis of an arbitrary structure; whereas this problem is usually addressed by decoupling longitudinal from lateral/directional axes. Here, a typical fly-by-wire FCS architecture is considered. The pilot provides inputs in terms of the commanded load factor Nz c , bank angle ϕ c , and sideslip β c . Control law feedback features a C and Y structure for longitudinal and lateral/directional control, respectively, for which the structure is provided in [START_REF] Favre | Fly-by-Wire for Commercial Aircraft: The Airbus Experience[END_REF]. More precisely, the following holds true:

1) The C structure is composed of the load factor Nz and pitch rate q feedback, together with an integrator for a zero steady-state tracking error and a direct feedthrough gain. The output of the law is an equivalent elevator order δm equi .

2) The Y structure features lateral/directional state feedback, namely, sideslip β, yaw rate r, bank angle ϕ, and roll rate p. An integrator is added to keep zero steady-state sideslip, as well as a bank angle order direct feedthrough gain. Outputs of this law are the equivalent aileron and rudder order δl equi and δn equi , respectively.

A general overview of the control law structure is visible in Fig. 8. As already mentioned, control law outputs are the equivalent elevator, aileron, and rudder orders, which are independent from the control surface's architecture. These equivalent orders are then converted into control surface deflections, thanks to a control allocation module, which is described in the next section.

B. Control Allocation Model

Control allocation is the problem of converting equivalent orders, computed by the control law, into control orders when there are more effectors than axes to control. For a comprehensive survey of control allocation methods, please refer to the work by Johansen and Fossen [START_REF] Johansen | Control Allocation-A Survey[END_REF]. In our study, a control allocation module needs to be incorporated in order to convert equivalent elevator, aileron, and rudder deflections δm equi ; δl equi ; δn equi into actual control surfaces deflections δm i;i1; : : : ;10 ; δn.

Mathematically, the control allocation problem is that of finding a deflections vector u satisfying the following:

2 4
Cm δm 1 : : : Cm

where B 1 η is the matrix of the elevon's gradients in pitch, roll, and yaw, respectively, which all depend on the parameter η.

Cm δm equi Cl δl equi Cn δn equi T is a vector of equivalent gradients as seen by the control law. These values may be set arbitrarily without loss of generality; we chose equivalent values of one on all axes. Then, a classical solution of Eq. ( 19) is the Moore-Penrose pseudoinverse [START_REF] Bodson | Evaluation of Optimization Methods for Control Allocation[END_REF]:

u K alloc η 2 4 δm equi δl equi δn equi 3 5 (20)
with

K alloc η B T 1 B 1 B T 1 -1 (21)
As comprehensively discussed in Sec. IV.C, our process features two steps. In the first step, three-axis gains are computed for a fixed η value; in this step, the pseudoinverse control allocation from Eq. ( 21) is used. During the second step, three-axis control law gains and the η parameter are simultaneously optimized. In this step, K alloc is no longer fixed but is a variable for the optimization. By doing this, the design space is widened, and a truly optimal strategy with respect to the imposed handling qualities constraints can be chosen by the optimizer.

More precisely, in order to limit the number of variables for the optimization, K alloc is parameterized as follows:

K alloc 2 6 6 4 K pitch alloc K roll alloc 0 K pitch alloc -K roll alloc 0 0 0 K yaw alloc 3 7 7 5 (22)
This means we impose symmetrical and antisymmetrical deflections of the elevons for a pitch and roll order, respectively; and a yaw order is allocated to the rudder only. By incorporating this physical knowledge of the control allocation structure, the number of variables within K alloc is reduced from 33 to 11 variables.

C. Simultaneous Three-Axis Control Law Synthesis

Integrated design and control of the BWB presented in this paper follows a two-step scheme:

1) A first control law's synthesis computes gains for an arbitrarily fixed η value. The problem consists of minimizing the difference between a reference model and the closed-loop aircraft. The output of this step is the optimum value of the H ∞ criterion. This section is devoted to describing this first step.

2) The output of step 1 is used to put a constraint on the maximal value of the H ∞ criterion in order to guarantee satisfactory closedloop behavior while optimizing the elevon's size η. This step is described extensively in the next section.

As stated previously, simultaneously optimizing the control surface's size η and control law gains requires setting a constraint, in the optimization sense, that ensures an adequate closed-loop behavior of the optimal solution. This constraint was set as a maximal admissible value of an H ∞ criterion: a value that must be computed through a first synthesis. The H ∞ criterion and its optimal value computation are described now.

A three-channel model-reference tracking scheme was used. This scheme consists of minimizing the difference between a reference dynamics model and the closed-loop aircraft, from the H ∞ norm point of view [START_REF] Sima | Algorithms for Linear-Quadratic Optimization[END_REF]. If the reference model is perfectly matched by the closed loop in the whole frequency domain, then the optimal H ∞ value is zero. However, this is practically infeasible due to physical limitations; as a consequence, the optimal value is always above zero.

Here, reference-model tracking is expressed by minimizing the H ∞ norm of a three-input three-output transfer function between pilot inputs Nz c ; ϕ c ; β c and outputs z 1 ; z 2 ; z 3 the differences between reference dynamics outputs and actual closed-loop signals Nz; ϕ; β. For a proper definition of signals, please refer to Fig. 8. In our case, multichannel transfer has several advantages over multiple single-input single-output (SISO) transfers. First, offdiagonal terms are implicitly set to zero. Hence, the resulting control law will totally decouple all three axes (namely, longitudinal from lateral/directional, but also lateral from directional), turns are performed with zero sideslip (and directional from lateral), and "pedal" inputs imply no bank. Couplings between all axes are explicitly taken into account by the control law, which a SISO approach would not. Then, this formulation is virtually independent from the open-loop model and structure of control laws. Finally, it allows for an elegant formulation of the optimization problem of Sec. IV.D: a single constraint on the H ∞ norm of this multichannel transfer ensures an adequate closedloop behavior on all three axes. An equivalent formulation with SISO transfers would require nine constraints.

Reference models on longitudinal, lateral, and directional axes are set as follows:

Nz ref Nz c ω 2 i s 2 2ξ i ω i s ω 2 i ( 23
)
ϕ ref ϕ c 1 1 τ rp s1 τ sp s (24)
β ref β c ω 2 dr s 2 2ξ dr ω dr s ω 2 dr ( 25
)
In our study, the closed-loop reference values are fixed, and they are given in Table 1. Ideally, these values should depend on the open-loop dynamics, but this is left for future work.

The optimization problem for simultaneous three-axis control law synthesis simply reads as follows: with K being a vector containing all control law gains defined in Sec. IV.A. The additional constraints ensure that closed-loop poles have a damping of at least 0.5 and a real part of at least 0.2. Even though those constraints may seem redundant with the reference model tracking objective, it was found that it helps the optimizer to converge.

To solve this optimization problem, the systune routine [START_REF] Apkarian | Tuning Controllers Against Multiple Design Requirements[END_REF][START_REF] Gahinet | Automated Tuning of Gain-Scheduled Control Systems[END_REF] from the MATLAB Robust Control Toolbox [START_REF]Robust Control Toolbox™[END_REF] was used. This routine allows tuning of fixed-order structured controllers, so it can handle physical parameter optimization combined with control gain computation. Couplings between the control and the design problem are therefore taken into account directly in a single optimization. Moreover, it is well suited to mathematical particularities of the optimization problem in Eq. ( 26), namely, the nonsmooth behavior of the H ∞ norm. Optima found by the algorithm are only local; consequently, several initializations should be performed in order to ensure the globality of the solution.

An example of optimized three-axis control law shown in Figs. 9 and 10 through the frequency domain and temporal simulation responses, respectively. The chosen flight point is M:35, H 3300 ft, and m 300 T. The optimal value for the H ∞ criterion γ ∞ for this flight point, as well as the normalized constraints at the optimum gBest, are as follows:

γ ∞ T Nz c ;ϕ c ;β c →z 1 ;z 2 ;z 3 K opt ∞ (27)
0.107 gBest 0.9997 [START_REF] Haghighat | Aeroservoelastic Design Optimization of a Flexible Wing[END_REF] where gBest < 1 at the optimum means the constraints are fulfilled.

To visualize the multiaxes behavior of the resulting control law, the simulation shown in Fig. 10 features the following maneuvers:

1) At t 0 s, a pilot pull up is commanded through a ΔNz c 1g step order. As a result, right and left control surfaces move in a symmetric way.

2) At t 5 s, a step with a bank angle of ϕ c 30 deg is commanded. Elevons start moving from their equilibrium position antisymmetrically. The Rudder is also actuated in order to maintain zero sideslip.

3) At t 12 s, a β c 5 deg sideslip order is commanded.

The initial elevon's deflection used to trim the aircraft is not represented on Fig. 10; only variations around their initial positions are shown. To conclude this section, we emphasize the fact that the desired outcome of the synthesis presented in this is only the optimal value γ ∞ of the H ∞ criterion, which will be used as a constraint for the optimization procedure described in the next section. In particular, control law gains are not kept; they are computed again in the combined optimization of control laws and parameter η.

D. Definition of the Optimization Problem

Once the optimum γ ∞ of the H ∞ criterion is computed, it is used as a constraint on the H ∞ norm of the multichannel transfer for the combined optimization problem. More precisely, the integrated design and control problem of this study is that of finding a minimal η such that 1) deflections and deflection rates in response to maneuvers do not exceed prescribed limits, and 2) closed-loop behavior is optimal.

The latter point is solved by ensuring that the H ∞ norm of the multichannel transfer kT Nz c ;ϕ c ;β c →z 1 ;z 2 ;z 3 ηk ∞ is kept under its optimal value (that is, equal to this value) whatever η. Once the optimum value γ ∞ for a fixed η is known from the optimization described in Sec. IV.C, the optimization problem of finding the best possible closed-loop behavior translates into a single constraint satisfaction problem. This elegant formulation allows minimizing of another objective (η in our case) while ensuring a satisfactory closed-loop behavior on all axes.

Constraints on maneuverability are cast as constraints on the H ∞ norm of adequate transfer functions, which is similar to the work of Niewhoener and Kaminer [START_REF] Niewoehner | Integrated Aircraft-Controller Design using Linear Matrix Inequalities[END_REF]. Therefore, these constraints are root mean square and not temporal: no guarantee can be obtained in timedomain simulations. When an H ∞ handling qualities constraint is saturated, the only time-domain interpretation that can be made is that the maximum deflection is achieved for a sinusoidal input at the frequency for which the maximum H ∞ norm was computed. The reader interested in an extensive comparison between the frequency constraints and the associated time responses may refer to [START_REF] Denieul | Preliminary Design of Control Surfaces and Laws for Unconventional Aircraft Configurations[END_REF]. Having in mind these limitations, we believe method still delivers useful information for sizing at a future project level. Moreover, it is widely used in the literature (see, for instance, [START_REF] Silvestre | Plant-Controller Optimization with Applications to Integrated Surface Sizing and Feedback Controller Design for Autonomous Underwater Vehicles (AUVs)[END_REF][START_REF] Niewoehner | Integrated Aircraft-Controller Design using Linear Matrix Inequalities[END_REF][START_REF] Yang | An Optimization Approach to Integrated Aircraft-Controller Design[END_REF]). Research has been done on time-domain output bounding (see, for instance, [START_REF] Chambon | Time-Response Shaping Using Output to Input Saturation Transformation[END_REF]); the integration of this work to our process is left for future work.

The following constraints are ensured:

1) The first constraint includes the maximum deflections and deflection rate in response to a pilot pull up. This is ensured through a constraint on the following H ∞ norms:

1 Δδm max i T Nz c →u ΔNz max c ∞ ≤ 1 and 1 _ δm max i T Nz c → _ u ΔNz max c ∞ ≤ 1 respectively, with Δδm max i δm max i -δm e .
2) The second constraint includes the maximum deflections and deflection rate in response to severe longitudinal turbulence. This is ensured through a constraint on the following H ∞ norms:

2 Δδm max i T e z →u ∞ ≤ 1 and 2 _ δm max i T e z → _ u ∞ ≤ 1 respectively.
3) The third constraint includes the maximum deflections and deflection rate in response to the ϕ max bank angle order. This is ensured through a constraint on the following H ∞ norms:

1 Δδm max i T ϕ c →u ϕ max ∞ ≤ 1 and 1 _ δm max i T ϕ c → _ u ϕ max ∞ ≤ 1
respectively. Chosen values for the sizing are as follows:

1) The first value is δm max 25 deg. A 5 deg margin versus deflection stops at 30 deg was chosen to avoid entering the nonlinear part of control authority. Please also note that, for the maximal deflection allowed, the deflection used for trim δm e was taken into account into the computation of Δδm max : the more deflections are needed in order to balance the aircraft, the less authority remains available for control.

2) The second value is _ δm max 60 deg ∕s. 2.

The process developed in this paper was extensively described in [START_REF] Denieul | Preliminary Design of Control Surfaces and Laws for Unconventional Aircraft Configurations[END_REF].

V. Results

In this section, the results of the integrated design and control process are analyzed.

B. Results on the Entire Flight Envelope

Integrated design and control with tunable allocation is now applied on the whole flight domain described in Sec. III. Figure 12 shows the resulting surface η opt fM; H, as well as sizing constraints associated with each optimum. The maximum value is η opt 0.5996: the aircraft can adequately be controlled on the whole flight envelope, without reaching its control stops, with only 60% of the initial control surface span. Then, the sizing values for η opt (i.e., maximum values over the flight envelope) are achieved for low dynamic pressure cases: that is, low speed and high altitude. It was also shown in Sec. II.B that these cases correspond to the maximum longitudinal instability. Interestingly, those sizing points are limited by a combination of turbulence, and longitudinal and lateral maneuvers. As expected, strong stabilization to an exogenous input is definitely sizing for some specific flight points. Moreover, these results highlight the importance of the control allocation module for a correct sizing of the aircraft: it has a direct influence on the design.

VI. Conclusions

A new method for sizing the control surfaces of an unstable blended-wing-body using closed-loop handling quality criteria was presented. This method consists of simultaneously optimizing the longitudinal and lateral control laws, as well as a control allocation module, while minimizing the control surface areas under handling quality constraints. From a sizing perspective, we have shown that the studied Airbus blended-wing-body configuration can adequately be controlled on three axes, on the whole flight envelope, with only 60% of the initial control surface's span, with normal laws and full control authority, with reasonable hypotheses on the actuators, sensors, and delay chain. Future work concerning the sizing of this aircraft should investigate the control authority under failure cases.

From a methodological perspective, the benefit of a simultaneous design and control approach is demonstrated: in a single step, the sizing is guaranteed with respect to the handling quality criteria, taking into account control laws. Further work should investigate temporal criteria instead of frequency (root mean square) criteria. Finally, alternative optimizers could be considered for solving this coupled problem; given the nonsmooth nature of the mathematical problem, simulation-based routines such as genetic algorithms could be worth examining.

Appendix

Developing the coefficients in state-space matrices of Sec. II.A gives the following:

x V -

4 Fig. 1

 41 Elevon sizes for different values of parameter η.

Fig. 3

 3 Fig.3Mach-altitude flight envelope.

Fig. 2

 2 Fig. 2 Poles of aircraft dynamics for aft CG (PR denotes pure roll, PH denotes phugoid mode, and SPI denotes spiral mode).

Fig. 5 Fig. 6

 56 Fig. 5 Comparison of AVL outputs with reference aircraft aerodynamic data (left) and with calibrated data (right).

Fig. 7

 7 Fig. 7 LFR polynomial approximations of elevon pitch and roll gradients.

Fig. 8

 8 Fig.8Closed-loop representation for integrated design and control optimization.

Fig. 9

 9 Fig. 9 Bode diagrams for three-axis reference models and optimized closed loop, for η 1.

Fig. 10

 10 Fig. 10 Temporal responses for η 1 (dotted lines) and η opt 0.3885 (solid lines).

3 )

 3 The third value is ΔNz max c 1.5g. 4) The fourth value is ϕ max c 45 deg. The combined optimization problem is finally summarized in Table

T 1 Maximum deflection rate in response to bank order 5 T 1

 151 ϕ c → _ u ϕ max ∞ ≤ Nz c ;ϕ c ;β c →z 1 ;z 2 ;z 3 ⋅ 1 γ ∞ ∞ ≤ Optimal closed-loop performance 1 ∀ p ∈ C; ppole of Ps: Closed-loop poles location 1 Rep ≤ -MinDecay; Rep ≤ -MinDamping:jpj ----K internally stabilizes Pη ----

Fig. 11

 11 Fig. 11 Bar diagram of normalized constraint values after optimization with tunable allocation: η opt 0.3885.

  cos α y p sin α g cos θ

			0 0	0 0		0 0	0 0	ñβ lβ	ñr lr	ñp lp	V e 0 0	3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5	2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4	δV δα q δθ β r p	3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
			0 | {z } 0 0 0 0 tan θ 1 0	ϕ | {z }
		2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4	x δx x δm i 0 -z δm i 0 m δm i 0 0 0 y δm i y δn 0 0 0 0 0 ñδm i ñδn 0 lδm i lδn	3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 7	Controls U 2 6 6 4 Δδx Δδm i δn 3 7 7 5 |{z}	State matrix A 2 0 Gust disturbance matrix B w 3 | {z } 0 6 6 6 6 6 6 6 6 4 -z α V e 0 . . . 7 7 7 7 7 7 5 7 7	w z			States X	(4)
		0 | {z } 0 0
		Control effectiveness matrix B
		2
	X δNz | {z } Outputs Y	6 6 6 6 6 6 6 6 6 4

  δm 10 Cm δn Cl δm 1 : : : Cl δm 10 Cl δn Cn δm 1 : : : Cn δn 10 Cn δn Cm δm equi δm equi Cl δl equi δl equi Cn δn equi δn equi

	3	2	3
	5 u	4	5
	| {z }
	B 1 η

Table 1

 1 Values for reference model Nz c ;ϕ c ;β c →z 1 ;z 2 ;z 3 Pη init ; K; K init alloc

	Parameter	Value
	ω i	1 rad ⋅ s -1
	ξ i	0.7
	τ rp	1.6 s
	τ sp ω dr	1.9 s 0.4 rad ⋅ s -1
	ξ dr	0.7

K T

Table 2

 2 Integrated design and control optimization problem

		Function/variable				Description	Quantity
	minimize η							Outer elevon's total span	--
	with respect to K						Control law gains	16
	K alloc						Control allocation matrix	11
	η							Outer elevon's total span	1
	subject to	1 Δδm max i	T Nz c →u Δα	V e g	z α	∞	≤ 1	longitudinal order Maximum deflection in response to	5
		1
		_ δm max

Acknowledgments

This work is part of a Ph.D. thesis in cooperation between Institut Supérieur de l'Aéronautique et de l'Espace (ISAE)-SUPAERO, ONERA-The French Aerospace Lab, and the Future Projects Office of Airbus Operations SAS.

A. Integrated Design and Control on a Single Flight Point Integrated design and control tunable allocation applied to the flight point of M.35, H 3300 ft, and m 300T gives the following results: η opt 0.3885 [START_REF] Niewoehner | Integrated Aircraft-Controller Design using Linear Matrix Inequalities[END_REF] gBest 0.9998 (30)

A 61% decrease in the control surfaces span is achieved as compared to the initial layout while still satisfying all handling quality constraints, i.e., gBest < 1. More precisely, Fig. 11, gathering all handling qualities constraints at the optimum, indicates that, on this flight point, the roll constraint is limiting. As a consequence, the optimal allocation is to distribute equally among all controls the roll order; hence, all roll criteria are equally saturated. As pitch is not limiting on this flight point, only two controls are used for pitch control. Such a strategy can also be seen on the temporal response of Fig. 10: all five controls are equally used for the roll maneuver, resulting in equal deflection rates for all controls. This behavior is confirmed by looking at the optimized allocation matrix: