Highlights

 23.9% of the French population will be 65 or over in 2030. Based on the French longterm care (LTC) indicator, i.e. the number of people aged 60 or more who receive Personal Autonomy Allowance (APA), the estimated number of recipients will increase from 1,265,000 in 2015 to 1,582,000 in 2030 and 2,235,000 in 2050, which is a growth rate of 76%.

 LTC public expenditure, representing 1.7% of GDP (currently evaluated at €23.7 billion -national data), constitutes a major, although fragmented, financial investment in France, and could reach 1.9% of GDP by 2030. In addition, the cost of informal care provided by 3.9 million carers has an estimated value of between €7 billion and €18 billion.

 Improving access to and the affordability of LTC services is a major concern that is managed by reducing out-of-pocket expenditure and developing coordination between the health and social care sectors.

 Despite real investment in developing a specific social care employment sector, a key challenge facing France today is the reorganisation of the LTC workforce.

 Since the Act on adapting society to an ageing population of December 2015, no comprehensive LTC reforms have taken place in France. However, the major guidelines of a deep-seated reform were presented in the Libault Report (2019), which includes a recommended investment of almost €10 billion in LTC by 2030.

1 Description of main features of the long-term care system(s)

Demographic trends

In 2019, the old age dependency ratio in France was 32.5 and is set to reach almost 40% in 2030 and 49,3 % in 2050, according to EU estimations of fertility rates and life expectancy at birth1 . This scenario is somewhat debatable. For example, it considers a stable fertility rate of 2 children per woman for the next 30 years, which is not the trend observed at national level (a slow but regular decrease from 2.03 in 2010 to 1.87 in 2017). 2 In 2019, 13.5 million people were aged 65+ (7.7 million women and 5.8 million men), representing one in five of the total population (20.1%). Projections for the next decades are as follows: in 2030, the 65+ population would represent 23.9% of the total population (16.4 million -9.2 million women and 7.2 million men); in 2070, the figure would be 19.1 million (10.7 million women and 8.4 million men) representing more than one quarter (26.6%) of the total population. These calculations take into account the evolution of life expectancy at 65. In 2019, life expectancy at 65 was 21.3 years (23.8 years for women and 19.7 for men). For the next few decades, EU estimations are as follows: for women, 24.6 years in 2030, 26.1 in 2050 and 27.5 in 2070; for men, 20.8 years in 2030, 22.5 in 2050 and 24.0 in 2070. According to these estimations, the gender gap in terms of life expectancy, observed over many decades, is shrinking slightly, from more than 4 years in 2019 to 3.5 years in 2070. This gap also concerns healthy life years: in 2019, healthy life expectancy is 11.3 years for women and 10.2 for men.

In addition to these estimations, two other trends are of interest: the first one is a slowdown in the gain in life expectancy at birth over the last decade, notably due to three major influenza epidemics from 2014 to 2019, each of which generated around 20,000 registered deaths; the second important issue concerns socioeconomic inequalities in life expectancy. For example, the difference in life expectancy at birth between the 5% richest and 5% poorest men is 13 years (8 years for women)3 .

Concerning the very old, the share of over-75-year-olds in the French population is growing much more rapidly: from 9.4% in 2019 to 12.5% in 2030 and 16.3% in 2050. In the light of demographic trends, the estimated number of potentially dependent elderly people ("self-perceived longstanding limitation in activities because of health problems, for at least the last 6 months") will grow by 46%, i.

Governance and financial arrangements

Traditionally characterised by a familialist approach to elderly care, with the legal obligation for families to care for their older parents, since the 1990s' France has developed several LTC policy measures and evolved towards a mixed model, combining public measures and family care5 . LTC policy in France cuts across different policy sectors -health, social and medico-social6 -and involves several levels of governance: the state, regions, départements and municipalities. Regional administrations execute national health policies defined at the central level by the government, whereas the decentralised French local authorities -départements -are responsible for social policy. They have a key role in LTC regulation: they define local policy orientations in their territory; finance and manage the national APA; and regulate care services. Though some territorial disparities exist, national regulations remain in social care policy through the definition of the different schemes by national legislation and the follow up of the territorial variations by the government (Libault report, 2019). In addition, municipalities can develop specific voluntary support measures.

Along with this territorial organisation, two major institutional actors are involved: the CNSA (national solidarity fund for autonomy -caisse nationale de solidarité pour l'autonomie), created in 2004, which is a national institution responsible for implementing policy measures aimed at older and disabled people 7 ; and the ARSs (regional health agencies -agences régionales de santé), introduced in 2009. They are the regional representative of central government, extending the regional agencies' traditionally healthsector-only intervention to the social care sector.

The LTC policy involves a wide range of funding (social security system, the départements, the CNSA and central government) for wide-ranging expenditure, combining health insurance system and tax-based system for the funding of the APA.

Social protection provisions

Different schemes can be distinguished, each with its own characteristics depending on the system (health or social care system) it relies on.

A first group of schemes corresponds to "social assistance schemes" which are managed by départements and/or pension funds -Social assistance for accommodation (Aide sociale à l'hébergement -ASH): paid out by the départements to old people aged 65+ in institutions with low income to complement the part of expenditure for accommodation which cannot be paid by the resident 10 -Financial support to pay for home helps: granted by the départements and the pensions funds depending on conditions of age and income of old people with the lowest level of dependency.

The corner stone of the social care policy is based on a specific policy measure: the Personal Autonomy Allowance (APA)

Introduced in the late 1990s and focused on situations of 'dependency', the benefit aimed at meeting the needs of elderly people who were not covered by health insurance, by helping them identify their needs and pay for social care services. It was reformed in 2002 and became the APA. Managed by the départements, the APA is paid -at home or in institutions -to any person aged 60 or over who needs assistance to accomplish everyday activities or who needs to be continuously watched over. Each level of dependencyaccording to the AGGIR Grid 11 -gives access to a maximum amount, which is then adjusted according to the recipient's needs and level of income. In 2019, for a GIR 1 the maximum amount of the allowance is €1713; €1375 for GIR 2; €931 for GIR 3 and €662 for GIR 4.

At home, the allowance is paid to finance a specific 'care plan' elaborated by a multidisciplinary team (health and social professionals from the départements) after an .The allowance is delivered to people with an income (houselholds' income + can also include the income of close relatives, depending on the départements) below the cost of accomodation. According to the DREES (https://drees.solidarites-sante.gouv.fr/IMG/pdf/16-17.pdf), in 2017, there was 122,000 recipients of the ASH, for an amount of around 870€/month. 92% of the recipients lived alone and 50% of them had an income below 900€/month. 11 AGGIR Grid distinguishes 6 levels of dependency from GIR1 the Highest to GIR 6, the lowest.

assessment of needs. The use of the benefit is controlled and the multidisciplinary teams are in charge of following up on the situation. The APA represents over €5 billion of expenditure, of which 65% comes from the département and 35% from the CNSA (CNSA, 2019). The APA was allocated to 1,285,500 elderly people in December 2016 (7.6% of people aged over 60), of whom 60% in the home and 40% in institutions.

Employing a care worker in the home also opens up the right to fiscal deductions and since 2017 tax credits.

A final category of schemes concerns informal carers, with two different leaves:

-a carer's leave: allocated for 3 months or a part-time period, renewable up to 1 year. Vote of a financial compensation October 2019 (between €42 and €55 per day from October 2020) -a family solidarity leave to assist a dying relative: allocated for 3 months, renewable once and be used for a part-time period. Daily allowance of €55 for a max. of 21 days.

Supply of services

A wide range of services in the social and health care sectors propose solutions for both home care and residential care, as well as supplementary options. They depend on different regulations (related to health or social care systems), which create financial and administrative complexity.

Home care nursing services (services de soins infirmiers à domicile, SSIAD) and Home social and healthcare services (services polyvalents d'aide et de soins à domicile): proposing 125,733 places in 2019 (CNSA, 2019). Health care Services are provided by salaried nurses and auxiliary nurses paid on a fee-for-service basis. Total expenditure on this type of care rose by 2.7% in 2017f, amounting to about € 1.6 billion [START_REF] Drees | Les dépenses de santé en 2017[END_REF] .

Home help and support services (services d'aide et d'accompagnement à domicile, SAAD) which constitutes a highly complex sector [START_REF] Khomri | Plan de mobilisation nationale en faveur de l'attractivité des métiers du grand âge [national plan to make work for the elderly more attractive[END_REF] including non-profit organisations and public social care services requiring quality certification by the départements. Based on the analysis of the APA recipients, their number was estimated at 7000. Private for-profit organisations in the personal services sector, which require specific quality certification. Their prices are established freely. In 2008, they represented only 4% of social care workers for elderly people [START_REF] Marquier | Les intervenants au domicile des personnes fragilisées en 2008 [workers providing support in the home for vulnerable people in 2008[END_REF] .

Residential homes (établissements d'hébergement pour personnes âgées dépendantes, EHPAD):

France features 7,438 residential homes (EHPADs), offering 98 places per 1,000 people aged 75 and over [START_REF] Bazin | Le recrutement en EHPAD [recruitment in nursing homes[END_REF] . In 2015, these institutions cared for 10% of elderly people aged 75 or more and one-third of those aged 90 or more. The average cost of EHPAD accommodation ranges from €51 to €71 per day [START_REF] Muller | résidents en établissements d'hébergement pour personnes âgées en[END_REF] . These institutions take different forms (Libault, 2019): private for-profit EHPADs (22% of places), private not-forprofit EHPADs (28% of places) and public EHPADs (50% of places). The latter may or may not be hospitals. Despite an increase in staff-to-patient rates from 2011 to 2015 (from 59.7 to 62.8 full-time employment (FTE) for 100 places (Bazin et al., 2018), the number of professionals remains insufficient to ensure quality care.

Supplementary options developed between home-based care and institutions:

-Housing facilities: concern old people who are mostly autonomous, and propose small apartments adapted to minimise the risk of fall. 2 Assessment of the long-term care challenges in the country

Access and affordability

In France, 4.8% of the over-65s were cared for in institutions compared to 6.2 in their homes. Out-of-pocket spending varies according to whether care is provided at home or in an institution, as well as according to the level of dependency of the elderly people concerned. One month's accommodation in an EHPAD for a person with a radical loss of autonomy costs on average €2,450 20 with on average €1,850 remaining to be paid by the user (after allowances and tax reduction) (Libault, 2019). While the APA covers 100% of healthcare costs and about two-thirds of "dependency" costs, accommodation services are the responsibility of families (except very low income; see Section 1.3).

The question of cost is more difficult to value in the home. Depending on the income of the old person, the remainder to be paid can range from €2,500 to €4,050 in highly dependent situations, when a permanent presence is required (Mutualité Française, 2018). However, these figures should be viewed differently for old people benefiting from support from an informal carer in their homes. According to DREES data, the remainder to be paid in this case depends on the level of dependency, and ranges from €117 for a high level of dependency (GIR 1) to €37 for an average level (GIR 4). This explains the average figure of €60 remaining to be paid by old dependent people (after allowances and tax reduction) presented in the Libault Report. While this amount may seem acceptable, it raises the question of the significant share of invisible, unpaid work carried out by informal carers to enable this lower financial cost through maintenance in the home.

Fragmentation is a main characteristic of the French LTC field, with the separation between health care and social care sectors. Improving the coordination of organisations, institutions and professionals in order to facilitate the access and affordability of services and schemes has been high on the political agenda during the last decade. Since 2010, three different schemes have been developed with the objective of facilitating relations between the different actors at the local level: the MAIA (Méthode d'Action pour l'Intégration des services d'aide et de soin dans le champ de l'Autonomie-action for integrated health and support services in the autonomy field), introduced in 2010 and concerning 60+ elderly people with complex needs; the PAERPA (Personne Agée En Risque de Perte d'Autonomie -old people at risk of losing their independence), created in 2014, which is a specific health pathway combining a range of tools in a prevention perspective targeting elderly people aged 75+; the PTA (Plateforme territoriale d'appui-territorial support platform), which is not population-based, and more recently the DAC (Dispositif d'Appui à la Coordination -coordination support measure), which aims at gathering all of these existing schemes into one (see Section 3).

Quality

In France, no formal, comprehensive definition of LTC quality has been produced by national or local public authorities. Nevertheless, the Act of 2 January 2002 reforming social care sector describes the different components of quality. Taking into account recent evolutions (ASV Act), three main dimensions can be identified:

(1) The obligation for social care institutions to carry out a double evaluation: internal evaluation carried out by the institution and focused on quality improvement propositions; and external evaluation (which guarantees renewed authorisation) carried out by an external body approved by the National Authority for Health (HAS) 21 , and which concerns their activities and the quality of the services they deliver.

(2) The respect of different basic user rights: respect for dignity, integrity, private life, intimacy and security; free choice between services at home or in institution; personalised, quality care and support that respects informed consent; confidentiality of user data; access to information; direct participation in the definition of the "care and support project" -as well as tools enabling the exercise of these rights: the existence of a welcome booklet; respect for the charter of rights and freedoms of the person hosted; recognition of a qualified individual; visible operating regulations of the establishment or service; the existence of a "community life council" which encourages user participation; and the production of an establishment or service project.

(3) Multiannual contracts (5 years) of objectives and means (CPOM) -which were made general to all social care facilities by the 2015 ASV Act -are signed between social care institutions and pricing authorities (ARS and départements).

The double evaluation process was questioned in a recent report 22 which identified different elements for improving quality approach: a better articulation between the two evaluations, a harmonization of the external bodies which produce evaluation and a standardization of the different indicators used for the evaluation.

This issue is related to the employment issue and the very low appeal of the LTC labour sector. Recent solutions were proposed by the Libault report (see box 1); for example, creating indicators to measure the quality of services available in residential homes.

Employment (workforce and informal carers)

It is difficult to estimate the number of professionals working with elderly people -both health professionals (nurses and assistant nurses) and social care workers (also called personal carers) -because no statistics precisely list professionals in this highly fragmented labour sector. In 2016 France had 2.3 LTC workers for 100 people aged 65+, (91% of these LTC workers are women). In terms of national data, the Libault Reportwhich raises LTC workforce as a key issue of the French LTC policy (see box 1) -makes the following estimate for 2018: about 830,000 FTEs work with dependent elderly people, including 430,000 FTEs in institutions, 270,000 FTEs in home social care services, and 130,000 in home health care services.

Qualification is a key issue concerning LTC workforce. As recalled in a recent OECD report [START_REF]Ensuring an Adequate LTC workforce[END_REF] , in France qualifications are required for nurses (specific training after bachelor's degree), and there is the possibility for personal workers to obtain a specific diploma (created in 2002).Continuing training programs were also developed. In spite of this, the LTC care work sector remains unattractive [START_REF] Bihan | The development of an ambiguous care work sector in France. Between professionalization and fragmentation[END_REF] . The OECD report underlines the difficult working conditions of LTC workers, with France having the highest shares of LTC workers reporting accidents and work-related health problems. The difference between LTC workers and hospital workers is also highlighted, with the first category earning less than the second one. The average wage of home carers is €832 per month, but monthly earnings vary significantly according to whether care workers are employed full time (€1,190) or part time (€717), and whether the structure they belong to is public or private [START_REF] Nahon | Les salaires dans le secteur social et médico-social en 2011. Une comparaison entre les secteurs privé et public [pay in the social and medico-social sector in 2011 -comparison between private and public sectors[END_REF] . Employment situation is another main issue. LTC workers work in shifts and work during weekends. The employment stability given by the existence of permanent contracts is only apparent because of the high prevalence of part-time work: 23% of nurses, 31% of personal carers and 42% of LTC workers holding a part time position at home (26% in institutions) (OECD, 2019). Besides, the employment of a great proportion of temporary agency workers in institutions can also be underlined (OECD, 2019).

Finally, it should be noted that the specific social care work sector developed in France to provide personal care (as a complement to the health care delivered by nurses and assistant nurses) in the home is part of the larger sector of 'personal services' (services à la personne), which includes any person providing services to individuals. Focusing on the volume of workers in order to reduce unemployment, it is not always matched by quality (Le Bihan et al.2018).

In France, a large majority of elderly people currently receive informal support from families and friends -21% of the 60+ population, representing 3 million elderly people (Brunel et al., 2019). 10,2% of the population provide informal care, with 12,2% women and 8,1% men. According to national statistics, 3.9 million informal carers [START_REF] Besnard | Les proches aidants des seniors et leur ressenti sur l'aide apportée, résultats des enquêtes CARE auprès des aidants (2015-2016) [family caregivers of the elderly and their experience of support, results of CARE surveys of carers] Les dossiers de la DREES[END_REF] perform a great variety of concrete tasks (ADL provision, meal preparation, personal care, etc.) and coordinate activities. Evidence shows that informal carers carry out more diverse tasks than professional care workers, who focus on instrumental tasks (Brunel et al. 2019). A recent survey (Besnard et al. 2019) shows that half of these carers are children, and a quarter are spouses or partners. Aged on average 73 for the latter and 52.2 for the former, most of them are women (59.9%). Four carers in ten are in employment. Considering the increase in the labour market participation of women aged 50 to 64 27 , the issue of worklife balance is particularly acute for women.

In France, public intervention aimed at informal carers is based on "supportive measures" defined as measures to assist carers in their role 28 : alongside the existing training available since the 2009 Health Law and the national web platform created in 2013, the 2015 ASV Act has introduced a right to respite, and created special centres for the elderly and their carers (see Section 3). Until recently, financial measures aimed at compensating informal carers remained marginal with the possibility to use the APA to pay a relative (except the spouse), (8% only of APA beneficiaries pay a relative as their home carers 29 ; the recent vote of a financial compensation to pay the existing carer's leave marks a turning point in the type of measures implemented (see Section 3).

Financial sustainability

Different factors can be identified to analyse the financial sustainability of LTC policy. One component is the demographic determinant (see Section 1.1). However, expenditure also varies according to the level and type of coverage of the needs of the population; in particular whether France relies on formal or informal care, at home or in institutions 30 .

According to data from the Economic Policy Committee (EPC) Working Group on Ageing Populations and Sustainability (AWG), public expenditure on LTC is projected to increase in France by 0.8 GDP percentage points (or 46%) between 2016 and 2070 (from 1.7 in 2016 to 2.5 in 2070) in the "demographic" scenario, based on the impact of an ageing population on public LTC expenditure (0.5 pp below the EU27 level of increase at 1.3). This increase still varies according to different demographic calculations: from 0.6 in the healthy ageing scenario to 1.1 in the high life expectancy scenario.

Nevertheless, these projections depend on different policy-change options. The EPC's Working Group on Ageing considers a variety of scenarios: a "shift from informal to formal care" scenario, which considers a shift to formal care to progressively replace the informal care and cash-for-care option (the change between 2016 and 2070 is estimated at 1,3 percentage point or an increase of 76%). EPC working group also consider a "AWG risk" scenario, which is based on the assumption that half of the future gains in life expectancy are spent with no care-demanding disability as in the reference scenario. In this scenario public expenditure on LTC will grow from 1.7 in 2016 to 4.5% of GDP in 2070, which means a 2.8 percentage points of difference or a growth by 160%.

By breaking down the impact of drivers on the difference in spending growth (2016-2060) and taking into account the real change between the 2015 and 2018 ageing reports, EPC establishes a relatively stable situation for France, with a zero base-year effect, which represents the difference between the variation in spending between the 2018 and 2015 ageing reports and the sum of the different sources of change, including those linked to methodology and policy reforms. 30 These elements generally appear in terms of level of coverage of the "potentially dependent population" in terms of access to home care, institutional care and cash benefits. The potentially dependent population refers to EU-SILC data on "self-perceived longstanding limitation in activities because of health problems [for at least the last 6 months]" (EU 2018 Ageing Report, op cit.).

National analyses on the cost of dependency have also been produced. Roussel (2017) estimates LTC costs at €30 billion in 2014 (1.4 point of GDP), which comprises €23.7 billion public expenditure (79% of the total) and €6.3 billion covered by households. Expenditure is split as follows: €12.2 billion devoted to healthcare expenditure (€12.1 billion public funding, €0,1 billion by households); €10.5 billion devoted to coverage of loss of autonomy (€8.3 billion public funding); €7.1 billion devoted to institutional accommodation (including €3.1 billion public expenditure and €3.8 billion by households). To this can be added €4.4 billion for "board and lodging" (the cost of food, housing, and insurance paid for by households in institutions that they would have to cover in the home). However, this figure does not include the cost of informal care which is evaluated at between €7 billion and €18 billion. As argued by several studies, the expenditure related to the elderly care is in fact equally shared between households and public sector. Based on these data, the Libault Report calls for a necessary 35% increase in the share of national wealth devoted to the loss of autonomy, which corresponds to public expenditure of about 1.6% of GDP. A specific funding plan is proposed in the report (see box 1).

Country-specific challenges regarding LTC for other age groups in need of care

Specific key challenges can be identified regarding care for children and disabled people. An important issue arises from the policy to facilitate inclusion for disabled people in a "normal environment"31 . This "inclusive turn or shift", which involves reducing institutionalisation, implies the development of suitable conditions in the so-called "normal environment". It concerns both encouraging employment of disabled people in public and private sectors and facilitating schooling for children with disabilities. In this perspective, the professionalisation of special needs assistants who support children at school every day is a key dimension of this policy orientation. The shift was initiated in 2013 with the introduction of a professional status -assistants for pupils with disabilities (Accompagnants des élèves en situation de handicap -AESH) requiring a higher level of education to be selected (at least baccalauréat or equivalent) and receive specific training. The professionalisation was then pursued with the creation of a state educational and social assistance diploma (Diplôme d'Etat d'Accompagnement Educatif et Social -DEAES). Another important step is the establishment of a personalised plan for each child included in a specialised unit (ULIS -Unité localisée pour l'inclusion scolaire, local unit for inclusion in schools) which is linked to an "ordinary" school. Another issue in the field of LTC care concerns the reform of the pricing of institutions and medico-social services working with disabled people. This vast project was initiated in 2015 and is still underway.

3

Reform objective and trends

The major guidelines of a deep-seated reform were presented in a report dating from March 2019 -the Libault Report (see box 1). Initially announced for 2019, the Old Age and Autonomy Plan was delayed. Yet, in the context of the COVID-19 crisis and considering the strong impact on elderly people at home and in institutions32 , it was confirmed as a policy priority in May 2020.

It is also worth noting the special attention paid to family caregivers in 2019: -The Act of 22 May 2019 aimed at facilitating taking leave and securing carers' rights.

-The vote in October 2019 of an allowance for recipients of the carer's leave which aims at encouraging carers to make use of the leave that was in little use at that point33 . -A national strategy to encourage support for carers (see box 1).

In addition, there was a specific measure with a view to improve quality, along with access and affordability: the Act of 24 July 2019 on the transformation of the health system, establishing the integration of existing coordination measures into complex pathways (MAIA, PTA, PAERPA, networks) within a unique support measure (DAC).

The ASV Act (implemented in January 2016) is the latest general reform to date. While criticised for the insufficient financial resources it opened up 34 , it has led to steps forward:

Concerning access and affordability, it is worth mentioning measures to upscale the APA and reduce the number of people subject to co-payment. The ASV Act has thus led to a +6.5% increase in APA expenditure (Libault, 2019). It also highlighted prevention and coordination, with the introduction of a special body -the funders' conference to prevent elderly people's loss of autonomy -to manage coordination at the départements level.

Concerning informal care, the ASV Act has had a real impact: the proposal of a formal definition of carers including friends and neighbours; the introduction of the right to respite (with financial support of up to €500 per year), and the creation of special centres for the elderly and their carers (Maisons des aînés et des aidants), a kind of one-stop shop that provides information, administrative support for users, and also support to organise the coordination of social and health care responses.

Concerning quality, the ASV Act made multiannual contracts (CPOM) general to regulate all social care facilities.

Box 1: Planned reforms and on-going legislative process and debates The 2019 Libault Report

While maintaining older people in their homes and increasing freedom of choice in the organisation of care are put forward as priorities, the 175 measures presented in the Report concern both the development of quality support in the home and care in institutions 35 . Concretely, three main strands can be identified:

The first of these consists of a series of measures aimed at reorganising the various types of existing financial support, which include: creating a new home-based cash benefit (to replace the current APA with three components: personal assistance, technical support and respite (for carers)); merging healthcare and social care expenditure in residential homes to reduce the remaining amount which is payable by residents.

The second strand concerns the development of services in the home and in residential homes through: renovating residential homes and making them more open to the outside world(a renovation plan worth €3 billion has been announced); increasing the supervision rate in residential homes by 25%, by recruiting 80,000 employees at an estimated cost of €1.2 billion; creating 60,000 places in residential homes; developing alternatives to residential homes or home-based care, i.e. temporary accommodation and collective housing; creating indicators to measure the quality of services available in residential homes.

Finally, the third strand aims to increase support to the 3.9 million informal carers who provide care to older relatives. This will be done by simplifying procedures and access to information, providing financial aid to support informal carers, and facilitating informal carers' work/life balance.

The set of measures announced in the report will require massive public funding with an estimated additional amount of €9.2 billion by 2030. The Report argues in favour of financing LTC policy through national solidarity, by recognising "loss of autonomy" as a genuine social protection risk, and including this in social security funding legislation. The favoured scenario is that an existing pay deduction (the Contribution to Reducing the Social Debt [CRDS]), which will have been fully paid by 2024, will be converted into funding for loss of autonomy. Recourse to private funding is presented as additional to public funding. One possibility could be to take account of a share of property assets when calculating the level of the benefit received, in order to support funding of home-based and residential care. Alternative scenarios, such as creating new mandatory pay deductions, or extending working time by cancelling a national holiday, have so far been ruled out. Besides this financing challenge, the governance of LTC policy needs to be clarified, as it involves two main institutional actors -regional health agencies and local councils -who can have a tense relationship36 .

Two documents have been produced in 2019 to confirm the chosen direction: the El Khomri report, which outlines measures and steps for the reorganisation of the LTC workforce; and the National strategy for mobilisation and support to carers which concerns all carers (whatever the age of the cared-for). Following in the vein of the ASV Act and the Libault Report, it stresses the need to open up new rights for carers and anticipate their exhaustion and isolation, and to diversify and increase the reception capacities of respite places.

4

Main opportunities for addressing LTC challenges

The announcement of a comprehensive Old age and Loss of Autonomy Plan, based on the Libault Report, is the next step awaited by all professional and institutional actors. Confirmed as a priority by the impact of the COVID 19 crisis, it represents main opportunities for addressing LTC challenges in relation with demographic evolutions: :  The recognition of "loss of autonomy" as a genuine social protection right, with its inclusion in social security funding legislation and the confirmation of the €9.2 billion by 2030.

 The reorganisation of the LTC workforce at home with a a reform of the complex pricing system; and an improvement in working conditions for care workers to make the sector more attractive  The recruitment of qualified and recognised health and social care professionnals in institutions.

 Further development of coordination efforts in order to simplify the existing schemes, avoid fragmentation or overlapping measures and facilitate continuity of care for the old person.

 Building on what has already been done in terms of compensation, conciliation or supportive measures for informal carers, the development of policy measures to support informal care combined with the improvement of services at home and in institutions in order to enlarge choice for informal carers.

 .

  The development of this type accommodation renamed autonomous residences (Résidences autonomie) is a priority of the 2015 ASV Act. They offered 110,000 places in 2018 17 . -Autonomous residences and nursing homes offered 11,900 day care places and 15,500 in temporary accommodation in 2018 (CNSA, 2019). In 2015, 728,000 elderly people lived in residential care, an increase of 4.8% compared with 2011 (Muller et al. 2017). The recent Capacités, Aides et Ressources des seniors (CARE) survey 18 estimates the number as being between 0.4 million (including only highlevel dependency cases) and 1.5 million (also including mid-level dependency cases). Concerning the balance between formal and informal carers, the CARE survey established that almost 50% of the elderly in need of care receive support from their relatives and 20% only professional support. A third get both formal and informal support 19 .

  /www.ccomptes.fr/sites/default/files/EzPublish/20160712-maintien-domicile-personnes-agees.pdf

	27 Up to up to 61.2% in 2017 according to INSEE, (2019), Les tableaux de l'économie française [French economy
	tables], INSEE.
	28 Le Bihan B. Lamura G., Marczak J., Fernandez J.L, Johansson L. and Sowa-Kofta A., (2019), Policy measures
	to support unpaid care across Europe, in Enhancing the sustainability of Long Term Care, Eurohealth, vol 25/4,
	p. 10-14.
											old
	dependent	people	in	their	homes],	Court	of	Auditor	report,	downloaded	from
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29

Court of Auditors (2016), Le maintien à domicile des personnes âgées en perte d'autonomie [Maintaining

All data used in the text come from Annex 1, Background Statistics unless explicitly stated otherwise.

This report does not take into account the impact of the current COVID-19 pandemic which will surely lead to important measures.

Blanpain, N., 2018. « L'espérance de vie par niveau de vie : chez les hommes, 13 ans d'écart entre les plus aisés et les plus modestes », Insee Première, n°1687, février.

Libault D. (2019), Grand âge et autonomie [old age and autonomy], Report commissioned by the ministry of solidarities and health. Downloaded from https://solidaritessante.gouv.fr/IMG/pdf/rapport_grand_age_autonomie.pdf

5 Le Bihan B., Da Roit B. and Sopadzhiyan A.,(2019), The turn to optional familialism through the market: Longterm care, cash-for-care, and caregiving policies in Europe, Social Policy and Administration, Special Issue Cash for Care in Europe, p. 579

-5946 A specifically French sector along with the health and social sectors, whereas common English usage only distinguishes 'health care' and 'social care'

sectors.7 The funds of the CNSA combined different sources (CNSA, 2019): a transfer of part of the sickness branch of the social security system (€20,4 billion); taxes (€2,7 billion); a social contribution -(€2.4 billion).

Created in 2009, the French National agency for evaluating the quality of social care institution and services (ANESM) is now included in the National authority for health (HAS).

Vidal A. (2018), Rapport d'information sur l'évolution de la démarche qualité au sein des EHPAD et de son dispositif d'évaluation [information report on the evolution of quality in residential care], Assemblée nationale, n° 1214.

This expression "environnement ordinaire" means inclusion of disabled people in all sectors and activities.

The current COVID-19 pandemic has dramatically impacted the older-old population. In April 2020, more than 25,000 people in nursing homes (EHPAD) had been infected by Covid-19 in France and more than 8,000 deaths had been registered in nursing homes. The mortality rate due to Covid-19 is closely linked to age and gender. 50% of patients with Covid-19 admitted into intensive care were aged 65 or over and 60% of them were men.

Sirven, N., Naiditch, M., and Fontaine, R., (2015), Etre aidant et travailler, premiers résultats d'une enquête pilote [combining caregiving and working, first results of our pilot survey], Université Paris-Descartes. Downloaded from http://www.aveclesaidants.fr/wp-content/uploads/2015/12/RapportFinalMACIF-JNA-2015.pdf

Le Bihan B., (2016), France anticipates ageing society through new piece of legislation, European social policy network, ESPN Flash report 2016/18.

Le Bihan B., and Sopadzhiyan A., (2019), Future trends in French LTC Policy: the Libault Report, European social policy network, ESPN Flash Report, 2019/25.

Le Bihan B. and Sopadzhiyan A., (2017), The development of integration in the elderly care sector: a qualitative analysis of national policies and local initiatives in France and Sweden, Ageing and Society, Published online by Cambridge University Press: 26 December 2017, pp. 1022-1049 Print publication: May 2019 doi:10.1017/S0144686X17001350

Annex 1 Background statistics Annex 2 National statistics

Loss of autonomy for elderly people are measured in the following ways:

-By adding up the number of people targeted by the public measures that concern them (APA). "Dependent" people are considered to be those who come under GIR 1 to 4. The GIR ranking is determined by a team of professionals from the health and social care sectors who visit the old person at home to assess his/her level of dependency and needs (there are 6 levels of GIR from the highest, GIR 1 to the lowest GIR 6).

Although relatively limited, this conventional definition of loss of autonomy has the advantage of corresponding to the eligibility criteria for the Personal Autonomy Allowance (APA). This definition, on which many statistics are elaborated in France, is limited because it uses an administrative definition of loss of autonomy and consequently excludes people who do not claim the benefit even though they might need it. -By evaluating people's functional status through tests. This measure has the advantage of being more reliable, but it is difficult to implement as part of a statistical survey on a broad cross-section of people. -Lastly, by directly questioning people about the difficulties that they come up against, covering a broad range of people to ensure that the results represent the entire group. This is the approach taken by the Handicap-Santé (2008) and CARE-Ménages (2015) surveys by carried out by the DREES which is referred to in this report.

Source: https://solidarites-sante.gouv.fr/affaires-sociales/personnes-agees/concertationgrand-age-et-autonomie/article/rapport-de-la-concertation-grand-age-et-autonomie


Anh-Tu Nguyen

Member, IEEE Philippe Chevrel

Fabien Claveau

LPV Static Output Feedback for Constrained Direct Tilt Control of Narrow Tilting Vehicles

Keywords: Narrow tilting vehicles, direct tilt control, static output feedback control, LPV control, vehicle dynamics, linear matrix inequality (LMI)

This paper presents a new direct tilt control (DTC) design to improve the lateral stability and the driving comfort of narrow tilting vehicles. To this end, a conceptual model is constructed from the vehicle dynamics, a simplified model of the driving environment, and the vehicle perceived acceleration. This latter is considered as the main performance variable of the related H2 control problem. The conceptual model is then represented in a polytopic linear parameter varying (LPV) form for control purposes. To avoid the use of costly vehicle sensors while favoring the simplest control structure for real-time implementation, a new LPV static output feedback (SOF) control method is proposed. Thanks to Lyapunov stability arguments, physical constraints on both system states and DTC actuator are explicitly considered in the design procedure to improve the safety and the comfort of passengers. Moreover, a parameter-dependent Lyapunov function is exploited for theoretical developments. In this way, the finite bounds on vehicle speed and acceleration are effectively taken into account in the control design to reduce the conservatism. The H2 control design is recast as an LMI-based optimization which can be easily solved with numerical solvers. The resulting robust DTC controller is evaluated with realistic driving scenarios.

I. INTRODUCTION

Narrow tilting vehicles (NTVs) have been increasingly studied as a promising solution to traffic congestion, pollution, and parking issues in urban areas [START_REF] Chiou | Modeling and verification of a diamond-shape narrow-tilting vehicle[END_REF]- [START_REF] Sindha | Automatic stability control of three-wheeler vehicles-recent developments and concerns towards a sustainable technology[END_REF]. Moreover, the reduced dimensions of NTVs also result in their high fuel efficiency [START_REF] Kidane | Development and experimental evaluation of a tilt stability control system for narrow commuter vehicles[END_REF], [START_REF] Ataei | Reconfigurable integrated stability control for four-and three-wheeled urban vehicles with flexible combinations of actuation systems[END_REF]. Due to their special features, NTVs are characterized by a high center of gravity (c.g.) leading to the major issue on roll stability [START_REF] Mourad | Direct and steering tilt robust control of narrow vehicles[END_REF]- [START_REF] Piyabongkarn | Active direct tilt control for stability enhancement of a narrow commuter vehicle[END_REF]. Indeed, to maintain the vehicle stability, NTVs should lean while cornering as motocycles to compensate the effects of the centrifugal force. Hence, effective tilt control systems are essential elements in any narrow vehicle system design, which greatly improves the acceptability of NTVs with respect to standard vehicles [START_REF] Rajamani | Dynamics of narrow tilting vehicles[END_REF]. Up to now, a few NTV prototypes have been developed in the automotive industry, e.g., F-300 Life Jet by Mercedes-Benz (1997), Brink Dynamics by Carver (2003), SMERA by Lumeneo (2008), Land Glider by Nissan (2009), see [START_REF] Kidane | Development and experimental evaluation of a tilt stability control system for narrow commuter vehicles[END_REF], [START_REF] Maakaroun | Modeling and simulating a narrow tilting car using robotics formalism[END_REF] for more details.

Two main types of control systems are available for vehicle tilting [START_REF] Rajamani | Dynamics of narrow tilting vehicles[END_REF]: direct tilt control (DTC), and steering tilt control (STC). STC relies on a steering actuator to control the vehicle tilt in the same way as a moto rider [START_REF] Corno | Single-track vehicle dynamics control: state of the art and perspective[END_REF]. This control strategy has some drawbacks: (i) it is not suitable at low speeds, nor on slippery roads [START_REF] Kidane | Development and experimental evaluation of a tilt stability control system for narrow commuter vehicles[END_REF], (ii) it may require a large countersteering to tilt the vehicle which leads to a significant trajectory deviation and ride discomfort [START_REF] Corno | Single-track vehicle dynamics control: state of the art and perspective[END_REF]. Using a dedicated tilt actuator, DTC strategy can directly control the vehicle roll. Then, DTC could be used to address the major drawbacks of STC. However, the main drawback of DTC consists in its energy consumption and discomfort at the beginning of curve taking [START_REF] Mourad | Direct and steering tilt robust control of narrow vehicles[END_REF]. It has been shown that the integrated approach, i.e., the combination of STC and DTC, can lead to the best control performance [START_REF] Kidane | Development and experimental evaluation of a tilt stability control system for narrow commuter vehicles[END_REF], [START_REF] Hibbard | Twenty first century transportation system solutions-a new type of small, relatively tall and narrow active tilting commuter vehicle[END_REF],

A.-T. Nguyen is with the LAMIH laboratory UMR CNRS 8201, Université Polytechnique Hauts-de-France, Valenciennes, France.

P. Chevrel and F. Claveau are with the IMT Atlantique, UBL, and the LS2N laboratory UMR CNRS 6004, Nantes, France. This work was done when A.-T. Nguyen was with the LS2N laboratory UMR CNRS 6004.

* Corresponding author. E-mail: nguyen.trananhtu@gmail.com. [START_REF] Chatterjee | A dynamic stability control for electric narrow tilting three wheeled vehicle using integrated multivariable controller[END_REF]. Unfortunately, this solution requires both a tilt actuator and a steer-by-wire system, which increases the cost of NTVs [START_REF] Corno | Single-track vehicle dynamics control: state of the art and perspective[END_REF].

This work is the continuity of our collaboration with Lumeneo company in developing control strategies for the SMERA vehicle [START_REF] Mourad | Direct and steering tilt robust control of narrow vehicles[END_REF], [START_REF] Maakaroun | Modeling and simulating a narrow tilting car using robotics formalism[END_REF]. This four-wheeled NTV has two seats in tandem and an DTC actuator for tilt control, see Fig. 1. As in our previous studies [START_REF] Mourad | Direct and steering tilt robust control of narrow vehicles[END_REF], a direct regulation/minimization of the perceived acceleration will be considered for the control design. This allows avoiding both the online computational complexity and the residual tilt torque, thus saving the energy consumption. The latter is caused by approximation errors of the tilt angle reference in tracking control approaches largely discussed in the literature, see [START_REF] Chiou | Modeling and verification of a diamond-shape narrow-tilting vehicle[END_REF]- [START_REF] Kidane | Development and experimental evaluation of a tilt stability control system for narrow commuter vehicles[END_REF], [START_REF] Sindha | Automatic stability control of three-wheeler vehicles-recent developments and concerns towards a sustainable technology[END_REF], [START_REF] Piyabongkarn | Active direct tilt control for stability enhancement of a narrow commuter vehicle[END_REF], [START_REF] Hibbard | Twenty first century transportation system solutions-a new type of small, relatively tall and narrow active tilting commuter vehicle[END_REF], [START_REF] Roqueiro | Sliding mode controller and flatness based set-point generator for a three wheeled narrow vehicle[END_REF] and references therein. Note that most of DTC methods have considered a constant vehicle speed to ease the design task, see [START_REF] Mourad | Direct and steering tilt robust control of narrow vehicles[END_REF], [START_REF] Corno | Single-track vehicle dynamics control: state of the art and perspective[END_REF] for short surveys. However, the corresponding linear controllers cannot provide a satisfactory closed-loop performance under various driving circumstances [START_REF] Snider | Automatic steering methods for autonomous automobile path tracking[END_REF]. Moreover, these methods are usually based on a state feedback control scheme which requires full vehicle sensor information for real-time implementation. This paper presents a systematic DTC method which can effectively address the above technical and practical control issues. The main contributions can be summarized as follows.

• Using an SOF control scheme, the proposed DTC controller is easily realized from engineering viewpoint without needing an important vehicle sensor to measure the lateral speed. Differently from [START_REF] Mourad | Direct and steering tilt robust control of narrow vehicles[END_REF], the new control method does not require any online estimation of the vehicle states which can lead to online computational burden, approximation errors, and especially an algebraic loop for real-time implementation. Moreover, similar to [START_REF] Mourad | Direct and steering tilt robust control of narrow vehicles[END_REF], the new method can be easily adapted to STC and integrated control approaches.

• LPV paradigm has been demonstrated as an effective tool for modeling and control in a variety of engineering applications, see [START_REF] Mohammadpour | Control of Linear Parameter Varying Systems with Applications[END_REF]- [START_REF] Briat | Linear Parameter-Varying and Time-Delay Systems: Analysis, Observation, Filtering & Control[END_REF] and related references. Here, the time-varying nature of the vehicle speed is tackled with a novel LPV control framework.

Especially, the finite bounds on both vehicle speed and acceleration are explicitly considered in the H2 control via a parameterdependent Lyapunov function to reduce the design conservatism.

• The new DTC method can explicitly take into account the physical constraints on both the tilt torque and the system states in the H2 control procedure to improve not only the energy consumption of DTC actuator but also the safety and the comfort of passengers. This major issue for lateral control of NTVs has not been well addressed in the open literature [START_REF] Piyabongkarn | Active direct tilt control for stability enhancement of a narrow commuter vehicle[END_REF]. The closed-loop performance of the LPV vehicle system is theoretically proved using Lyapunov stability theory, which is not the case of most existing methods [START_REF] Mourad | Direct and steering tilt robust control of narrow vehicles[END_REF]. In particular, the design conditions are formulated as an LMI optimization which can be effectively solved with numerical solvers [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. Note that LMI-based H2 static output control for LPV systems still remains an open research topic [START_REF] Sadabadi | From static output feedback to structured robust static output feedback: A survey[END_REF]. The proposed theoretical results go beyond the scope of the considered application and can be applied to a larger class of LPV systems.

The paper is organized as follows. Section II presents the nonlinear model of NTVs used for simulation purposes and formulates the control goals. In Section III, we define an LPV standard model of NTVs used for lateral control purposes. Section IV presents a new robust SOF method allowing to achieve several control specifications of NTVs. The effectiveness of the new DTC method is demonstrated in Section V. Conclusions are given in Section VI.

Notation. ΩN denotes the number set {1, 2, . . . , N }. I denotes the identity matrix of appropriate dimension. For a matrix X, X indicates its transpose, and X (i) denotes its ith row. For any square matrix X, X > 0 indicates a symmetric positive definite matrix, and He X = X + X . diag(•) denotes a block diagonal matrix formed by the blocks given in the parenthesis. The ith element of a vector u is denoted by ui, and stands for matrix blocks deduced by symmetry. co{S} denotes the convex hull of the set S. The time argument is dropped when convenient.

II. VEHICLE MODELING AND PROBLEM DEFINITION

Several models of NTVs have been proposed for both simulation and control purposes, see for instance [START_REF] Rajamani | Dynamics of narrow tilting vehicles[END_REF], [START_REF] Maakaroun | Modeling and simulating a narrow tilting car using robotics formalism[END_REF], [START_REF] Chatterjee | A dynamic stability control for electric narrow tilting three wheeled vehicle using integrated multivariable controller[END_REF], [START_REF] Tang | Design and analysis of an integrated suspension tilting mechanism for narrow urban vehicles[END_REF], [START_REF] Furuichi | Switching dynamic modeling and stability analysis of three-wheeled narrow tilting vehicle[END_REF]. This section presents the nonlinear model of the SMERA vehicle used for our simulation study. Then, the related control problem is formulated. The vehicle nomenclature is given in Table I.

A. Nonlinear Model of Narrow Commuter Vehicles

Fig. 2 depicts the four degrees of freedom (DOF) of the SMERA vehicle: the longitudinal and lateral positions (x, y) of the vehicle, the tilt angle θ, and the yaw angle ψ. The corresponding dynamics model was developed in collaboration with Lumeneo company by inspiring from two nonlinear vehicle models. The first one focusses on the dynamics of the vehicle lateral position y, the tilt angle θ, and the vehicle yaw ψ while considering a constant vehicle speed [START_REF] Rajamani | Dynamics of narrow tilting vehicles[END_REF]. The second bicycle model represents both lateral and longitudinal dynamics [START_REF] Fuchshumer | Nonlinear vehicle dynamics control -A flatness based approach[END_REF]. Then, the SMERA dynamics is described by

m vx = F sf sin(β -δ) + F lf cos(β -δ) + Fsr sin β + F lr cos β -F θ sin β mvx β = F sf cos(β -δ) -F lf sin(β -δ) + Fsr cos β -F lr sin β -mvx ψ -F θ cos β Iz ψ = l f (F sf cos δ + F lf sin δ) -lrFsr Ix θ = mgh sin θ -mh 2 θ2 cos θ sin θ -mh 2 θ sin 2 θ -(F sf cos δ + F lf sin δ + Fsr)h cos θ + Mt (1)
where F θ = m(h θ cos θ -h θ2 sin θ), Ix = Ix + mh 2 sin 2 θ and vy = vx sin β. The nonlinear lateral tire forces are modeled as

Fsi = Fsi(αi) + 2λiθ (2)
for i ∈ {f, r}. The nonlinear forces Fsi(αi) in (2) are modeled following the well-known Pacejka's magic formula [START_REF] Pacejka | Tire and Vehicle Dynamics[END_REF]:

Fsi(αi) = Di sin (Ci arctan [(1 -Ei)Biαi + Ei arctan (Biαi)])
where Bi, Ci, Di and Ei, for i ∈ {f, r}, are the Pacejka parameters. The sideslip angles of the front/rear tires are given as follows [START_REF] Fuchshumer | Nonlinear vehicle dynamics control -A flatness based approach[END_REF]:

α f = δ -arctan vx sin β+l f ψ vx cos β , αr = -arctan vx sin β-lr ψ vx cos β .

B. Available Sensors for Real-Time Control Implementation

As most of NTVs, SMERA is equipped with a tilt angle sensor and an inertial measurement unit (IMU), which provide the measurements of the vehicle states θ, θ and ψ. An odometer is used to measure the vehicle speed vx whereas the steering angle δ and its derivative δ are obtained from an optical encoder. The IMU provides also the lateral acceleration aper at the gravity center G, defined as follows [START_REF] Rajamani | Dynamics of narrow tilting vehicles[END_REF]:

aper = ( vy + vx ψ) cos θ + h θ -g sin θ (3)
The acceleration aper plays a crucial role in the study of lateral stability of NTVs, and the comfort of passengers [START_REF] Mourad | Direct and steering tilt robust control of narrow vehicles[END_REF]- [START_REF] Piyabongkarn | Active direct tilt control for stability enhancement of a narrow commuter vehicle[END_REF].

The sideslip angle β and the lateral speed vy can be measured by a CORREVIT sensor. Unfortunately, due to the excessive cost of this optical sensor (about 15ke per unit), the measurements of β and vy are unavailable for control implementation [START_REF] Nguyen | Sensor reduction for driverautomation shared steering control via an adaptive authority allocation strategy[END_REF]. This paper presents a systematic method for multiobjective lateral control of NTVs modeled by (1) which can handle this major practical issue.

C. Problem Statement and Control Goals

Automatic DTC controllers aim to guarantee the lateral stability of NTVs, i.e., to regulate the perceived acceleration aper around zero during cornering [START_REF] Kidane | Development and experimental evaluation of a tilt stability control system for narrow commuter vehicles[END_REF]. Here, the following specifications are required for DTC control design.

• DTC controllers should be easily computed and implemented with only available vehicle sensors. The closed-loop control performance and robustness can be demonstrated using Lyapunov stability arguments.

• The integral of the perceived acceleration aper, denoted by a I per , should be directly regulated and minimized to: (i) avoid static errors during long curves, (ii) improve the comfort of passengers and the energy consumption [START_REF] Mourad | Direct and steering tilt robust control of narrow vehicles[END_REF].

• The constraints on both the system states and the tilt actuator should be explicitly taken into account in the control design to improve both the safety and the comfort of passengers.

To meet these specifications, we propose in Section IV a new LPVbased SOF control method in multiobjective setting.

III. STANDARD MODEL FORMULATION FOR H2 CONTROL

Based on an H2 control scheme, the proposed DTC method makes use of a standard model Σ composed of three generic elements: plant model Σp, environment model Σw, and model of regulated signals Σe, see Fig. 3. Next, these three elements are first defined in the similar way as in [START_REF] Mourad | Direct and steering tilt robust control of narrow vehicles[END_REF]. Then, we construct the corresponding controlbased LPV standard model.

A. Vehicle Plant Model

For lateral control purposes, a 3 DOF vehicle model is derived from (1). To this end, the following usual assumptions are considered [START_REF] Rajamani | Dynamics of narrow tilting vehicles[END_REF]. (i) The longitudinal dynamics is neglected. (ii) The lateral tire forces are proportional to the slip angles of each axle. (iii) The vehicle angles are small. It is important to note that these assumptions are relevant for normal driving conditions in urban areas which are the control focus of this paper. Then, the lateral dynamics of NTVs can be modeled as follows [START_REF] Piyabongkarn | Active direct tilt control for stability enhancement of a narrow commuter vehicle[END_REF]:

Σp : ẋp = Apxp + Bpuu + B δ δ (4)
where xp = vy ψ θ θ is the vehicle state vector, and the control input is the tilt torque from the DTC actuator u = Mt. The system matrices of (4) are given by

Ap =     a11 a12 a13 0 a21 a22 a23 0 0 0 0 1 a41 a42 a43 0     , Bpu =     -h Ix 0 0 1 Ix     , B = 2C f 1 m + h 2 Ix 2l f C f Iz 0 - 2hC f Ix , where a11 = -1 vx a m + h 2 a Ix , a12 = -1 vx b m + h 2 b Ix -vx, a13 = 2(λ f + λr) 1 m + h 2 Ix -mgh 2 Ix , a21 = -b Iz vx , a22 = -2(l 2 r Cr +l 2 f C f ) Iz vx , a23 = 2(λ f l f -λr lr ) Iz , a41 = ha Ixvx , a42 = hb Ixvx , a43 = mgh-2h(λ f +λr ) Ix , a = 2(Cr + C f ), b = 2(C f l f -Crlr).
The output equation of system (4) is given by

yp =   0 1 0 0 0 0 1 0 0 0 0 1   xp = Cpxp.

B. Environnement Model

The goal is to stabilize the lateral acceleration aper under the variation of the steering actions. Hence, the environnement model of Σp represents the a priori knowledge available on δ. The following model is used for the curved trajectory prediction [START_REF] Mourad | Direct and steering tilt robust control of narrow vehicles[END_REF]:

Σw : ẋw = Awxw + B1ww y1w = C1wxw, y2w = C2wxw (5)
with xw = δ δ , w is an impulse signal and

Aw = 0 1 -α1α2 -(α1 + α2) , C2w = 1 0 0 1 , B1w = 0 1 , C1w = 1 0 ,
where Aw is Hurwitz with α1 = 0.5 and α2 = 1. The following remark is reported for the environment model Σw.

Remark 1. The environnement model ( 5) represents the fact that the steering angle δ is band limited and can be truncated in sequences of asymptotically convergent signals. Note that the time constant of Σw is about 2s which is reasonable for road curvature prediction in most real-world situations. This cannot be considered as a driver model since its dynamics does not depend on that of the vehicle. However, the consideration of such a predictor model in the control design offers a significant performance improvement. Especially, for the case of NTVs, the introduction of δ in the output y2w of Σw improves the prediction capacity of DTC controller with respect to curved trajectories to avoid an excessive tilt torque [START_REF] Mourad | Direct and steering tilt robust control of narrow vehicles[END_REF].

C. Model of Regulated Signals

As stated above, the integral of aper should be directly controlled and minimized to improve the DTC performance. The model of the regulated signal a I per is given by

ȧI per = aper (6)
From ( 3), model ( 6) can be represented in the following form by considering the small angles assumption:

ȧI per = aper ≈ vy + vx ψ + h θ -gθ = (G1 + G2Ap)xp + G2B δ δ (7)
where G1 = 0 vx -g 0 and G2 = 1 0 0 h . Moreover, following the guideline in [START_REF] Lundström | Performance weight selection for H∞ and µ-control methods[END_REF], the performance vector z associated to model [START_REF] Rajamani | Dynamics of narrow tilting vehicles[END_REF] can be chosen as follows:

z = We(s)aper, We(s) = κs + M M κs (8)
where the parameters M and κ are positive. The weighting function We(s) is defined as above for two reasons. First, this guarantees that aper should be less than M in magnitude at high frequencies. Second, we integrate an integral action and require a response time better than about κ. From ( 6), ( 7) and ( 8), the model Σe can be represented as

Σe : ẋe = Aexe + Bexxp + Bewxw z = Dexxp + Dexe + Dewxw, ye = Cexe (9)
where xe = a I per and

Ae = 0, Bex = G1 + G2Ap, Bew = G2B δ 0 , Dex = 1 M Bex, De = 1 κ , Dew = 1 M Bew, Ce = 1.

D. Standard Model and Its Polytopic LPV Representation

From the definitions of Σp in (4), Σw in (5), and Σe in [START_REF] Maakaroun | Modeling and simulating a narrow tilting car using robotics formalism[END_REF], the corresponding standard model Σ(vx)1 for H2 control of the SMERA vehicle is constructed as follows:

Σ(vx) : ẋ = A(vx)x + Buu + Bww z = Cz(vx)x, y = Cyx (10)
where x = x p x e x w , y = y p y e y w and

A(vx) =   Ap 0 B δ C1w Bex Ae Bew 0 0 Aw   , Bu =   Bpu 0 0   , C z (vx) = D ex D e D ew , B w = 0 0 B 1w , Cy = diag(Cp, Ce, C2w).
Remark 2. Note that the dynamics matrix and the performance matrix of the standard model Σ(vx) depend explicitly on the timevarying speed which is measured and bounded

vmin ≤ vx ≤ vmax, vmin = 2 [m/s], vmax = 18 [m/s] (11)
Different methods using different LPV representations, e.g., linear fractional transformation (LFT) form, polytopic descriptor form, etc. [START_REF] Briat | Linear Parameter-Varying and Time-Delay Systems: Analysis, Observation, Filtering & Control[END_REF], can be applied for the control design of LPV system [START_REF] Corno | Single-track vehicle dynamics control: state of the art and perspective[END_REF]. However, a polytopic LPV formulation has been chosen here since it leads to a simple characterization of the LPV controller for NTVs. As shown later, the proposed control solution is also of reasonable complexity for real-time implementation.

There are two dependently varying parameters involved in the dynamics of LPV system [START_REF] Corno | Single-track vehicle dynamics control: state of the art and perspective[END_REF], i.e., ω (t) = vx

1 vx
. These parameters form a convex hull Pω with four vertices

ωv1 = vmin 1 v min , ωv2 = vmin 1 vmax , ωv3 = vmax 1 vmax , ωv4 = vmax 1 v min .
Observe in Fig. 4 that the parameter polytope Pω with four vertices leads to design conservatism and numerical complexity. This is due to the fact that vx and 1 vx are separately considered despite its strong dependency. Note that vx and 1 vx only evolve on the curve C * , and the vertices ωv2 and ωv4 are unreachable for any value of vx. Here, to take into account this strong dependency and to reduce significantly the numerical complexity of the control structure, we make use of the following variable change:

vx = v0v1 v1 + v0ω ⇔ 1 vx = 1 v0 + 1 v1 ω (12)
where the new parameter ω used for gain scheduling satisfies

ωmin ≤ ω ≤ ωmax, ωmin = -1, ωmax = 1 (13)
The two constants v0 and v1 in [START_REF] Chatterjee | A dynamic stability control for electric narrow tilting three wheeled vehicle using integrated multivariable controller[END_REF] are given by

v0 = 2vminvmax vmin + vmax , v1 = 2vminvmax vmin -vmax .
It is easy to verify that vx = vmin for ω = ωmin and vx = vmax for ω = ωmax. Hence, ω can be used to describe the variation of vx between its lower and upper bounds. Moreover, using the Taylor's approximation as in [START_REF] Nguyen | Driver-automation cooperative approach for shared steering control under multiple system constraints: Design and experiments[END_REF], the vehicle speed can be also approximated from the second expression in (12) by vx v0 1 -v 0 v 1 ω . As a consequence, the parameter curve C * is approximated by the straight line C, see Fig. 4. Remark that the expressions of both vx and 1/vx are now linearly dependent on ω. Substituting these expressions into [START_REF] Corno | Single-track vehicle dynamics control: state of the art and perspective[END_REF], it can be easily observed that the corresponding standard model of the SMERA vehicle

Σ(ω) : ẋ = A(ω)x + Buu + Bww z = Cz(ω)x, y = Cyx ( 14
)
depends linearly on the new time-varying parameter ω. Using the sector nonlinearity approach [27, Chapter 2], Σ(ω) defined in ( 14) can be exactly represented in the polytopic LPV form

Σ(ω) :            ẋ = 2 i=1 ηi(ω)(Aix + B u i u + B w i w) z = 2 i=1 ηi(ω)C z i x, y = Cyx ( 15
)
where the membership functions (MFs), also called barycentric coordinates, and the state-space matrices are given by It is clear now that using the variable change ( 12) together with the above Taylor's approximation, the number of vertices is reduced from four to two. From the practical viewpoint, note that the induced approximation error is expected to be small over the whole system [START_REF] Corno | Single-track vehicle dynamics control: state of the art and perspective[END_REF] since only a part of the element a12 of matrix Ap in ( 4) is affected by this approximation. This is also justified by experimental results on vehicle control presented in [START_REF] Nguyen | Driver-automation cooperative approach for shared steering control under multiple system constraints: Design and experiments[END_REF].

η1(ω) = 1 -ω 2 , η2(ω) = 1 -η1(ω), A1 = A(ωmin), B u 1 = B u 2 = Bu, B w 1 = B w 2 = Bw, A2 = A(ωmax), C z 1 = Cz(ωmin), C z 2 = Cz(ωmax).
Remark 3. Besides the bounds on the vehicle speed [START_REF] Hibbard | Twenty first century transportation system solutions-a new type of small, relatively tall and narrow active tilting commuter vehicle[END_REF], those of the vehicle acceleration ax = vx are also given

amin ≤ ax ≤ amax, amax = -amin = 3.5 [m/s 2 ] ( 16
)
These physical acceleration bounds allow to limit the theoretical kinematic centripetal acceleration of the vehicle [START_REF] Snider | Automatic steering methods for autonomous automobile path tracking[END_REF]. From ( 12) and ( 16), it follows that

amin a0 ≤ ω ≤ amax a0 , with a0 = - v 2 0 v1 . (17)
As proved later, an explicit consideration of the bounds on both vehicle speed and acceleration via ( 13) and [START_REF] Briat | Linear Parameter-Varying and Time-Delay Systems: Analysis, Observation, Filtering & Control[END_REF] in the control design allows reducing further the design conservatism.

IV. LPV MULTIOBJECTIVE STATIC OUTPUT CONTROL

This section presents new conditions to design a multiobjective SOF controller that will be used for the SMERA vehicle.

A. Preliminaries

Let us consider LPV system [START_REF] Mohammadpour | Control of Linear Parameter Varying Systems with Applications[END_REF] in the general form

ẋ = N i=1 ηi(ω)(Aix + B u i u + B w i w) z = N i=1 ηi(ω)(C z i x + D z i u), y = Cyx (18)
where x ∈ R nx is the state, u ∈ R nu is the control input, w ∈ R nw is the disturbance, z ∈ R nz is the performance output, and y ∈ R ny is the system output. The scheduling vector ω ∈ R nω is measured and assumed to be confined to a compact set Pω ⊂ R nω , defined by the following convex hull:

Pω = co {ωv1, ωv2, . . . , ωvN } (19)
where the vertices ωvi, i ∈ ΩN , are determined by all combinations of the upper and lower bounds ωi min and ωi max of individual scheduling parameters. The constant matrices Ai, B u i , B w i , C z i , Cy, i ∈ ΩN , are of adequate dimensions. The MFs in [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] satisfy the property

N i=1 ηi(ω) = 1, N i=1 ηi(ω) = 0 ηi(ω) ≥ 0, φi1 ≤ ηi(ω) ≤ φi2 (20)
where φi1 ≤ φi2 are known lower and upper bounds of ηi(ω).

Remark 4. Given the bounds of ω and its rate of variation ω, the values of φi1 and φi2, for i ∈ ΩN , can be easily obtained, e.g., it follows from ( 13) and ( 17) that

φ11 ≤ η1(ω) ≤ φ12, φ21 ≤ η2(ω) ≤ φ22, where φ11 = -amax 2a0 , φ12 = -amin 2a0 , φ21 = amin 2a0 , φ22 = amax 2a0 .
We consider the parameter-dependent SOF controller

u = N i=1 ηi(ω)Kiy = K(ω)y (21)
From ( 18) and ( 21), the closed-loop system is rewritten as

Σ cl (ω) : ẋ = Â(ω)x + Bw(ω)w z = Ĉz(ω)x (22)
where Â(ω) = A(ω) + Bu(ω)K(ω)Cy, Ĉz(ω) = Cz(ω) + Dz(ω)K(ω)Cy. This paper proposes a constructive solution for the following control problem.

Problem 1. Determine the parameter-dependent matrix gain K(ω) such that the SOF controller (21) stabilizes LPV system (18) while minimizing the H2 norm Σ cl (ω) 2 of the closed-loop system [START_REF] Fuchshumer | Nonlinear vehicle dynamics control -A flatness based approach[END_REF] with ω ∈ Pω, ∀t > 0, and the compact set Pω is defined in [START_REF] Sadabadi | From static output feedback to structured robust static output feedback: A survey[END_REF].

Note that the H2 control performance is considered to achieve a robust regulation of the lateral acceleration aper under the variation of the steering actions. Hence, this allows improving the DTC performance of the SMERA vehicle during bend taking. For stability analysis and control design, we consider the following parameterdependent Lyapunov function (PDLF):

V (x) = x N i=1 ηi(ω)Qi -1 x = x Q(ω) -1 x ( 23
)
where Qi > 0, for ∀i ∈ ΩN . The following result is standard in the H2 control framework, see for example [START_REF] Souza | Gain-scheduled H 2 controller synthesis for linear parameter varying systems via parameter-dependent Lyapunov functions[END_REF].

Lemma 1. Consider LPV system Σ cl (ω) in [START_REF] Fuchshumer | Nonlinear vehicle dynamics control -A flatness based approach[END_REF] with ω ∈ Pω, ∀t > 0. If there exist a positive definite matrix Q(ω) ∈ R nx×nx , a matrix Z(ω) ∈ R nw ×nw , and a positive scalar γ such that

He Â(ω)Q(ω) -Q(ω) Ĉz(ω)Q(ω) -I < 0 (24)
Z(ω) Bw(ω) Q(ω) > 0 (25) trace(Z(ω)) < γ 2 (26)
Then, it follows that Σ cl (ω) 2 < γ and the associate Lyapunov function of LPV system ( 22) is given in [START_REF] Pacejka | Tire and Vehicle Dynamics[END_REF].

Remark 5. The design conditions in Lemma 1 depend explicitly on both ω and ω (via the term Q(ω)). Moreover, the control gain K(ω) appears nonlinearly in the expressions of Â(ω) and Ĉz(ω) in [START_REF] Fuchshumer | Nonlinear vehicle dynamics control -A flatness based approach[END_REF]. Hence, it is not trivial to obtain an effective solution from such conditions. On the basis of Lemma 1, we propose hereafter new tractable conditions to design an SOF controller [START_REF] Furuichi | Switching dynamic modeling and stability analysis of three-wheeled narrow tilting vehicle[END_REF].

In the framework of polytopic systems, many control design conditions can be represented in the following form [START_REF] Sala | Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya's theorem[END_REF]:

Υ(ω) = N i=1 N j=1 ηi(ω)ηj(ω)Υij < 0 ( 27
)
where matrices Υij are linearly dependent on the decision variables and ω ∈ Pω. To convert the parameter-dependent condition (27) into a finite set of LMIs while avoiding excessive computational burden of parameter-gridding algorithms, the MFs have to be dropped out.

Without involving slack variables, the following lemma leads to a good tradeoff between numerical complexity and conservatism [START_REF] Tuan | Parameterized linear matrix inequality techniques in fuzzy control system design[END_REF].

Lemma 2. Let Υij, i, j ∈ ΩN , be symmetric matrices of appropriate dimensions, and {ηi} i∈Ω N be a family of functions satisfying [START_REF] Tang | Design and analysis of an integrated suspension tilting mechanism for narrow urban vehicles[END_REF]. Condition ( 27) holds if

Υii < 0, 2 N -1 Υii + Υij + Υji < 0 ( 28
)
for i, j ∈ ΩN , and i < j.

Using slack variables, relaxation results with different degrees of design conservatism and/or computational complexity can be found in [START_REF] Sala | Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya's theorem[END_REF].

B. LMI-Based H2 Static Output Feedback Control

The following theorem provides LMI-based conditions to design an H2 SOF controller [START_REF] Furuichi | Switching dynamic modeling and stability analysis of three-wheeled narrow tilting vehicle[END_REF] for LPV system [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF].

Theorem 1. Given an LPV system [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] with ω ∈ Pω, ∀t > 0.

If there exist symmetric positive definite matrices Qi ∈ R nx×nx , matrices Mi ∈ R nu×ny , X ∈ R ny ×ny , Zi ∈ R nw ×nw , for i ∈ ΩN , and positive scalars , γ satisfying the following optimization:

minimize γ 2 subject to Zi B w i Qi > 0 ( 29
) trace (Zi) < γ 2 (30)
Φ klm ii < 0, 2 N -1 Φ klm ii + Φ klm ij + Φ klm ji < 0 (31)
for i, j, k, l ∈ ΩN , i < j, k = l, and m ∈ Ω2. The quantity Φ klm ij is defined as follows:

Φ klm ij = He   Φ klm [11]ij 0 B u i Mj D z i MjCy + C z i Qj -I/2 D z i Mj CyQj -XCy 0 -X   (32) with Φ klm [11]ij = AiQj + B u i MjCy -φ km (Q k -Q l )/2.
Then, the SOF controller [START_REF] Furuichi | Switching dynamic modeling and stability analysis of three-wheeled narrow tilting vehicle[END_REF] solves Problem 1. Moreover, the control feedback gains in [START_REF] Furuichi | Switching dynamic modeling and stability analysis of three-wheeled narrow tilting vehicle[END_REF] are given by

Ki = MiX -1 , i ∈ ΩN (33)
Proof. Multiplying (29) by ηi(ω) ≥ 0 and summing up for all i ∈ ΩN , we obtain [START_REF] Lundström | Performance weight selection for H∞ and µ-control methods[END_REF]. In the same fashion, (30) implies [START_REF] Nguyen | Driver-automation cooperative approach for shared steering control under multiple system constraints: Design and experiments[END_REF]. Note from the definition of Φ klm ij in (32) that if [START_REF] Nguyen | Gain-scheduled static output feedback control for saturated LPV systems with bounded parameter variations[END_REF] holds, it follows that X + X > 0. This guarantees the nonsingularity of X, thus the validity of the gain expression in [START_REF] Agulhari | LMI relaxations for reduced-order robust H∞ control of continuous-time uncertain linear systems[END_REF].

Exploiting the constraint N i=1 ηi(ω) = 0 in (20), it can be easily deduced that

Q(ω) = ηl (ω)Q l + N k=1 k =l ηk (ω)Q k = N k=1 k =l ηk (ω)(Q k -Q l ) (34)
For any φ k1 ≤ ηk (ω) ≤ φ k2 , we can rewrite

ηk (ω) = χ k1 (ω)φ k1 + χ k2 (ω)φ k2 , k ∈ ΩN ( 35
)
where

χ k1 (ω) = φ k2 -ηk (ω) φ k2 -φ k1 , χ k2 (ω) = ηk (ω) -φ k1 φ k2 -φ k1 .
Note that χ kl (ω) ≥ 0, 34) and [START_REF] Löfberg | Yalmip: A toolbox for modeling and optimization in Matlab[END_REF], Q(ω) can be exactly represented as follows:

2 l=1 χ kl (ω) = 1, for k ∈ ΩN . From (
Q(ω) = N k=1 k =l 2 m=1 χ km (ω)φ km (Q k -Q l ) ( 36
)
Using expressions [START_REF] Ebihara | S-Variable Approach to LMI-Based Robust Control[END_REF] and (36), condition [START_REF] Nguyen | Gain-scheduled static output feedback control for saturated LPV systems with bounded parameter variations[END_REF] can be equivalently represented by [START_REF] Souza | Gain-scheduled H 2 controller synthesis for linear parameter varying systems via parameter-dependent Lyapunov functions[END_REF], where

Υij = He   Υ [11]ij 0 B u i Mj D z i MjCy + C z i Qj -I/2 D z i Mj CyQj -XCy 0 -X   ,
and

Υ [11]ij = AiQj + B u i MjCy -Q(ω)/2.
By Lemma 2, condition (28) implies clearly that

Υ(ω) = He   Υ [11] (ω) 0 Bu(ω)M (ω) Υ [21] (ω) -I/2 Dz(ω)M (ω) Υ [31] (ω) 0 -X   < 0 ( 37
)
where Υ [START_REF] Hibbard | Twenty first century transportation system solutions-a new type of small, relatively tall and narrow active tilting commuter vehicle[END_REF] (ω

) = A(ω)Q(ω) + Bu(ω)M (ω)Cy -Q(ω)/2, Υ [21]
(ω) = Dz(ω)M (ω)Cy + Cz(ω)Q(ω) and Υ [START_REF] Nguyen | Gain-scheduled static output feedback control for saturated LPV systems with bounded parameter variations[END_REF] (ω) = CyQ(ω) -XCy. Pre-and postmultiplying (37) with I 0 Bu(ω)M (ω)X -1 0 I Dz(ω)M (ω)X -1 and its transpose, we obtain (24) after some simple but tedious algebraic manipulations. By Lemma 1, the proof of Theorem 1 can be now concluded.

Remark 6. The information on both ω and ω is explicitly considered in the control design by exploiting the bounds φ kl , for k ∈ ΩN , l ∈ Ω2, see Remark 4. This allows using the PDLF [START_REF] Pacejka | Tire and Vehicle Dynamics[END_REF] to reduce the design conservatism. Indeed, if condition ( 31) is feasible for arbitrarily high variation of the MFs, i.e., φ k1 → -∞ and φ k2 → +∞, ∀k ∈ ΩN , then the only possible solution is [START_REF] Ebihara | S-Variable Approach to LMI-Based Robust Control[END_REF]. On the other hand, if one imposes that Qi = Q, for ∀i ∈ ΩN , in [START_REF] Pacejka | Tire and Vehicle Dynamics[END_REF], then the common quadratic Lyapunov function V (x) = x Q -1 x is recovered. This discussion shows that the result of Theorem 1 includes precisely that of quadratic approches. A similar conclusion for the case of statefeedback control of affine LPV systems can be found in [START_REF] Souza | Gain-scheduled H 2 controller synthesis for linear parameter varying systems via parameter-dependent Lyapunov functions[END_REF], [START_REF] Nguyen | Gain-scheduled static output feedback control for saturated LPV systems with bounded parameter variations[END_REF].

Q1 ≈ • • • ≈ QN to minimize the effects of φ km (Q k -Q l ) involved in

C. Constraints on Control Input and System States

It has been shown that even the NTV is at an equilibrium in a perfectly coordinated turn, i.e., only a small amount of torque is required for small deviations, tilting the vehicle into a turn at high speeds may require excessive torque values [START_REF] Piyabongkarn | Active direct tilt control for stability enhancement of a narrow commuter vehicle[END_REF]. Hence, to improve the energy consumption of the tilt actuator and also the ride qualities, the amplitude-limitation |u(t)| ≤ ū, ∀t ≥ 0, where ū is the actuator saturation level, should be imposed for control design. Moreover, as shown in [START_REF] Nguyen | Driver-automation cooperative approach for shared steering control under multiple system constraints: Design and experiments[END_REF] and in the sequel, many safety and comfort criteria can be mathematically translated into state constraints of the form

x ∈ Dx = x ∈ R nx : |H (m) x| ≤ 1, m ∈ Ωq (38)
where the given matrix H ∈ R q×nx characterizes the state domain Dx. Hence, taking into account explicitly the state constraints (38) in the control design is crucial to improve both the safety and the comfort of passengers. Such design constraints can be represented in terms of matrix inequalities as shown in the following theorem.

Theorem 2. Given an LPV system [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] with ω ∈ Pω, ∀t > 0, and |x(0)| ≤ φ, for φ > 0. If there exist symmetric positive definite matrices Qi ∈ R nx×nx , matrices Mi ∈ R nu×ny , X ∈ R ny ×ny , Zi ∈ R nw ×nw , for i ∈ ΩN , and positive scalars , γ satisfying the following optimization:

minimize γ 2
subject to ( 29), ( 30), [START_REF] Nguyen | Gain-scheduled static output feedback control for saturated LPV systems with bounded parameter variations[END_REF] and the following conditions:

Qi ≥ φ 2 I (39)   Qi (MiCy) (l) ū2 l CyQi -XCy -M i (X + X )   ≥ 0 (40) Qi H (m) Qi 1 ≥ 0 ( 41
)
where i ∈ ΩN , l ∈ Ωn u , m ∈ Ωq. Then, the SOF controller [START_REF] Furuichi | Switching dynamic modeling and stability analysis of three-wheeled narrow tilting vehicle[END_REF] with the control gains given in [START_REF] Agulhari | LMI relaxations for reduced-order robust H∞ control of continuous-time uncertain linear systems[END_REF] solves Problem 1. Moreover, the system constraints |u(t)| ≤ ū and x(t) ∈ Dx are enforced for ∀t ≥ 0.

Proof. Multiplying (39) by ηi(ω) ≥ 0 and summing up for all i ∈ ΩN , we obtain N i=1 ηi(ω)Qi ≥ φ 2 I, which implies that

x(0) Q(ω) -1 x(0) ≤ 1 φ 2 x(0) x(0) ≤ 1 (42)
for ∀x(0) ∈ R nx such that |x(0)| ≤ φ. This means that condition (39) guarantees V (x(0)) ≤ 1. Now, multiplying (40) by ηi(ω) ≥ 0 and summing up for all i ∈ ΩN yields

  Q(ω) (M (ω)Cy) (l) ū2 l CyQ(ω) -XCy -M (ω) (X + X )   ≥ 0 (43)
Pre-and postmultiplying (43) with I 0 0 0 I M (ω)X -1 and its transpose, we obtain (44) after some simple manipulations.

Q(ω) (K(ω)CyQ(ω)) (l) ū2 l ≥ 0 ⇔ Q(ω) -1 (K(ω)Cy) (l) ū2 l ≥ 0 (44)
By Schur complement lemma, this latter is equivalent to

x Q(ω) -1 x ≥ 1 ū2 l x (K(ω)Cy) (l) (K(ω)Cy) (l) x (45)
for ∀x ∈ R nx . Since (see also (42))

x(t) Q(ω) -1 x(t) ≤ x(0) Q(ω) -1 x(0) ≤ 1, ∀t ≥ 0 (46)
it follows from (45) that

x (K(ω)Cy) (l) (K(ω)Cy) (l) x ≤ ū2 l , l ∈ Ωn u , which means that |u(t)| ≤ ū, ∀t ≥ 0.
Applying Schur complement lemma to (41), then multiplying the result by ηi(ω) ≥ 0 and summing up for all i ∈ ΩN , we obtain Q(ω) -1 ≥ H (m) H (m) , for ∀m ∈ Ωq. Combining with (46), this latter condition implies clearly that

1 ≥ x Q(ω) -1 x ≥ x H (m) H (m) x, ∀m ∈ Ωq,
which guarantees that x ∈ Dx, see [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]Chapter 5]. The rest of the proof follows directly from the result of Theorem 1.

Remark 7. By a judicious introduction of the scalar and the slack variable X into the design conditions in Theorems 1 and 2, the complex couplings between Lyapunov matrices, control feedback gains and state-space matrices can be avoided. This enables an LMIbased formulation with a line search over a scalar for SOF control without explicitly requiring any matrix equality constraint and/or matrix rank condition as in most of existing works, see [START_REF] Sadabadi | From static output feedback to structured robust static output feedback: A survey[END_REF] for a recent survey. The proposed SOF control design belongs to the class of S-variable approach discussed in [START_REF] Ebihara | S-Variable Approach to LMI-Based Robust Control[END_REF]. Some other notable SOF control results for linear systems based on S-variable approach can be found in [START_REF] Nguyen | Gain-scheduled static output feedback control for saturated LPV systems with bounded parameter variations[END_REF], [START_REF] Agulhari | LMI relaxations for reduced-order robust H∞ control of continuous-time uncertain linear systems[END_REF], [START_REF] Dong | Robust static output feedback control synthesis for linear continuous systems with polytopic uncertainties[END_REF] and related references.

Remark 8. The design conditions in Theorems 1 and 2 are a set of LMIs with a line search over . The control gains Ki, i ∈ ΩN , can be easily computed with YALMIP toolbox and SDPT3 solver [START_REF] Löfberg | Yalmip: A toolbox for modeling and optimization in Matlab[END_REF] performing a line search for over a logarithmic scale in 10 -6 , 10 6 . Remark 9. For multiobjective control design, the proposed control approach offers a possibility to use V (x) defined in [START_REF] Pacejka | Tire and Vehicle Dynamics[END_REF] to guarantee the H2 control performance as in Theorem 1 and a new Lyapunov

function V (x) = x N i=1 ηi(ω) Qi -1
x, where Qi > 0, for ∀i ∈ ΩN , to enforce the constraints |u(t)| ≤ ū and x(t) ∈ Dx, for ∀t ≥ 0. Although it could help to reduce further the design conservatism, this may lead, however, to additional computational burden to compute the SOF controller [START_REF] Furuichi | Switching dynamic modeling and stability analysis of three-wheeled narrow tilting vehicle[END_REF] with the line search described above. As shown in the next section, Theorem 2 can provide an effective SOF controller for direct tilt control of the SMERA vehicle with a reasonable numerical complexity.

V. SIMULATION RESULTS AND DISCUSSIONS

The nonlinear model (1) of the SMERA vehicle is used to simulate the performance of the SOF controller [START_REF] Furuichi | Switching dynamic modeling and stability analysis of three-wheeled narrow tilting vehicle[END_REF] designed in Section IV. It is important to note that a validation study of model [START_REF] Chiou | Modeling and verification of a diamond-shape narrow-tilting vehicle[END_REF] with experimental data collected from the SMERA vehicle was carried out to identify the vehicle parameters. Moreover, due to industrial confidentiality reasons, all the result figures are slightly scaled so that the vehicle characteristics are not revealed. The following state constraints are considered for the control design:

|θ(t)| ≤ 0.35 [rad], |aper(t)| ≤ 1 [m/s 2 ] ( 47
)
Note that the first state constraint aims to prevent the vehicle rollover while cornering, and the second one is directly related to the comfort of passengers. It follows from [START_REF] Rajamani | Dynamics of narrow tilting vehicles[END_REF] that

aper(vx) = G(vx)xp + G2B δ δ,
where G(vx) = G1(vx) + G2Ap(vx). Using the variable change of scheduling parameter as described in Section III-D together with the sector nonlinearity approach [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] yields

aper(ω) = 2 i=1 ηi(ω)Gixp + G2B δ δ,
where G1 = G(ωmin) and G2 = G(ωmax

and γ = 27.3, with = 0.012. The computation time to perform the line search described in Remark 8 is about 197 seconds. Observe that the two control gains corresponding to LPV model [START_REF] Mohammadpour | Control of Linear Parameter Varying Systems with Applications[END_REF] are significantly different. This also justifies a posteriori the interest of using LPV controller [START_REF] Furuichi | Switching dynamic modeling and stability analysis of three-wheeled narrow tilting vehicle[END_REF] to improve the closed-loop performance since the vehicle speed is time-varying. Solving the same design conditions while imposing Q1 = Q2 = Q (i.e., quadratic approach), we obtain γ quad = 34.4 > γ, with quad = 0.003 and a line search computation time of about 124 seconds. Thus, the quadratic approach leads to more than 20% of degradation in terms of H2 performance for this application. This confirms the interest of considering the physical limitations of both vehicle speed and acceleration into the control design to reduce the conservatism, see Remark 6.

A. Scenario 1: Vehicle Circular Trajectory

This scenario represents a situation for which the SMERA vehicle takes a medium-sized roundabout with a constant speed vx = 6 [m/s]. As depicted in Fig. 5 (a), the vehicle starts turning at t = 2s and the steering angle δ gradually attains its absolute maximal value 0.22 [rad] at t = 8s. Then, the vehicle performs a circular trajectory with a constant radius (R ≈ 19 [m]), see Fig. 5 (b). Note that this test scenario is much challenging compared to those tested in [START_REF] Piyabongkarn | Active direct tilt control for stability enhancement of a narrow commuter vehicle[END_REF], [START_REF] Hibbard | Twenty first century transportation system solutions-a new type of small, relatively tall and narrow active tilting commuter vehicle[END_REF] with a constant radius about 500 [m]. Observe in Fig. 5 (c) that the perceived acceleration aper(t) is quite small in the transient phase. In particular, this acceleration is also perfectly regulated during the circular trajectory. Fig. 5 (d) shows that the tilt torque Mt is within the physical limitations of DTC actuator during the whole test scenario. Fig. 6 depicts the closed-loop behavior of the vehicle obtained with LPV controller given by Theorem 1. It is stressed that without any guarantee on the control input amplitude, the value of the designed input u(t) becomes excessively large during roundabout taking, which causes a saturation of DTC actuator as depicted in Figs. 6 (a) and 2 For real-time implementation, the hard constraint on the physical limitation of the tilt torque is guaranteed by the "Saturation" block in Simulink. (b), respectively. As a result, the perceived acceleration cannot be regulated in this case, see Fig. 6 (c). Moreover, the vehicle variables tend to be saturated with excessive amplitudes when the vehicle takes the roundabout as shown in Figs. 6 (c),(d), (e) and (f). It is important to note also that a similar closed-loop behavior of the SMERA vehicle is obtained when using the gain-scheduling controller proposed in [START_REF] Mourad | Direct and steering tilt robust control of narrow vehicles[END_REF]. This clearly demonstrates the interest of considering the DTC actuator limitations in the H2 control design. Furthermore, the gain scheduling technique used in [START_REF] Mourad | Direct and steering tilt robust control of narrow vehicles[END_REF] can lead to a tedious and timeconsuming design procedure without any rigorous guarantee of the closed-loop performance. These facts emphasize the contributions of the proposed control method compared to the existing literature. Mt remains within its saturation limits for both SOF controllers during the whole test as shown in Fig. 7 (d). Note that the considered test scenario is much more challenging than those used in other DTC control contexts of NTVs [START_REF] Mourad | Direct and steering tilt robust control of narrow vehicles[END_REF], [START_REF] Piyabongkarn | Active direct tilt control for stability enhancement of a narrow commuter vehicle[END_REF], [START_REF] Hibbard | Twenty first century transportation system solutions-a new type of small, relatively tall and narrow active tilting commuter vehicle[END_REF], [START_REF] Furuichi | Switching dynamic modeling and stability analysis of three-wheeled narrow tilting vehicle[END_REF], i.e., driving scenarios with low and/or constant vehicle speed and very large bends. It is important to note also that for this challenging scenario, the gain-scheduling controller proposed in [START_REF] Mourad | Direct and steering tilt robust control of narrow vehicles[END_REF] also provides an unstable closed-loop behavior, which is not shown here for brevity.

VI. CONCLUDING REMARKS

This paper proposes a new LPV control method for automatic DTC of NTVs. For real-time control implementation, only common sensors available on commercial vehicles are required and the robust SOF controller is of the simplest structure. The proposed DTC method is formulated as an LMI-based optimization in multiobjective setting. Based on the use of a PDLF, the information on both speed and acceleration limitations is incorporated into the design of robust DTC controllers to reduce the conservatism. The effectiveness of the proposed method is clearly demonstrated with realistic driving scenarios. Our future works focus on the shared control between a human driver and a STC system (see for instance [START_REF] Nguyen | Sensor reduction for driverautomation shared steering control via an adaptive authority allocation strategy[END_REF]) in the presence of DTC control actions. Moreover, experimental validations of the proposed DTC method in real-world driving conditions should be also investigated.

Fig. 1 .

 1 Fig. 1. Photo of the SMERA vehicle.

Fig. 2 .

 2 Fig. 2. Four degrees of freedom of SMERA: (a) top view; (b) rear view.

Fig. 3 .

 3 Fig. 3. Structure of the standard model Σ used for H 2 control design.

Fig. 4 .

 4 Fig. 4. Four-vertices polytope Pω of ω .

Fig. 5 .

 5 Fig. 5. Scenario 1: Closed-loop behavior of SMERA when actuator limitations are taken into account in the H 2 control. (a) steering angle; (b) vehicle trajectory; (c) perceived acceleration; (d) tilt torque.

Fig. Scenario 1 :

 1 Fig. Scenario 1: Closed-loop behavior of SMERA when actuator limitations are not taken into account in the H 2 control. (a) control input; (b) tilt torque; (c) perceived acceleration; (d) yaw rate; (e) tilt angle; (f) tilt rate.

Fig. 7 .

 7 Fig. 7. Scenario 2: Control performance comparison. (a) steering angle; (b) vehicle speed; (c) perceived acceleration; (d) tilt torque; (e) lateral speed; (f) yaw rate; (g) tilt angle; (h) tilt rate.

TABLE I

 I

		VEHICLE NOMENCLATURE
	Symbol	Description
	vx	Vehicle longitudinal speed
	vy	Vehicle lateral speed
	β	Sideslip angle
	ψ	Vehicle yaw angle
	θ	Vehicle tilt angle
	δ	Vehicle steering angle
	F lf /F lr	Front/rear longitudinal force
	F sf /Fsr	Front/rear lateral force
	l f /lr	Distance from c.g. G to front/rear wheels
	Iz/Ix	Tilting/yaw moment of inertia of vehicle
	C f /Cr	Front/rear cornering stiffness
	λ f /λr	Front/rear camber stiffness
	α f /αr	Front/rear tire sideslip angle
	g	Gravitational constant
	h	Height of c.g. G from ground
	m	Mass of vehicle
	Mt	Tilt torque from DTC actuator

  ). Now, the state constraints (47) can be easily put in the form (38) for control design. The control input constraint ū = 40 [N m] is the physical limitation of the DTC actuator2 . Solving the optimization problem in Theorem 2 leads to the following result:

	K1 = [3059.4 -46142 -5772.7 937.16 31480 5693.5]
	K2 = [3447.1 -38962 -3848.6 827.72 35565 3864.9]

The notation Σ(vx) is simply to make clear that the system dynamics depends explicitly on the time-varying parameter vx.