Angeladine Kenne Malaha

Clémence Thébaut

Dayna Achille

Pierre-Marie Preux

PhD Maëlenn Guerchet
email: maelenn.guerchet@ird.fr

COSTS OF DEMENTIA IN LOW-AND MIDDLE-INCOME COUNTRIES: A SYSTEMATIC REVIEW

Keywords: Dementia, Alzheimer's disease, Cost of illness, Disease Burden

Background

The proportion of people living with dementia in low-and middle-income countries (LMICs) is expected to reach 71% by 2050. Appraising the economic burden of the disease may contribute to strategic policy planning.

Objective

To review studies conducted on the costs of dementia in LMICs, describe their methodology and summarize available costs estimates.

Methods

Systematic review, including a search of health, economics and social science bibliographic databases. No date or language restrictions were applied. All studies with a direct measure of the costs of dementia care were included.

Results

Of the 6,843 publications reviewed, 17 studies from 11 LMICs were included. Costs of dementia tended to increase with the severity of the disease. Medical costs were greater in the mild stage, while social and informal care costs were highest in the moderate and severe stages. Annual cost estimates per patient ranged from PPP$131.0 to PPP$31,188.8 for medical costs; from PPP$16.1 to PPP$10,581.7 for social care services and from PPP$140.0 to PPP$25,798 for informal care. Overall, dementia care can cost from PPP$479.0 to PPP$66,143.6 per year for a single patient.

Conclusion

Few studies have been conducted on the costs of dementia in LMICs, and none so far in Africa. There seems to be a need to provide accurate data on the burden of disease in these countries to guide public health policies in the coming decades.

Systematic review; Low-Income; Middle-Income.

47

INTRODUCTION

An estimated 57.4 million people worldwide were living with dementia in 2019. This number is expected to increase to 152.8 million by 2050, mainly due to population growth and ageing [1]. According to the World Health Organization (WHO), much of this increase will be in low-and middle-income countries (LMICs) where 60% of people with dementia (PWD) already live. This figure could rise to 71% in 2050 [1,2].

In the latest Global status report on the public health response to dementia, the annual global cost of dementia has been estimated at US$ 1.3 trillion in 2019, up from US$818 billion in 2015. This figure is expected to reach US$1.7 trillion by 2050 [3,[START_REF] Wimo | Alzheimer's Disease International[END_REF]. In LMICs, these costs amount to about US$341.0 billion, divided into US$59.4 billion for direct medical costs, US$61.0 billion for direct social sector costs and US$220.6 billion for informal care. For high-income countries (HICs), these costs are US$972.3 billion, US$153.9 billion, US$387.7 billion and US$430.8 billion respectively [3]. Therefore, informal care costs account for half (49.6%) of the total cost of dementia worldwide, with 64.7% in LMICs and 44.3% in HICs, while evidence show that about 84% of PWD worldwide live with their families [3,[START_REF] Wimo | Global estimates of informal care, Alzheimer's Disease International[END_REF].

Cost-of-illness (COI) studies aim to assess the economic burden of health problems on the whole population [START_REF] Bloom | Cost of illness of Alzheimer's disease: how useful are current estimates[END_REF][START_REF] Larg | Cost-of-illness studies: a guide to critical evaluation[END_REF][START_REF] Céilleachair | Cost comparisons and methodological heterogeneity in costof-illness studies: the example of colorectal cancer[END_REF]. They are useful for determining the current status of a disease care system and its costs, as well as for assessing changes over time or between diseases, to assist in public health decision-making [START_REF] Céilleachair | Cost comparisons and methodological heterogeneity in costof-illness studies: the example of colorectal cancer[END_REF][START_REF] Bloom | Usefulness of US costof-illness studies in healthcare decision making[END_REF]. However, evaluating the costs of dementia remains a complex exercise. It requires distinguishing between costs categories, identifying costs attributable to the disease, managing different data sources, and conducting accurate analyses [START_REF] Kenigsberg | Impact socio-économique de la maladie d'Alzheimer et des maladies apparentées en Europe[END_REF][START_REF] Colucci | Alzheimer's disease costs: what we know and what we should take into account[END_REF]. A few studies have estimated the global costs of dementia [START_REF] Wimo | Alzheimer's Disease International[END_REF][START_REF] Wimo | Global estimates of informal care, Alzheimer's Disease International[END_REF][START_REF] Kenigsberg | Impact socio-économique de la maladie d'Alzheimer et des maladies apparentées en Europe[END_REF][START_REF] Colucci | Alzheimer's disease costs: what we know and what we should take into account[END_REF][START_REF] Wimo | World Alzheimer Report 2010: The Global Economic Impact of Dementia, Published by Alzheimer's Disease International (ADI)[END_REF]. The 2010 and 2015 reports on the economic impact of Alzheimer's disease [START_REF] Wimo | Alzheimer's Disease International[END_REF][START_REF] Wimo | World Alzheimer Report 2010: The Global Economic Impact of Dementia, Published by Alzheimer's Disease International (ADI)[END_REF] used the annual cost of dementia per person in each country, where available, computed with the projected number of PWD living in the country. Analyses also included imputations and a set of assumptions for countries with missing or insufficient data. For LMICs with very little primary data, authors reported using data from neighboring or comparable countries, gross domestic product, and health care expenditure per capita [START_REF] Wimo | Alzheimer's Disease International[END_REF][START_REF] Wimo | World Alzheimer Report 2010: The Global Economic Impact of Dementia, Published by Alzheimer's Disease International (ADI)[END_REF], which led to less reliable or accurate estimates for these regions.

As mentioned in the Alzheimer's Disease International report on dementia in sub-Saharan Africa, the lack of original cost data remains the weakest link in assessing the burden of dementia in the region [START_REF] Guerchet | Dementia in sub-Saharan Africa: Challenges and opportunities[END_REF]. Therefore, this review was conducted to identify costs of dementia studies conducted to date in LMICs with the aim of describing their methods, summarizing the available data and identifying the challenges of such studies in resource-limited settings.

MATERIALS AND METHODS

We conducted a systematic review on the costs of dementia in LMICs and the methodologies used for their evaluation. All original research articles from studies conducted in LMICs (according to World Bank classification on January 1 st 2020) and presenting a cost evaluation of any aspect of dementia care were considered. Disease cost categories are generally related to direct medical (i.e related to the medical care system), social care (i.e formal services provided outside of the medical care system) and informal care (related to the time spent by non-paid relatives on care) [START_REF] Wimo | World Alzheimer Report 2010: The Global Economic Impact of Dementia, Published by Alzheimer's Disease International (ADI)[END_REF].

Search strategy

Bibliographic databases on health, health economics and social sciences were searched: Pubmed/Medline, Embase, Global health, PsycINFO, PsycARTICLES (via OVID), Web of science and Econlit. The main grey literature resources in the health field were also searched (Greylit, ProQuest, Georgetown University resource and Google scholar). No language or date restrictions were applied. Search strategies consisted of combinations of different keywords related to the disease (dementia, Alzheimer's, cognitive disorder / impairment), costs (cost, valuation, expenses, and burden) and settings (low-income, middle-income, resource limited, developing country and, the list of the 137 LMICs countries (according to the World Bank on January 1 st , 2020 [START_REF]Low & middle income countries | Data[END_REF])). Specific equations for each database as searched on March 2 nd 2020 are presented in Supplementary material 1. An updated search using the same strategy was conducted on December 28 th 2021 to identify potential new studies of interest.

Studies selection

Results obtained from the searches on the different bibliographic databases were imported and processed using the Rayyan QCRI tool [START_REF] Ouzzani | Rayyan-a web and mobile app for systematic reviews[END_REF]. In a first step, duplicates were removed. Then, following a blind selection method, two investigators (AKM and DA) assessed the eligibility of the selected articles based on titles and abstracts. The proportion of agreement between the two investigators was measured; conflicting articles at this stage were all retained for the next stage. Full texts of selected references were retrieved, and the corresponding authors were contacted to request the manuscript when those were not available. Full-texts were then evaluated by the same two investigators (AKM and DA) for the final inclusion decision. Conflicting articles were discussed and the inclusion decision was made with a third investigator (MG or CT).

Protocols, reviews and meta-analyses were not included, but were examined to identify any relevant study for consideration. In addition, references of selected articles were checked. For publications in languages other than English and French (those in Spanish, Portuguese and Chinese), collaborators were asked to provide a translation and/or description of the studies in order to make a decision on eligibility and to extract the data of interest.

During the selection process, articles were excluded according to the Population, Intervention, Comparator and Outcome (PICO) framework. Laboratory experiments (i.e fundamental research on Alzheimer's Disease / dementia, including in-vitro studies and studies in non-human populations), studies not addressing dementia care, and studies conducted outside LMICs were considered as wrong population. Wrong publication types included narrative review, meta-analysis, overview article, letter to the editor and commentaries. Studies without a direct measure of dementia cost were accounted as wrong outcomes. Studies for which the full-text was not retrieved despite multiple searches and contact with the authors were also excluded.

Quality assessment

Although various guidelines have been published for assessing the quality of economic evaluation in health care [START_REF] Drummond | Methods for the Economic Evaluation of Health Care Programmes[END_REF][START_REF] Husereau | CHEERS Task Force (2013) Consolidated Health Economic Evaluation Reporting Standards (CHEERS) statement[END_REF], they mainly focus on cost-effectiveness analyses. In the absence of a standard instrument to evaluate cost-of-illness studies, we assessed the quality of articles using a checklist, derived from the guide proposed by Larg and Moss [START_REF] Larg | Cost-of-illness studies: a guide to critical evaluation[END_REF] and the checklist developed by Kline-Budde et al. [START_REF] Kleine-Budde | Cost of illness for bipolar disorder: a systematic review of the economic burden[END_REF] for their review on bipolar disorders. Full description of the quality assessment criteria used is provided in Supplementary material 2.

Data processing and analysis

Full-texts were reviewed and data extracted using a standardized extraction grid developed in Epi Info 7 and Microsoft Excel®. Data were collected on study design, population characteristics and costing methods. Particular attention was paid to the costing perspective (i.e the point of view from which costs are considered). Studies may measure costs to a health care system, a third-party payers, commercial sectors, government or participants and their families [START_REF] Jo | Cost-of-illness studies: concepts, scopes, and methods[END_REF]. In this review, the perspective was collected as specified in the original article. Instruments used to collect and valuate health resource utilization were also explored, to check the use of standard validated tools such as the Resource Utilization in Dementia (RUD) instrument [START_REF] Wimo | Evaluation of the resource utilization and caregiver time in Anti-dementia drug trialsa quantitative battery[END_REF] or questionnaires specifically designed by researchers for the purpose of their study.

Costing approaches used to value informal care time were also assessed. Various methods for the valuation of informal care are available in the literature. On the one hand, there are revealed preference methods that are based on real-life decision data:

1) the opportunity cost method, which aims to estimate the cost of the opportunity to earn income or partake in leisure activities that an individual forgoes when engaging in informal care; 2) the replacement cost, or proxy good method, which assigns costs based on the market value that the care services would cost if they were provided by a professional provider. On the other hand, stated preference studies obtain value by asking respondents directly what economic value they attach to informal care through contingent valuation or conjoint analysis [START_REF] Koopmanschap | An Overview of Methods and Applications to Value Informal Care in Economic Evaluations of Healthcare[END_REF][START_REF] Oliva-Moreno | The Valuation of Informal Care in Cost-of-Illness Studies: A Systematic Review[END_REF].

When information was not presented in the original article as required for extraction, available data were converted to the accurate value and unit. For example, all sex ratios and gender figures were converted to male percentage. When data was not available or presented in a non-transformable format, it was considered missing. When summarizing data, weighting was done using the representativeness of each category in the total sample. For example, to calculate the total medical costs of a given study presented by disease severity, the proportion of participants in each stage was taken into account. Monetary data were converted into local currency unit (LCU) using the exchange rate provided by the authors. Where these were not available, we used exchange rates provided by the World Bank for the country in the year of the study cost [START_REF]Official exchange rate (LCU per US$, period average) | Data[END_REF]. Costs were then updated to their 2020 value using country-specific deflators provided by the World Bank [START_REF]Inflation, GDP deflator (annual %) | Data[END_REF], and converted to 2020 dollar purchasing power parities (PPP) using official conversion factors [START_REF]PPP conversion factor, GDP[END_REF].

All costs were collected under the original items as provided by studies. They were then grouped into a dozen of health resource services and finally into the three main cost-of-illness categories (i.e. direct medical care, social care and informal care). Some studies presented costs results directly in the three main categories which does not allow for a desegregated presentation; these are presented only in total costs descriptions.

The protocol of this systematic review was registered in PROSPERO (registration number CRD42020169898) before data extraction began.

RESULTS

From the initial search, 11,168 publications were identified and extracted from the databases. Details are presented in Supplementary material 3. Figure 1 shows the articles selection flow chart according to updated PRISMA guidelines [START_REF] Page | The PRISMA 2020 statement: an updated guideline for reporting systematic reviews[END_REF]. After eliminating duplicates, 6,843 publications were assessed for eligibility. Of the 91 eligible articles, 74 full-text were retrieved and evaluated for final inclusion. Of these, 56 were excluded for the following reasons: not a cost-of-illness study (n = 10); not an original research (n = 13); modelling studies (n= 8) and study conducted in high-income countries (n = 25). The latest updated search identified two recent studies that were included in the analyses.

Finally, 20 articles representing 17 individual studies were selected for this review.

Butman et al. [START_REF] Butman | Impacto económico de la enfermedad de Alzheimer: Resultados preliminares en Buenos Aires[END_REF] presented preliminary results of a study subsequently published by Allegri et al. [START_REF] Allegri | Economic impact of dementia in developing countries: an evaluation of costs of Alzheimer-type dementia in Argentina[END_REF] ; similarly, the pilot study conducted by Ferretti et al. [START_REF] Ferretti | Indirect cost with dementia: A Brazilian study[END_REF] in Brazil was completed and final results published by the same authors [START_REF] Ferretti | An assessment of direct and indirect costs of dementia in Brazil[END_REF]. In both cases, only the final publications [START_REF] Allegri | Economic impact of dementia in developing countries: an evaluation of costs of Alzheimer-type dementia in Argentina[END_REF][START_REF] Ferretti | An assessment of direct and indirect costs of dementia in Brazil[END_REF] were considered for the analyses. Jia et al. [START_REF] Jia | The cost of Alzheimer's disease in China and re-estimation of costs worldwide[END_REF] and Yan et al. [START_REF] Yan | Associated Factors of Total Costs of Alzheimer's Disease: A Cluster-Randomized Observational Study in China[END_REF] presented different costs for the same cluster-randomized observational study in China, so their characteristics and results were merged.

Characteristics of included studies

Characteristics of the 17 included studies are summarized in Table 1. Out of 137 lowand middle-income countries, only 11 were represented in this review, including six studies in China [START_REF] Jia | The cost of Alzheimer's disease in China and re-estimation of costs worldwide[END_REF][START_REF] Yan | Associated Factors of Total Costs of Alzheimer's Disease: A Cluster-Randomized Observational Study in China[END_REF][START_REF] Mould-Quevedo | The burden of caring for dementia patients: caregiver reports from a crosssectional hospital-based study in China[END_REF][START_REF] Wang | Economic impact of dementia in developing countries: an evaluation of Alzheimer-type dementia in Shanghai, China[END_REF][START_REF] Wang | Caregiver Time and Cost of Home Care for Alzheimer's Disease: A Clinic-based Observational Study in Beijing, China[END_REF][START_REF] Zhang | Costs of Hospitalization for Dementia in Urban China: Estimates from Two Urban Health Insurance Scheme Claims Data in Guangzhou City[END_REF][START_REF] Xiaojuan | Investigation on the economic burden and its influencing factors of Alzheimer's disease in Xi'an[END_REF]; 2 in Brazil [START_REF] Ferretti | An assessment of direct and indirect costs of dementia in Brazil[END_REF][START_REF] Soares | Economic impact and prevalence of Alzheimer's Disease in a Brazilian capital[END_REF]; and one in each of the following countries: Argentina [START_REF] Allegri | Economic impact of dementia in developing countries: an evaluation of costs of Alzheimer-type dementia in Argentina[END_REF], Colombia [START_REF] Prada | Actual expense associated with patients with Alzheimer's disease in Colombia[END_REF], India [START_REF] Malapur | Cost of Illness of Major Neurocognitive Disorders in India[END_REF], Iran [START_REF] Aajami | Direct and indirect cost of managing Alzheimer's disease in the Islamic Republic of Iran[END_REF], Peru [START_REF] Custodio | Cost-of-illness study in a retrospective cohort of patients with dementia in Lima, Peru[END_REF], Philippines [START_REF] Dominguez | Dementia Incidence, Burden and Cost of Care: A Filipino Community-Based Study[END_REF], Romania [START_REF] Onetiu | Cost of dementia in Romania: a cross-sectional Cost-of-Illness study undertaken in Bucharest[END_REF], Thailand [START_REF] Kongpakwattana | A Real-World Evidence Analysis of Associations Among Costs, Quality of Life, and Disease-Severity Indicators of Alzheimer's Disease in Thailand[END_REF] and Turkey [START_REF] Zencir | Cost of Alzheimer's disease in a developing country setting[END_REF]. Our dataset covered all epidemiological regions of the Global Burden of Disease, with the exception of Africa. Regional coverage was as follow: 1 from Eastern Mediterranean [START_REF] Aajami | Direct and indirect cost of managing Alzheimer's disease in the Islamic Republic of Iran[END_REF]; 2 from South-East Asia [START_REF] Malapur | Cost of Illness of Major Neurocognitive Disorders in India[END_REF][START_REF] Kongpakwattana | A Real-World Evidence Analysis of Associations Among Costs, Quality of Life, and Disease-Severity Indicators of Alzheimer's Disease in Thailand[END_REF]; 2 from Europe [START_REF] Onetiu | Cost of dementia in Romania: a cross-sectional Cost-of-Illness study undertaken in Bucharest[END_REF][START_REF] Zencir | Cost of Alzheimer's disease in a developing country setting[END_REF]; 5 from the Americas [START_REF] Allegri | Economic impact of dementia in developing countries: an evaluation of costs of Alzheimer-type dementia in Argentina[END_REF][START_REF] Ferretti | An assessment of direct and indirect costs of dementia in Brazil[END_REF][START_REF] Soares | Economic impact and prevalence of Alzheimer's Disease in a Brazilian capital[END_REF][START_REF] Prada | Actual expense associated with patients with Alzheimer's disease in Colombia[END_REF][START_REF] Custodio | Cost-of-illness study in a retrospective cohort of patients with dementia in Lima, Peru[END_REF] and 7 from Western Pacific [START_REF] Jia | The cost of Alzheimer's disease in China and re-estimation of costs worldwide[END_REF][START_REF] Yan | Associated Factors of Total Costs of Alzheimer's Disease: A Cluster-Randomized Observational Study in China[END_REF][START_REF] Mould-Quevedo | The burden of caring for dementia patients: caregiver reports from a crosssectional hospital-based study in China[END_REF][START_REF] Wang | Economic impact of dementia in developing countries: an evaluation of Alzheimer-type dementia in Shanghai, China[END_REF][START_REF] Wang | Caregiver Time and Cost of Home Care for Alzheimer's Disease: A Clinic-based Observational Study in Beijing, China[END_REF][START_REF] Zhang | Costs of Hospitalization for Dementia in Urban China: Estimates from Two Urban Health Insurance Scheme Claims Data in Guangzhou City[END_REF][START_REF] Xiaojuan | Investigation on the economic burden and its influencing factors of Alzheimer's disease in Xi'an[END_REF][START_REF] Dominguez | Dementia Incidence, Burden and Cost of Care: A Filipino Community-Based Study[END_REF]. Studies were conducted between 2003 and 2020. All studies performed a retrospective collection of data in the previous 1-12 months, except studies from Turkey [START_REF] Zencir | Cost of Alzheimer's disease in a developing country setting[END_REF] and Philippines [START_REF] Dominguez | Dementia Incidence, Burden and Cost of Care: A Filipino Community-Based Study[END_REF], which added a 15-day and 5-year prospective data collection pe riod on time spent in informal care, respectively.

The majority of studies focused on Alzheimer's Disease only (12/20), 5 studies included all types of dementia and the remaining four did not specify dementia subtypes in their study population. In almost all studies, case identification was based on a previous dementia diagnosis. A few updated or verified the diagnosis before inclusion in the studies. Disease definition followed standard criteria including ICD-10, DSM-IV-TR and/or NINCDS -ADRDA, plus various clinical or imaging investigations. Some studies excluded patients with mild cognitive impairment [START_REF] Ferretti | Indirect cost with dementia: A Brazilian study[END_REF][START_REF] Ferretti | An assessment of direct and indirect costs of dementia in Brazil[END_REF], early onset dementia [START_REF] Aajami | Direct and indirect cost of managing Alzheimer's disease in the Islamic Republic of Iran[END_REF] or limited functional independence [START_REF] Custodio | Cost-of-illness study in a retrospective cohort of patients with dementia in Lima, Peru[END_REF]. Only four studies reported the mean disease duration: 2.7 years, 51.0 months, 60.1 months and 5.1 years in China [START_REF] Wang | Economic impact of dementia in developing countries: an evaluation of Alzheimer-type dementia in Shanghai, China[END_REF], Romania [START_REF] Onetiu | Cost of dementia in Romania: a cross-sectional Cost-of-Illness study undertaken in Bucharest[END_REF], Brazil [START_REF] Ferretti | An assessment of direct and indirect costs of dementia in Brazil[END_REF] and Thailand [START_REF] Kongpakwattana | A Real-World Evidence Analysis of Associations Among Costs, Quality of Life, and Disease-Severity Indicators of Alzheimer's Disease in Thailand[END_REF] respectively.

The majority of studies examined all three categories of costs (direct medical, social and informal care costs). Three studies [START_REF] Zhang | Costs of Hospitalization for Dementia in Urban China: Estimates from Two Urban Health Insurance Scheme Claims Data in Guangzhou City[END_REF][START_REF] Soares | Economic impact and prevalence of Alzheimer's Disease in a Brazilian capital[END_REF][START_REF] Prada | Actual expense associated with patients with Alzheimer's disease in Colombia[END_REF] investigated only medical costs, while Wang et al. [START_REF] Wang | Caregiver Time and Cost of Home Care for Alzheimer's Disease: A Clinic-based Observational Study in Beijing, China[END_REF] only looked at informal care costs. Zencir et al. [START_REF] Zencir | Cost of Alzheimer's disease in a developing country setting[END_REF] and Mould-Quevedo et al. [START_REF] Mould-Quevedo | The burden of caring for dementia patients: caregiver reports from a crosssectional hospital-based study in China[END_REF] did not collect data on social and informal care costs respectively.

Items included in each cost category varied widely. For direct medical costs, most studies included outpatient, inpatient and medication costs. Some reported only medication [START_REF] Ferretti | An assessment of direct and indirect costs of dementia in Brazil[END_REF][START_REF] Soares | Economic impact and prevalence of Alzheimer's Disease in a Brazilian capital[END_REF][START_REF] Prada | Actual expense associated with patients with Alzheimer's disease in Colombia[END_REF] or hospitalization [START_REF] Zhang | Costs of Hospitalization for Dementia in Urban China: Estimates from Two Urban Health Insurance Scheme Claims Data in Guangzhou City[END_REF] costs. Regarding medication costs for instance, Soares et al. [START_REF] Soares | Economic impact and prevalence of Alzheimer's Disease in a Brazilian capital[END_REF] in Brazil reported the costs of dementia-specific drugs dispensed by a government dispensing center. Meanwhile in Colombia, Prada et al.

[39] considered direct insurance reimbursement for all-condition drugs of dementia patients. In one study from China, Wang et al. [START_REF] Wang | Economic impact of dementia in developing countries: an evaluation of Alzheimer-type dementia in Shanghai, China[END_REF] included the costs of official dementia drugs, but also those of traditional medicine. Social care costs included travel expenses and formal care services where appropriate, but also food expenses [START_REF] Jia | The cost of Alzheimer's disease in China and re-estimation of costs worldwide[END_REF][START_REF] Aajami | Direct and indirect cost of managing Alzheimer's disease in the Islamic Republic of Iran[END_REF].

With regard to informal care costs, most studies reported estimates of time spent on care, without much detail on the activities considered. Two studies considered the patient loss-of-productivity [START_REF] Ferretti | An assessment of direct and indirect costs of dementia in Brazil[END_REF][START_REF] Onetiu | Cost of dementia in Romania: a cross-sectional Cost-of-Illness study undertaken in Bucharest[END_REF] and one included the costs of health services for the treatment of the caregiver [START_REF] Ferretti | An assessment of direct and indirect costs of dementia in Brazil[END_REF]. Although the majority of studies valuated caregiver time by the replacement cost methods (which assigns a monetary value to caregiver's time based on what it would cost to replace informal care activities with formal care), the proxy used varied considerably, ranging from national minimum or average wage, to the salary of a nurse in a public institution, or to the caregivers' salary. Table 2 provides more details on the design characteristics of the included studies.

Results from the quality assessment of the studies are presented in Table 3. Almost all studies provided basic information on the study design and population characteristics, such as objectives, sample size and demographics. Eight out of the 17 studies clearly indicated the cost estimate perspective. Only Allegri et al. [START_REF] Allegri | Economic impact of dementia in developing countries: an evaluation of costs of Alzheimer-type dementia in Argentina[END_REF] and Custodio et al. [START_REF] Custodio | Cost-of-illness study in a retrospective cohort of patients with dementia in Lima, Peru[END_REF] included a group of healthy participants in their sample, although they did not use this information to calculate disease-specific net costs. Only the cluster-randomized study in China presented the results of a sensitivity analysis [START_REF] Jia | The cost of Alzheimer's disease in China and re-estimation of costs worldwide[END_REF][START_REF] Yan | Associated Factors of Total Costs of Alzheimer's Disease: A Cluster-Randomized Observational Study in China[END_REF]. Most studies (9/17) failed to present separate information on the number of services used before getting in their valuation. Very few studies performed sensitivity analyses to account for uncertainties in key indicators. Onetiu et al. [START_REF] Onetiu | Cost of dementia in Romania: a cross-sectional Cost-of-Illness study undertaken in Bucharest[END_REF] for instance estimated the cost of time spent on informal care by a replacement cost approach using the national minimum wage, the average national wage and the caregiver's salary. Finally, all studies presented and discussed methodological limitations.

Costs estimates

Annual costs of dementia per patient according to disease severity are summarized in Table 4. Annual domestic government health expenditures in 2020 for each country are also shown in this table. Overall, in all costs categories, dementia costs tend to increase with disease severity. Prada et al. [START_REF] Prada | Actual expense associated with patients with Alzheimer's disease in Colombia[END_REF] calculated the ratio of direct insurer expense to mild stage at 1.62 for moderate and 4.87 for severe stages; i.e. compared to a mild patient, a moderate stage patient will use 62% more resources and a severe stage patient 387% more resources. Medical costs are generally highest in the mild stage, while social costs (including institution fees) and informal care costs become more important in the moderate and severe stages. Regardless of the disease stage, we found a wide range of estimates of annual per patient costs in the studies. Medical costs ranged from PPP$131.0 in India [START_REF] Malapur | Cost of Illness of Major Neurocognitive Disorders in India[END_REF] to PPP$31,188.8 in China [START_REF] Kongpakwattana | A Real-World Evidence Analysis of Associations Among Costs, Quality of Life, and Disease-Severity Indicators of Alzheimer's Disease in Thailand[END_REF]; social costs from PPP$16.1 in Philippines [START_REF] Dominguez | Dementia Incidence, Burden and Cost of Care: A Filipino Community-Based Study[END_REF] to PPP$10,581.7 in China [START_REF] Jia | The cost of Alzheimer's disease in China and re-estimation of costs worldwide[END_REF][START_REF] Yan | Associated Factors of Total Costs of Alzheimer's Disease: A Cluster-Randomized Observational Study in China[END_REF]; and informal care costs from PPP$140.0 in India [START_REF] Malapur | Cost of Illness of Major Neurocognitive Disorders in India[END_REF] to PPP$25,798 in China [START_REF] Kongpakwattana | A Real-World Evidence Analysis of Associations Among Costs, Quality of Life, and Disease-Severity Indicators of Alzheimer's Disease in Thailand[END_REF]. Overall, dementia care can cost from PPP$479.0 in India [START_REF] Malapur | Cost of Illness of Major Neurocognitive Disorders in India[END_REF] to PPP$66,143.6 in China [START_REF] Jia | The cost of Alzheimer's disease in China and re-estimation of costs worldwide[END_REF][START_REF] Yan | Associated Factors of Total Costs of Alzheimer's Disease: A Cluster-Randomized Observational Study in China[END_REF] per year for a single patient.

DISCUSSION

Findings

Studies included in this systematic review comprised 17 cost-of-illness studies on dementia in 11 LMICs across all GDB regions except Africa. These studies were conducted from 2003 to 2020 and focused mostly on Alzheimer's disease. All estimates of dementia costs increase with disease severity and informal care costs tend to exceed medical costs.

Globally, very few studies have been conducted to assess dementia costs worldwide. About 10 years ago, Quentin et al. [START_REF] Quentin | Cost-ofillness studies of dementia: a systematic review focusing on stage dependency of costs[END_REF] reviewed cost-of-illness (COI) studies in Europe and North America, including 28 studies. A review published in 2012 by Knapp et al. [START_REF] Knapp | Dementia care costs and outcomes: a systematic review[END_REF] on the economic evaluation of dementia care identified only 29 individual studies.

Similarly, the international review of COI studies on dementia, by Schaller et al. in 2014 [START_REF] Schaller | The main cost drivers in dementia: a systematic review[END_REF] found 27 individual studies for the period 2003 to 2012. Our review focusing on LMICs included 17 studies. As reported by Hendriks et al. [START_REF] Hendriks | Step-by-step guideline for disease-specific costing studies in low-and middle-income countries: a mixed methodology[END_REF], beyond the complexity of COI studies, the overriding constraint in LMICs is data availability, including lack of accurate financial records and incomplete patient disease registers, as well as lack of expertise to conduct cost studies.

Costs of dementia are high and highly variable. They range from PPP$479.0 to PPP$66,143.6 per year for a single patient. These costs are colossal as families remain the main bearers of the burden. Indeed, according to the WHO, in 2016, half of the world's countries spent less than US$350 per person annually on health-related expenses, including most LMICs [START_REF] Xu | Public Spending on Health: A Closer Look at Global Trends[END_REF]. In 2018, the mean per capita general government domestic health expenditure was US$134.7 in LMICs and US$5,562.3 in HICs [START_REF]Domestic general government health expenditure per capita[END_REF]. In this study, the per-patient costs of dementia reached fifty times these public expenditures.

Dementia costs increase with disease severity; similar results have been reported in other studies and can be explained by the rise in care needs as dependence and impairment increase [START_REF] Quentin | Cost-ofillness studies of dementia: a systematic review focusing on stage dependency of costs[END_REF][START_REF] Jönsson | The cost of dementia in Europe: a review of the evidence, and methodological considerations[END_REF][START_REF] Luppa | Direct costs associated with mild cognitive impairment in primary care[END_REF]. Evidence from the United States suggests that informal care costs decrease with disease severity as patients need more attention and are therefore institutionalized [START_REF] Yaffe | Patient and caregiver characteristics and nursing home placement in patients with dementia[END_REF]. This was not observed in this review, where informal care costs remained the highest in most studies. Actually, in LMICs, about 95% of patients with dementia live with their families, compared to 69% in HICs [START_REF] Wimo | Global estimates of informal care, Alzheimer's Disease International[END_REF].

Thus, in these countries where universal health coverage is not achieved, families bear the burden [START_REF] Ferri | Dementia in low-income and middle-income countries: Different realities mandate tailored solutions[END_REF].

Limitations

The majority of included studies reported convenience sampling and/or small sample size as a study limitation in their publications. Unfortunately, these two limitations restrict further use of the results as they are difficult to generalize. Half of the included studies (9/17) focused on Alzheimer's Disease only, as this is the main pathology of dementia. It should be noted that, the disease considered for cost evaluation and its definition criteria strongly influence costs estimates. In fact, Custodio et al. reported that fronto-temporal dementia costs were higher than Alzheimer's Disease or vascular dementia ones [START_REF] Custodio | Cost-of-illness study in a retrospective cohort of patients with dementia in Lima, Peru[END_REF].

Considering there is no standard methodology for cost-of-illness studies and no reference tool for quality assessment, available checklists [START_REF] Larg | Cost-of-illness studies: a guide to critical evaluation[END_REF][START_REF] Kleine-Budde | Cost of illness for bipolar disorder: a systematic review of the economic burden[END_REF] were adapted to assess the studies included in this review, which appeared to be of satisfactory quality. However, very few studies included a detailed description of service utilization and a clear definition of items considered in each cost category. The way in which services are defined and valuated is of crucial value in understanding and comparing the resulting estimates, particularly in the context of LMICs where health care and health financing systems vary. The lack of methodological standard in COI studies is a common and challenging issue to address, as described in other reviews on dementia [START_REF] Bloom | Cost of illness of Alzheimer's disease: how useful are current estimates[END_REF][START_REF] Quentin | Cost-ofillness studies of dementia: a systematic review focusing on stage dependency of costs[END_REF][START_REF] Jönsson | The cost of dementia in Europe: a review of the evidence, and methodological considerations[END_REF]. It has also been described in other diseases, including colorectal cancer [START_REF] Céilleachair | Cost comparisons and methodological heterogeneity in costof-illness studies: the example of colorectal cancer[END_REF] and diabetes [START_REF] Pagano | Costs of diabetes. A methodological analysis of the literature[END_REF].

Collection and valuation of time spent by caregivers on care is rather difficult. First, many activities and services (activities of daily living and instrumental activities of daily living) are done for the older people regardless of their health status and are not considered by caregivers as an additional workload, even when the person can no longer perform them themselves. This is reinforced by the fact that many people still consider dementia to be a normal part of ageing [START_REF] Cations | What does the general public understand about prevention and treatment of dementia? A systematic review of population-based surveys[END_REF]. Secondly, a joint effect should be considered, that of achieving many activities simultaneously, such as shopping for the PWD and caregiver themselves or supervising the PWD while cooking for the entire household. According to monetary valuation, a good number of methods exist, with the difficulties of disaggregating time spent (paid work time, domestic work and leisure time) and defining the unit cost to be applied to each category [START_REF] Oliva-Moreno | The Valuation of Informal Care in Cost-of-Illness Studies: A Systematic Review[END_REF][START_REF] Oliva-Moreno | The economic value of time of informal care and its determinants (The CUIDARSE Study)[END_REF]. In the included studies, both opportunity cost and proxy good approaches were used, but were not sufficiently robust and/or their methods not well described. The quality of diseasespecific costs estimates strongly rely on the accuracy of data collected and therefore of the collection tools. Therefore, harmonizing the costs items definition and collection is strongly recommended to ensure accuracy and comparability with other studies.

Standard collection tools exist, including the Resource Utilization in Dementia (RUD)

Instrument [START_REF] Wimo | Evaluation of the resource utilization and caregiver time in Anti-dementia drug trialsa quantitative battery[END_REF]. This is a dementia-specific tool that has been widely used and validated in global settings [START_REF] Wimo | cognitive impairment, cost of illness, cost of care, low and middle income countries List of all 137 countries considered as LMICs[END_REF]. Otherwise, a minimum set of items should be collected, such as outpatient, inpatient and medication for medical costs, transport and accommodation for social costs. Attention should be paid to informal care time too, specifying which ADL, IADL or supervision activities were carried out, and how caregiving time was valued in money.

In addition, it was not always clearly stated in all studies which perspective was considered for the costs or whether these were dementia-specific or total patient costs. Key distinctions that may affect the interpretation and use of costs. The majority of studies presented estimates directly in US$ or local currency. It was therefore necessary to know the precise cost reference date, exchange rate, country-specific inflators and purchasing power parities to convert all estimates into international dollars. This process allowed relevant comparisons of dementia costs between the studies included in this review. However, it limits comparisons with studies from other settings or global estimates which usually present the results in US dollars. Due to the wide heterogeneity of study designs, methodological and costing-related limitations cited above, disease costs summaries provided in this review should be taken with caution.

In this review, modelling studies were excluded. Modelling is a good way to appraise the reality without investing huge resources in complex field surveys. However, the likelihood of a model relies on the proxies used, for example, utilization of health services, market price of health services or reference wage for workers. These data are often neither standardized in resource-limited settings nor available [START_REF] Hendriks | Step-by-step guideline for disease-specific costing studies in low-and middle-income countries: a mixed methodology[END_REF]. As a result, many modelling studies may not provide an accurate picture of reality. Thus, in order to identify accurate methods for calculating dementia costs in LMICs, and to summarize primary costs estimates, we considered only studies with field collection of health services use and costs. Similarly, studies with a top-down cost approach were excluded as they did not include a direct measurement of dementia costs. Although top-down studies are highly valuable, they mainly capture disease-specific medical costs from the state or third party payer perspective, with very little information on social and informal care costs. They also require a number of indicators like the number of dementia cases or heath resource utilization, which may not be available in the countries of interest, leading to the use of additional proxies and imputations.

Our search strategy included the most plausible sources of publications for the topic.

The high rate of duplicates bears out the exhaustiveness of this review. In addition, no language or date restrictions were applied to the searches or results. Moreover, all studies that included cost estimation of any component of dementia care were included and all the costs estimates were converted from US dollars or local currency to purchasing power parity dollars to allow for cost comparisons between studies.

CONCLUSION

To the best of our knowledge, this is among the first reviews to focus on the cost of dementia in LMICs. It is valuable regarding appraisal of methodological constraints when evaluating the costs of dementia in resource-limited settings. Findings highlighted once more the scarcity of valuable cost-of-illness studies in these countries, particularly in Africa where original research for primary data collection is essential.

The costs of dementia were found to be high, placing an enormous burden on families.

These results should serve to raise the alarm on the urgency of dementia and to advocate for dementia-oriented programs and policies.

ACKNOWLEDGEMENTS

Figure 1 : 8 Studies

 18 Figure 1: Articles selection flow diagram

Table 1 :

 1 General characteristics of included studiesIf data were collected from or estimated for a period longer than one year, costs were adjusted for differential timing and the inflation rate/discount rate was mentioned : Larg A, Moss JR (2011) Cost-of-illness studies: a guide to critical evaluation. PharmacoEconomics 29, 653-671. & Kleine-Budde K, Touil E, Moock J, Bramesfeld A, Kawohl W, Rössler W (2014) Cost of illness for bipolar disorder: a systematic review of the economic burden. Bipolar Disord 16, 337-353.

	Study	Country	Study period	Settings	Dementia subtype	Sample size	Mean age (years)	Male percentage	Disease severity criteria	Disease severity stages (%) Mild Moderate Severe
	Aajami et al., [41]	Iran	2017	Urban only	Alzheimer's disease	300	80.0	48.0	MMSE	33.3	33.3	33.3
	Allegri et al., [28]	Argentina NS	Urban only	Alzheimer's disease	125	74.6	19.2	MMSE	48.0	30.0	22
	Custodio et al., [42]	Peru	2012 -2013	Urban only	All types of dementia	106	70.0	39.7	None		NA
	Dominguez et al., [43]	Philippines 2015-2016	Urban & rural	All types of dementia	81	73.8	25.9	None		NA
	Ferretti et al., [30]	Brazil	2011 -2015	Urban only	Dementia not specified	156	72.9	41.7	FAST Scale	29.1	47.5	13.4
	Kongpakwattana et al., [45]	Thailand	2017 -2018	Urban only	Alzheimer's disease	148	80.1	29.1	MMSE	23.6	39.9	36.5
	Malapur et al., [40]	India	NS	Urban & rural	Major Neurocognitive Disorders	50	68.6	60.0	CDR	46		54
	Mould-Quevedo et al., [33] China	2009	Urban only	Dementia not specified	1,387	67.6	52.0	None		NA
	Onetiu et al., [44]	Romania	2013 -2014	Urban only	Dementia not specified	31	NS	NS	NS	29.0	45.2	25.8
	Prada et al., [39]	Colombia 2011 -2013	Urban & rural	All types of dementia	340	NS	43.0	Drug dispensed	36.1**	44.5**	12.9**
	Soares et al., [38]	Brazil	2014	Urban only	Alzheimer's disease	855	78.7	30.6	None		NA
	Wang et al., [34]	China	2005-2006	Urban only	Alzheimer's disease	66	74.0	34.8	MMSE	19.7	56.1	24.2
	Wang et al., [35]	China	2008 -2009	Urban only	Alzheimer's disease	79	NS	21.5	MMSE	25.4	60.6	14.0
	XiaoJuan et al., [37]	China	2016 -2018	Urban & rural	Alzheimer's disease	119	70.8	38.1	MMSE	7.6	67.23	25.2
	Jia et al., Yan et al., [31,32]	China	2015 -2016	Urban & rural	Alzheimer's disease	3,046	75.3	45.8	MMSE	18.7	49.3	32.0
	Zencir et al., [46]	Turkey	2003	Urban only	Alzheimer's disease	42	70.5	38.1	MMSE	42.9	16.7	40.4
	Zhang et al., [36]	China	2008 -2013	Urban only	All types of dementia	5,747	77.4	42.8	None		NA

**Study reported 6.5% of the population with an undefined disease severity stage; CDR: clinical dementia rating scale; FAST: Functional Assessment Staging Tool; MMSE: Mini Mental State Examination; NA: not applicable; NS: not specified. Source

The authors would like to thank fellow colleagues from the Global Brain Health Institute (Fernando Aguzzoli-Peres, Naiara Demnitz and Yaohua Chen) and the Inserm U1094, IRD U270 -EpiMaCT research unit (Daniells Erazo) for their help in translating abstracts and articles in Portuguese, Chinese and Spanish identified during this systematic review.

No funds were received for this study.

DISCLOSURE STATEMENT

All the authors have no conflict of interest to declare.

Lancet Public Health 7, e105-e125.

[2] World Health Organization, Dementia statistics, Last updated 20 September 2022, https://www.who.int/news-room/fact-sheets/detail/dementia, Accessed on 2022-10-27.

[3] World Health Organization (2021) Global status report on the public health response to dementia, World Health Organisation, Geneva, Switzerland. Allegri et al [START_REF] Allegri | Economic impact of dementia in developing countries: an evaluation of costs of Alzheimer-type dementia in Argentina[END_REF] Custodio et al, [START_REF] Custodio | Cost-of-illness study in a retrospective cohort of patients with dementia in Lima, Peru[END_REF] Dominguez et al., [START_REF] Dominguez | Dementia Incidence, Burden and Cost of Care: A Filipino Community-Based Study[END_REF] Ferretti et al, [START_REF] Ferretti | An assessment of direct and indirect costs of dementia in Brazil[END_REF] Kongpakwattana et al, [START_REF] Kongpakwattana | A Real-World Evidence Analysis of Associations Among Costs, Quality of Life, and Disease-Severity Indicators of Alzheimer's Disease in Thailand[END_REF] Malapur et al., [START_REF] Malapur | Cost of Illness of Major Neurocognitive Disorders in India[END_REF] Mould-Quevedo et al, [START_REF] Mould-Quevedo | The burden of caring for dementia patients: caregiver reports from a crosssectional hospital-based study in China[END_REF] Onetiu et al, [START_REF] Onetiu | Cost of dementia in Romania: a cross-sectional Cost-of-Illness study undertaken in Bucharest[END_REF] Prada et al, [START_REF] Prada | Actual expense associated with patients with Alzheimer's disease in Colombia[END_REF] Soares et al, [START_REF] Soares | Economic impact and prevalence of Alzheimer's Disease in a Brazilian capital[END_REF] Wang et al, [START_REF] Wang | Economic impact of dementia in developing countries: an evaluation of Alzheimer-type dementia in Shanghai, China[END_REF] Wang et al, [START_REF] Wang | Caregiver Time and Cost of Home Care for Alzheimer's Disease: A Clinic-based Observational Study in Beijing, China[END_REF] XiaoJuan et al [START_REF] Xiaojuan | Investigation on the economic burden and its influencing factors of Alzheimer's disease in Xi'an[END_REF] Jia et al, Yan et al, [START_REF] Jia | The cost of Alzheimer's disease in China and re-estimation of costs worldwide[END_REF][START_REF] Yan | Associated Factors of Total Costs of Alzheimer's Disease: A Cluster-Randomized Observational Study in China[END_REF] Zencir et al, [START_REF] Zencir | Cost of Alzheimer's disease in a developing country setting[END_REF] Zhang et al, [START_REF] Zhang | Costs of Hospitalization for Dementia in Urban China: Estimates from Two Urban Health Insurance Scheme Claims Data in Guangzhou City[END_REF] Aims (dement* OR Alzheim* OR "cognitive impairment" OR "cognitive dysfunction" OR "cognitive decline" OR "cognitive disorder" OR "memory loss" OR "memory disorder") AND (cost* OR econom* OR "cost of illness" OR "cost analysis" OR "valuation" OR "expenditure" OR "budget" OR financ* OR fund* OR "payment") AND ("Afghanistan" OR "Albania" OR "Algeria" OR "American Samoa" OR "Angola" OR "Armenia" OR "Azerbaijan" OR "Bangladesh" OR "Belarus" OR "Belize" OR "Benin" OR "Bhutan" OR "Bolivia" OR "Bosnia and Herzegovina" OR "Botswana" OR "Brazil" OR "Bulgaria" OR "Burkina Faso" OR "Burundi" OR "Cape Verde" OR "Cambodia" OR "Cameroon" OR "Central African Republic" OR "Chad" OR "China" OR "Colombia" OR "Comoros" OR "Congo Democratic" OR "Congo" OR "Costa Rica" OR "Cuba" OR "Djibouti" OR "Dominica" OR "Dominican Republic" OR "Ecuador" OR "Egypt" OR "El Salvador" OR "Equatorial Guinea" OR "Eritrea" OR "Ethiopia" OR "Fiji" OR "Gabon" OR "Gambia" OR "Georgia" OR "Ghana" OR "Grenada" OR "Guatemala" OR "Guinea" OR "Guinea Bissau" OR "Guyana" OR "Haiti" OR "Honduras" OR "Ivory Coast" OR "India" OR "Indonesia" OR "Iran" OR "Iraq" OR "Jamaica" OR "Jordan" OR "Kazakhstan" OR "Kenya" OR "Kiribati" OR "Korea" OR "Kosovo" OR "Kyrgyz Republic" OR "Lao PDR" OR "Lebanon" OR "Lesotho" OR "Liberia" OR "Libya" OR "Macedonia" OR "Madagascar" OR "Malawi" OR "Malaysia" OR "Maldives" OR "Mali" OR "Marshall Supplementary material 2: Explanation of quality criteria

Criterion Description

Aims and methods

Study objective

The objective(s) and research question(s) of the study was

Calculation of costs

Data source

The source of information on healthcare utilization or costs was reported and mentioned specifically

Perspective

The perspective of the cost calculation was reported (e.g., from payer, employer, or societal perspective)

Cost calculation

The method of cost calculation was clearly documented

Cost categories

The study estimated costs from the utilization of different kinds of healthcare services or areas, which meant that more than one category was given, in order to consider, at best, all costs that accrued from the disease under study Reference year (price) All costs were valued at the price level of a stated base year

Currency

The currency in which the costs were calculated was reported

Supplementary material 3: search results

Sources Number of results

Indexed databases

Econlit 25

Embase


Maitreyee Dutta

Antonio Loría

Elena Panteley

Srikant Sukumar

Backstepping leaderless dynamic consensus among nonlinear systems over directed networks: application to attitude synchronisation control

Keywords: Leaderless consensus, backstepping, Lyapunov stability theory, multi-agent systems

We present a distributed consensus controller for multi-agent homogeneous nonlinear systems over directed networks. The systems are assumed to be of second order and the control design relies on a standard backstepping approach. In that light, the control design also hinges on the ability to construct a strict Lyapunov function for the multi-agent nonlinear system interconnected over a directed graph for which the only assumption is that it is connected. That is, that there exists a directed spanning tree, but there is no requirement of conservative conditions such as strong or balanced connectivity. To construct a strict Lyapunov function, we use a generalised Lyapunov equation for the directed-graph Laplacian matrix, which characterises the spanning-tree-existence condition. Then, we establish exponential stability of the consensus manifold. In addition, we implement our dynamic consensus controller on a multiagent satellite system in the context of attitude synchronisation and demonstrate its efficacy in numerical simulations.

INTRODUCTION

Collation and synergisation among dynamical systems participating in an complex interconnected network can be broadly categorised as either leader-follower or leaderless consensus. As the name suggests, in leader-follower consensus, the response of the assigned singular leader dictates the desired response of the follower agents. If a directed network is a spanning tree, a singular node that has no incoming edge can directly or indirectly interact with all other nodes in the network; this is the leader of the network. In other cases, a virtual leader is assigned a priori which can act as a yardstick for the remaining systems. On the other hand, leaderless consensus is useful for applications where the nature of the consensus equilibrium is not as important as the interacting agents behaving synonymously [START_REF] Ren | Distributed leaderless consensus algorithms for networked euler-lagrange systems[END_REF]. Leaderless consensus is advantageous in applications exhibiting swarm behaviour, such as opinions coagulation in social networks, symbiotic relationship in biological systems, search and rescue, scientific reconnaissance, security, rendezvous or even attitude synchronisation of unmanned aerial vehicles, to name a few. By opting for leaderless configuration, the collected behaviour is not influenced by a single member and each agent can act independently to any adverse circumstance [START_REF] Rao | Sliding mode controlbased autopilots for leaderless consensus of unmanned aerial vehicles[END_REF]. In a directed network, this concept translates into confluence of behaviour of all the agents This work was supported by the CEFIPRA under the grant number 6001-A E. Panteley and A. Loría's work is also supported by the ANR (project HANDY, contract number ANR-18-CE40-0010).

that can transmit information to all the other agents in the communication topology to form the collective consensus behaviour.

Backstepping control has been proved to be efficacious in establishing leader-follower consensus over not only undirected-graph but also directed-graph networks in concatenation with different distributed control methods, such as finite-time consensus control [START_REF] Tian | Robust finitetime consensus control for multi-agent systems with disturbances and unknown velocities[END_REF][START_REF] Shahvali | Distributed finite-time control for arbitrary switched nonlinear multi-agent systems: an observer-based approach[END_REF], adaptive fuzzy control [START_REF] Zhao | Distributed adaptive output consensus tracking of nonlinear multi-agent systems via state observer and command filtered backstepping[END_REF][START_REF] Du | Adaptive fuzzy control for multi-agent systems with unknown measurement sensitivity via a simplified backstepping approach[END_REF], neuro adaptive control [START_REF] Shahvali | Bipartite consensus control for fractional-order nonlinear multi-agent systems: An output constraint approach[END_REF], event triggered adaptive neural network based sliding mode control [START_REF] Chen | Eventtriggered adaptive neural network backstepping sliding mode control of fractional-order multi-agent systems with input delay[END_REF], command filtered approach [START_REF] Cui | Distributed consensus tracking for non-linear multi-agent systems with input saturation: a command filtered backstepping approach[END_REF], non-smooth backstepping control [START_REF] Du | Robust consensus algorithm for second-order multi-agent systems with external disturbances[END_REF] to name a few. The so-called leader present in the leader-follower consensus scheme in the context of directed graphs generally implies a virtual leader or a single root node.

Some of the available works that highlight leaderless consensus among the participating systems which rendezvous through unilateral connections include eventtriggered approach for general linear [START_REF] Chen | Event-triggered coordination of multi-agent systems via a lyapunovbased approach for leaderless consensus[END_REF][START_REF] Wu | Leader-following and leaderless consensus of linear multiagent systems under directed graphs by double dynamic event-triggered mechanism[END_REF] or nonlinear systems [START_REF] Rehan | Leaderless consensus control of nonlinear multi-agent systems under directed topologies subject to input saturation using adaptive event-triggered mechanism[END_REF][START_REF] Ahmed | A novel exponential approach for dynamic event-triggered leaderless consensus of nonlinear multi-agent systems over directed graphs[END_REF][START_REF] Long | Adaptive leaderless consensus for uncertain high-order nonlinear multiagent systems with event-triggered communication[END_REF], distributed adaptive approach for general linear systems [START_REF] Mei | Consensus of linear multi-agent systems with fully distributed control gains under a general directed graph[END_REF][START_REF] Lv | Novel distributed robust adaptive consensus protocols for linear multi-agent systems with directed graphs and external disturbances[END_REF], [START_REF] Li | Distributed adaptive consensus and output tracking of unknown linear systems on directed graphs[END_REF] or Lipschitz nonlinear dynamics [START_REF] Liu | Leaderless consensus of multi-agent systems with lipschitz nonlinear dynamics and switching topologies[END_REF], finite time control among double integrators [START_REF] He | Distributed finite-time leaderless consensus control for double-integrator multiagent systems with external disturbances[END_REF] or second order nonlinear systems [START_REF] Wang | Designing distributed consensus protocols for second-order nonlinear multi-agents with unknown control directions under directed graphs[END_REF], matrix decomposition among linear systems [START_REF] Zhou | Leaderless consensus of linear multi-agent systems: matrix decomposition approach[END_REF].

Distributed adaptive output consensus among systems with strict feedback structure is covered in [START_REF] Huang | Distributed adaptive leader-follower and leaderless consensus control of a class of strict-feedback nonlinear systems: A unified approach[END_REF]. Some additional works that cover special cases include leaderless consensus among discrete time systems with first and second-order dynamics, as explored in [START_REF] Kim | Leaderless and leader-following consensus for heterogeneous multiagent systems with random link failures[END_REF], consensus among fractional order single and double integrator is proposed in [START_REF] Bai | Leaderless consensus for the fractional-order nonlinear multi-agent systems under directed interaction topology[END_REF]. However, several of these works rely on communication topology being strongly connected or balanced, which easens the control design and analysis at the price of certain conservatism-cf. [START_REF] Rehan | Leaderless consensus control of nonlinear multi-agent systems under directed topologies subject to input saturation using adaptive event-triggered mechanism[END_REF][START_REF] Ahmed | A novel exponential approach for dynamic event-triggered leaderless consensus of nonlinear multi-agent systems over directed graphs[END_REF][START_REF] Long | Adaptive leaderless consensus for uncertain high-order nonlinear multiagent systems with event-triggered communication[END_REF][START_REF] Mei | Consensus of linear multi-agent systems with fully distributed control gains under a general directed graph[END_REF][START_REF] Lv | Novel distributed robust adaptive consensus protocols for linear multi-agent systems with directed graphs and external disturbances[END_REF][START_REF] Li | Distributed adaptive consensus and output tracking of unknown linear systems on directed graphs[END_REF][START_REF] Liu | Leaderless consensus of multi-agent systems with lipschitz nonlinear dynamics and switching topologies[END_REF]. If the Lyapunov functions used for stability analysis is non-strict then this brings about the requirement of additional tools such as La Salle's invariance principle and Barbȃlat's Lemma-cf. [START_REF] Chen | Event-triggered coordination of multi-agent systems via a lyapunovbased approach for leaderless consensus[END_REF][START_REF] Wu | Leader-following and leaderless consensus of linear multiagent systems under directed graphs by double dynamic event-triggered mechanism[END_REF]. Moreover, some of the procedures like finite time distributed consensus rely on non-smooth control-cf. [START_REF] He | Distributed finite-time leaderless consensus control for double-integrator multiagent systems with external disturbances[END_REF][START_REF] Wang | Designing distributed consensus protocols for second-order nonlinear multi-agents with unknown control directions under directed graphs[END_REF][START_REF] Bai | Leaderless consensus for the fractional-order nonlinear multi-agent systems under directed interaction topology[END_REF].

Some strides made regarding leaderless consensus among agents colluding through directed network using the principles of backstepping control are detailed in [START_REF] Miao | Backstepping-based high-order nonlinear consensus for multi agent system via edge laplacian[END_REF][START_REF] Wang | Consensus algorithms for second-order nonlinear multi-agent systems using backstepping control[END_REF]. In [START_REF] Wang | Consensus algorithms for second-order nonlinear multi-agent systems using backstepping control[END_REF], each agent is assigned an individual reference system and it is ensured that the reference systems can reach concurrence among themselves using feedback which is dependent on global information, namely, the minimum eigenvalue of the Laplacian matrix. In [START_REF] Miao | Backstepping-based high-order nonlinear consensus for multi agent system via edge laplacian[END_REF], the leaderless consensus problem is converted into stabilization problem of edge based state by using edge Laplacian agreement. Here, the control input design is nonsmooth even in absence of any parameter uncertainties or disturbances.

In this manuscript, we propose a smooth distributed backstepping-based controller for leaderless state consensus among second-order nonlinear systems interacting over directed connected networks. It is assumed that each agent has nonlinear dynamics transformable into a strict feedback form. We establish global exponential stability of the synchronisation manifold and we provide a constructive proof, i.e., we provide a strict Lyapunov function.

Then, we apply our proposed control method to solve a problem of attitude synchronisation among rigid bodies. Attitude synchronisation refers to application of feedback control input which ensures that a multitude of 3-D rigid bodies have same orientation eventually [START_REF] Sarlette | Autonomous rigid body attitude synchronization[END_REF]. Attitude synchronisation is indispensable in formation of satellites. For instance, in the European space agency sponsored infrared space interferometry based Darwin mission, whose objective is to detect exo-planets and capture images of these astrophysical objects with unprecedented spatial resolution [START_REF] Fridlund | The darwin mission[END_REF], on-orbit self assembly and satellite swarms for coordinated observations [START_REF] Izzo | Equilibrium shaping: distributed motion planning for satellite swarm[END_REF].

The remainder of the paper is organised as follows: in Section 2 we give a detailed problem formulation and recall some technical statements, fundamental to our analysis, our the main result is presented in Section 3 and in Section 4 we propose a formation controller for rigid bodies. The paper is wrapped up with some concluding remarks in Section 5.

SYSTEM DESCRIPTION AND PROBLEM FORMULATION

We consider N identical nonlinear agents with strict feedback structure which is given by ẋ1i

= f 1 (x 1i ) + g(x 1i )x 2i , (1a) ẋ2i = f 2 (x 1i , x 2i ) + u i , ∀i = {1, • • • , N }, (1b) where x ji ∈ R n for j = {1, 2}, u i ∈ R n , f 1 : R n → R n , f 2 : R n × R n → R n
are locally Lipschitz and smooth functions, and g : R n → R n×n is a positive or negative definite matrix such that g(•) = 0. The control objective is to ensure that

lim t→∞ x 1i (t) -x 1j (t) → 0, (2a) lim t→∞ x 2i (t) -x 2j (t) → 0 ∀ j = i (2b)
under the standing assumption that the digraph G = (V, E) is connected, i.e., that it contains a directed spanning tree. The strength of a particular unilateral link E ij is captured by element a ij ≥ 0 whereby a ij > 0 if j th agent can transmit information to the i th agent else a ij = 0. The static communication graph can be portrayed concisely by means of the Laplacian matrix L ∈ R N ×N whose elements are l ii = j∈Ni a ij and l ij = -a ij ∀i = j.

For further development, we recall two important statements that hold for connected graphs. The first is a wellknown statement on the properties of the Laplacian matrix and the second provides a basis to construct strict Lyapunov functions for multi-agent systems interconnected over generic directed graphs. Lemma 1. [START_REF] Ren | Distributed leaderless consensus algorithms for networked euler-lagrange systems[END_REF]Beard (2005, 2008) If a directed network has a directed spanning tree, then the Laplacian matrix L = [l ij ] ∈ R N ×N has a singular zero eigenvalue and N -1 eigenvalues have strictly positive real part. That is,

σ 1 (L) = 0, {σ k (L)} > 0, k = {2, • • • , N }.
On the other hand, the right eigenvector of the zero eigenvalue is [START_REF] Panteley | Strict lyapunov functions for consensus under directed connected graphs[END_REF] Let us consider a directed graph G of order N containing a spanning tree and its Laplacian matrix is L ∈ R N ×N . Then, for any positive definite symmetric matrix Q L ∈ R N ×N and α ∈ R + , there exists another positive definite symmetric matrix

1 N = [1 1 • • • 1] and the left eigenvector v l satisfies N k=1 v l k = 1 and v l L = 0 N . Lemma 2.
P L ∈ R N ×N such that P L L + L P L = Q L -α[P L 1 N v l + v l 1 N P L ]. (3)
Now, to analyse the collective behaviour of the multiagent system we start by rewriting its dynamics in the compact form,

ẋ1 = F 1 (x 1 ) + G(x 1 )x 2 , (4a) ẋ2 = F 2 (x 1 , x 2 ) + u, (4b)
where

x j = x j1 • • • x jN for j = {1, 2}, F 1 (x 1 ) = f 1 (x 11 ) • • • f 1 (x 1N ) , F 2 (x 1 , x 2 ) = f 2 (x 11 , x 21 ) • • • f 2 (x 1N , x 2N ) G(x 1 ) =     g(x 11 ) 0 n×n • • • 0 n×n g(x 12 ) • • • . . . . . . 0 n×n • • • g(x 1N )     .
Then, the collective behaviour of ( 4) is assessed by two dynamical systems evolving in orthogonal spaces [START_REF] Panteley | Synchronization and dynamic consensus of heterogeneous networked systems[END_REF]. The first corresponds to a mean-field dynamics with a state defined, roughly speaking, as a weighted average defined by the left eigen-vector corresponding to the unique null eigenvalue of the Laplacian. The second corresponds to the difference between the individual systems' states and the mean-field state; this defines a synchronisation error. That is, we define

x 1m =(v l ⊗ I n )x 1 , e 1 = [(I N -1 N v l ) ⊗ I n ]x 1 , x 2m =(v l ⊗ I n )x 2 , e 2 = [(I N -1 N v l ) ⊗ I n ]
x 2 , where x 1m , x 2m ∈ R n , are the mean-field states and e 1 , e 2 ∈ R N n , are the synchronisation errors. Then, we say that the systems are (resp. achieve) in dynamic consensus if e i = 0 (resp. e i → 0). Indeed, on the synchronisation manifold M = {e ∈ R 2N n : e = 0}, (5) the individual systems behave according to the mean-field dynamics, given by ẋ1m

=(v l ⊗ I n )[F 1 (x 1 ) + G(x 1 )x 2 ], ẋ2m =(v l ⊗ I n )[F 2 (x 1 , x 2 ) + u].
In turn, the dynamics of the synchronisation errors are given by ė1 =ΠF 1 (x 1 ) + ΠG(x 1 )x 2 , (6a) ė2 =ΠF 2 (x 1 , x 2 ) + Πu, (6b) where we introduced the projection matrix Π := (I N -1 N v l ) ⊗ I n (7) for further development.

Thus the synchronisation control design consists in determining a control law u that stabilises the origin for (6). To that end, we design a backstepping distributed controller.

BACKSTEPPING CONTROL DESIGN

Under the standing assumption that the digraph is connected, let Lemma 2 generate a symmetric positive-definite matrix P L such that (3) holds. Then, consider the function V 1 : R N n → R ≥0 , defined as V 1 (e 1 ) = e 1 (P L ⊗ I n )e 1 and which is positive-definite and decrescent. Moreover, its total derivative along the trajectories of (6a) yields V1 (e 1 ) = 2e 1 (P L ⊗ I n )ΠF 1 (x 1 ) + 2e 1 (P L ⊗ I n )e w 2 , (8) where e w 2 := ΠG(x 1 )x 2 is regarded as a virtual control input. Indeed, if e w 2 = e w 2d := -ΠF 1 (x 1 ) -e 1 , V1 (e 1 ) = -2e 1 (P L ⊗ I n )e 1 .

(9)

Next, we introduce ẽ := e w 2 -e w 2d , which satisfies ẽ = Π[G(x 1 )x 2 + F 1 (x 1 ) + x 1 ] (10) and, therefore,

ė =Π ˙ G(x 1 )x 2 + G(x 1 )[F 2 (x 1 , x 2 ) + u] + ∂F 1 (x 1 ) ∂x 1 + I N n F 1 (x 1 ) + G(x 1 )x 2 . ( 11
)
Then, consider the control Lyapunov function candidate

V 2 (e 1 , ẽ) = e 1 (P L ⊗ I n )e 1 + γẽ (P L ⊗ I n )ẽ, γ > 0,
which is positive definite and decrescent for all (e 1 , ẽ) ∈ R N n × R N n . Its total derivative along the trajectories of (6a)-( 11) yields V2 (e 1 , ẽ) = 2e

1 (P L ⊗ I n )(ẽ -e 1 ) + 2γẽ (P L ⊗ I n )× Π ˙ G(x 1 )x 2 + G(x 1 )[F 2 (x 1 , x 2 ) + u] + ∂F 1 (x 1 ) ∂x 1 + I N n F 1 (x 1 ) + G(x 1 )x 2 .
We see that, defining

u = -F 2 (x 1 , x 2 ) + G(x 1 ) -1 - ˙ G(x 1 )x 2 -[L ⊗ I n ]ẽ - ∂F 1 (x 1 ) ∂x 1 + I N n F 1 (x 1 ) + G(x 1 )x 2 , (12) we obtain V2 (e 1 , ẽ) = -2e 1 [P L ⊗ I n ]e 1 + 2e 1 [P L ⊗ I n ]ẽ -γẽ [P L L + L P L ] ⊗ I n ẽ. ( 13
)
Now, after Lemma 2, the last term on the right-hand side of the previous expression equals to ẽ

[Q L -α(P L 1 N v l + v l 1 N P L )] ⊗ I n ẽ
for any α > 0 and Q L = Q L > 0. However, upon expanding the term ((P L 1 N v l ) ⊗ I n )ẽ, using ( 7), ( 10), v l 1 N = 1, and the properties of the Kronecker product, we obtain 14) Using the triangle inequality, we obtain

(P L 1 N v l (I N -1 N v l )) ⊗ I n [G(x 1 )x 2 + F 1 (x 1 ) + x 1 ] = P L (1 N v l -1 N v l ) ⊗ I n [G(x 1 )x 2 + F 1 (x 1 ) + x 1 ] = 0. We conclude that V2 (e 1 , ẽ) = -2e 1 [P L ⊗ I n ]e 1 + 2e 1 [P L ⊗ I n ]ẽ -γẽ [Q L ⊗ I n ]ẽ. (
V2 (e 1 , ẽ) ≤ -2p Lm - 1 ε |e 1 | 2 -γq Lm -εp 2 L M |ẽ| 2 , ( 15
) where ε > 0 and k m and k M denote lower and upperbounds for z Kz, for a square matrix K. It is clear that V2 is negative definite for appropriate values of ε and γ, for any given Q L and P L . Thus, from the previous developments we draw the following statement. Proposition 3. Consider the multi-agent system (4) in closed loop with ( 12), (10), and (7), under the standing assumption that the underlying graph is directed and connected. Then, the synchronisation manifold M in ( 5) is globally exponentially stable. Therefore, the multi-agent system (1) achieves dynamic consensus, that is, lim

t→∞ x 1i (t) -x 1m (t) → 0, lim t→∞ x 2i (t) -x 2m (t) → 0, so (2) hold.
Proof. For the system (6a)-( 11) in closed loop with ( 12), ( 15) holds. The right-hand side of this inequality is negative definite if, for any given Q L and corresponding

P L , > 1 2p Lm , γ > p 2 L M q Lm . ( 16
)
Since V 2 is positive definite and radially unbounded, it follows that {(e 1 , ẽ) = (0, 0)} is globally exponentially stable. On the other hand, note that e 1 = 0 implies that x 1 = (1 N ⊗ x 1m ), so, from (10) and using e 1 = Πx 1 , it follows that (e 1 , ẽ) = (0, 0) implies that Π[G(x 1 )x 2 + F 1 (x 1 )] = 0. On the other hand,

ΠF 1 (x 1 ) =Π    f 1 (x 11 ) . . . f 1 (x 1N )    = Π    f 1 (x 1m ) . . . f 1 (x 1m )    = [(I N -1 N v l ) ⊗ I n ][1 N ⊗ f 1 (x 1m )] = 0 N n , so Π[G(x 1 )x 2 +F 1 (x 1 )] = 0 is equivalent to ΠG(x 1 )x 2 = 0.
In turn, the latter is equivalent to

ΠG(x 1 )x 2 = Π    g(x 11 ) 0 n×n • • • . . . • • • 0 n×n g(x 1N )    x 2 = Π    g(x 1m ) 0 n×n • • • . . . • • • 0 n×n g(x 1m )    x 2 =Π(I N ⊗ g(x 1m ))x 2 = [(I N -1 N v l ) ⊗ g(x 1m )]x 2 = 0 ⇐⇒ [I N ⊗ g(x 1m
)]e 2 = 0 ⇐⇒ e 2 = 0. The last equation holds true as g(•) is either a positive or negative definite matrix as per the given assumption. The statement follows.

ATTITUDE SYNCHRONISATION OF RIGID BODIES

We consider now a multi-agent network in which each node consists of a rigid body-cf. [START_REF] Tsiotras | Stabilization and optimality results for the attitude control problem[END_REF]; [START_REF] Shuster | A survey of attitude representations[END_REF]. We are primarily concerned with synchronising the attitude (orientation) of such a network of spacecraft modelled as rigid bodies. This is critical in several space applications such as Earth observation (EO) [START_REF] Folta | Nasa's autonomous formation flying technology demonstration, earth observing-1 (eo-1)[END_REF][START_REF] Neeck | Nasa's small satellite missions for earth observation[END_REF], terrestrial planet finder (TPF) [START_REF] Lawson | The terrestrial planet finder[END_REF], space telescope assembly (STA), stellar imager (SI) and synthetic aperture imaging (SAI) [START_REF] Kang | Coordinated attitude and formation control of multi-satellite systems[END_REF]. This is primarily motivated by reduced size of each spacecraft and increased robustness in a multi-spacecraft formation. Euler's parameters are typically used to describe the rigid body rotation about an inertial (Earth Centered Inertial) axis. The frame rotating with the rigid body is called the 'bodyfixed frame'. The Euler's parameters can be uniquely reduced to any set of three parameters by using a suitable transformation. The Rodrigues parameters are one such commonly used set with the transformation from Euler's parameters briefly recounted below as per the details given in [START_REF] Schaub | Stereographic orientation parameters for attitude dynamics: A generalization of the rodrigues parameters[END_REF].

Let Θ be the principal rotation angle and n be the normal unit vector. Then, the Euler's parameters are

σ 0 = cos Θ 2 , σ k = n sin Θ 2 , k ∈ {1, 2, 3} (17)
σ σ = σ 2 0 + σ 2 1 + σ 2 2 + σ 2 3 = 1, (18)
where σ = [σ 0 σ 1 σ 2 σ 3 ] . The Euler's parameters σ k have to conform to the holonomic constraint as given in (18) which quite naturally describes a three dimensional unit sphere in R 4 . The transformation from the Euler's parameters to the Rodrigues parameters are defined as

ρ k = σ k σ 0 , k = 1, 2, 3. ( 19
)
It is important to note that the Rodrigues parameters encounter a singularity at σ 0 = 0 which corresponds to a principal rotation angle, Θ = ±180. In the multi-agent setting considered for simulations with five rigid bodies in a network, the kinematics equations in terms of the Rodrigues parameters

ρ i = [ρ 1i ρ 2i ρ 3i ] , i ∈ {1, • • • , 5},
which represents the attitude of the i-th rigid body, are given by ρi = (

I 3 + ρ × i + ρ i ρ i )ω i , (20)
where ω i ∈ R 3 is angular velocity of the i-th rigid body in a body-fixed frame. Then, the Euler's equations of rotational motion are

J ωi = -(ω × i Jω i ) + u i , i = {1, • • • , 5}, (21)
where J ∈ R 3×3 , such that J = J > 0, represents the moment of inertia for the rigid body, u i ∈ R 3 is the input torque, and for a vector a ∈ R 3 , a × represents the vector cross product. The input torque u is usually produced by external actuators such as gas jets in the spacecraft context.

In compact form, ( 20)-( 21) can be written as ρ = blkdiag

i∈{1,••• ,5} {R(ρ i )}ω = R(ρ)ω (22a) (I 5 ⊗ J) ω = -blkdiag i∈{1,••• ,5} col(ω × i Jω i ) + u (22b) where R(ρ i ) = I 3 + ρ × i + ρ i ρ i , ρ = [ρ 1 ρ 2 ρ 3 ρ 4 ρ 5 ] , ω = [ω 1 ω 2 ω 3 ω 4 ω 5 ] , u = [u 1 u 2 u 3 u 4 u 5 ] . Indeed the control objective in this setup is to ensure lim t→∞ ρ i (t)- ρ im (t) = 0 and subsequently lim t→∞ ω i (t) -ω im (t) = 0 ∀ i = {1, • • • , 5} such that ρ im (t)
, ω im (t) are the corresponding mean field values given by ρ im = (v l ⊗ I 3 )ρ and ω im = (v l ⊗ I 3 )ω. Comparing (20)-( 21) to (1) yields

x 1i := ρ i , x 2i := ω i , f 1 (x 1i ) := 0 3 , g(x 1i ) := R(ρ i ), f 2 (x 1 , x 2 ) := -ω × i Jω i . ( 23
) Then, based on derived feedback control input (12), the control torque that needs to be applied to each of the agent is given by

u i =(ω × i Jω i ) + JR(ρ i ) -1 -Ṙ(ρ i )ω i -R(ρ i )ω i -Σ j∈Ni a ij (R(ρ i )ω i + ρ i ) -(R(ρ j )ω j + ρ j ) .
(24) The interaction graph between the agents is represented by the graph in Figure 1.

1 2 3 5 4 L =      1 0 0 -1 0 -1 1 0 0 0 -1 0 1 0 0 0 0 -1 1 0 0 0 0 -1 1     
.

Fig. 1. Connected digraph and its corresponding Laplacian

Arbitrary and distinct initial conditions are assigned to the five agents. As per the given graph, the first, third, and fourth agents can transmit information to all the other nodes in the graph. Indeed, the eigenvector v l in this specific case is given by 1 3 0 1 3 1 3 0 . The simulation result can be seen in Figures 2 and 3 above. It can be seen that the ρ i , ω i , ∀i ∈ {1, • • • , 5} for each of the agents reaches the mean-field value which is given by ρ m = 1 3 (ρ 1 + ρ 3 + ρ 4 ) and ω m = 1 3 (ω 1 + ω 3 + ω 4 ) respectively. In the first subplot of Figure 2, the bold red, purple and navy blue plots represents the meanfield values specifically for ρ 1i , ρ 2i , ρ 3i ∀i ∈ {1, • • • , 5} respectively. In the second subplot of Figure 2, the angular velocity ω converges to zero as the attitude of individual rigid bodies synchronises with the other agents. Figure 3 displays the behaviour of the control torque applied to the interconnected system which complements the conclusion drawn from Figure 2.

CONCLUSION

We proposed a Lyapunov-based distributed controller for leaderless consensus of second-order nonlinear systems. The control design relies on the classical backstepping method. The difficulty involved in multi-agent systems is the construction of a strict Lyapunov function. Yet, our results apply to generic connected directed graphs. The efficacy of the method was demonstrated by designing an attitude synchronisation controller for multi-agent rigidbodies. For simplicity, in this paper the controller relies on a feedback-linearizing inner control loop, but we believe that the procedure may be used to design less stringent controllers, such as passivity-based. Also, the fact that a strict Lyapunov function is explicitly computed, sets the basis for further research on robust consensus control with respect to external disturbances, measurement noise, sensor attacks, etc. These topics are subject of future research.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Consensus among five agents with rigid body dynamics.