MD Gaétan Gentile
email: gaetan.gentile@univ-amu.fr

G Fréchard

A Dia

A Buzens

R Vives

M Jego

V Pommier De Santi

F Simon

2013) dans les armées françaises Incidence of acute respiratory tract infections (2006-2015) and influenza (2006-2013) among French armed forces

Keywords: infections de l'appareil respiratoire, grippe, prévention, armées, militaire respiratory tract infections, flu, prevention, armies, military

Objectives. We aimed to assess the incidence of respiratory tract infections in military settings between 2006 and 2015.

Patients and methods.

We performed a retrospective epidemiological study of the entire military population from 2006 to 2015. Comprehensive data was collected from all medical centers, operational medical units, naval services, and army training hospitals and provided by the epidemiological surveillance of the armies.

Results. The annual average population of the study was 331,394 soldiers. For acute respiratory tract infections (2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014)(2015), 22,818 cases were reported in metropolitan France, 3,211 cases in French overseas territories, 1,595 cases in the French Navy, and 1,318 cases in external military operations for a total of 28,942 cases. For influenza (2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013), 934 cases were reported in metropolitan France, 101 cases in French overseas territories, and 23 cases in external operations, for a total of 1,058 cases. The mean incidence rate of acute respiratory tract infections expressed as case number per 1,000 person-years (PY) was 8.7 PY (95% CI [8.6-8.8]) with an exceptional increased incidence rate in 2009 (25.9 PY, 95% CI [25.4-26.4]). The mean incidence rate of influenza was 0.35 PY (95% CI [0.33-0.37]) with a peak incidence rate of 1.9 PY in 2009.

Conclusion.

Acute respiratory tract infections are at the forefront of infectious episodes in the French armies. Although not necessarily severe, current prevention measures are not enough to reduce the incidence threshold of these infections and need to be improved.

INTRODUCTION

Acute febrile respiratory tract infections (AFRTIs) are the most frequent seasonal infections in industrialized countries [START_REF] Heikkinen | The common cold[END_REF]2]. They affect adults three times a year on average and are mainly due to poorly pathogenic viruses in healthy adults, but also to seasonal influenza or sometimes to pandemic influenza strains (A H1N1) [START_REF] Heikkinen | The common cold[END_REF]. Although most often mild, these viral infections may have a strong impact on health expenditure and organization of care [START_REF] Campbell | Acute respiratory infection: a global challenge[END_REF].

Bacterial infections are less frequent in young adults but when observed, pneumococcal infections and infections caused by intracellular bacteria such as mycoplasma are predominant [START_REF] Trémolières | Épidémiologie microbienne des infections respiratoires basses actualités[END_REF].

Members of the French armed forces are particularly affected by AFRTIs, as they are the second leading infectious episodes in terms of incidence [START_REF] Michel | Prevention of infectious diseases during military deployments: A review of the French armed forces strategy[END_REF]. This may be explained by the demographic profile of this population (young adults, community life) and by the related occupational specificities (missions in metropolitan France and French overseas territories, external operations, high exposure to stress of various origins and to a varied microbial environment) [START_REF] Korzeniewski | Respiratory tract infections in the military environment[END_REF]. AFRTIs may rapidly disturb military missions because of the associated temporary incapacity, mainly in cases of outbreaks.

The management and prevention of AFRTIs are a daily challenge for the French army health service [START_REF]Military Strength Ranking The complete Global Firepower list for 2018 puts the military powers of the world into full perspective: 2018 France Military Strength Current military capabilities and available firepower for the nation of France[END_REF], which strives to limit their impact on the operational capacity of soldiers and on the military network for primary care services [START_REF] Bailey | A brief history of British military experiences with infectious and tropical diseases[END_REF]. The French army health service thus mainly relies on the medical network of the French armed forces, made of military primary care physicians who directly support military units. Prevention measures have been implemented such as hand hygiene, but the most important measure is the three-year vaccination against influenza for members of the French armed forces [START_REF] Nicand | Prophylaxie de la grippe dans l'armée française : le schéma triennal de vaccination[END_REF]. Vaccination against influenza is included in the military immunization calendar. The vaccination is performed at enrollment for all members of the armed forces and is renewed every three years. This vaccination strategy was implemented in 1994 and aims at maintaining collective immunity to prevent outbreaks that would impact the operational capacity of the armed forces. Influenza may occur in vaccinated subjects. These influenza cases are due to the inadequacy between viruses circulating during seasonal outbreaks and valences. No data has however been systematically collected on that matter (only a few clusters of cases notified through alert messages and then investigated). A 2005 study (unpublished data) performed among members of the armed forces confirmed that individuals could only be protected against influenza if vaccinated within the previous 12 months. However, no influenza outbreak with an impact on the operational capacity of the army has so far been reported.

The three-year vaccination strategy is therefore not questioned. Annual vaccination of soldiers before deployment may be required for highly operational situations in an epidemiological context of influenza outbreak. Such strategy was for instance implemented in Afghanistan during the winter season. The French army health service must also anticipate and adapt its strategies depending on the progress of the mission and on local epidemiological contexts.

To better comprehend the impact of AFRTIs in a military context and implement adequate strategies, we performed a retrospective epidemiological study to measure the incidence of AFRTIs among members of the French armed forces between 2006 and 2015.

MATERIAL AND METHODS

Epidemiological surveillance of the French armed forces

The French center for epidemiology and public health of the armed forces (French acronym CESPA) performs an epidemiological surveillance of all soldiers in activity in France, abroad, or at sea. The surveillance focuses on approximately 60 disorders or diseases declared by the Army medical centers, the inter-army medical centers, the operational medical units, the navy medical services, and the military hospitals. CESPA collects and analyzes data to improve knowledge of these disorders and diseases, to inform the stakeholders of the surveillance network, to identify outbreaks, and to perform epidemiological surveys.

The epidemiological surveillance strategy is exhaustive and focuses on all 60 disorders or diseases mentioned above. AFRTI and influenza cases may be notified through alert messages (for outbreaks) and through weekly epidemiological messages (number of clustered cases in each declaring unit).

Population and study period

The target population was all members of the French armed forces over the period 2006-2015. We grouped regions into six categories for better interpretation of regional risks (Europe: metropolitan France and Kosovo; Africa and Indian Ocean: Uganda, Cameroon, Djibouti, Senegal, Niger, Chad, Ivory Coast, Guinea, Mali, Central African Republic, Gabon, Mayotte, and Reunion; Atlantic: Antilles and French Guyana; Middle East: United Arab Emirates, Jordan, Lebanon, Iraq, Qatar; Central Asia: Afghanistan, Tajikistan; Pacific: New Caledonia, French Polynesia). Regions were further subdivided by hemisphere for further assessment of AFRTIs and influenza outcome (Northern Hemisphere: Uganda, Cameroon, Djibouti, Senegal, Niger, Chad, Ivory Coast, Guinea, Mali, Central African Republic, Atlantic, Europe, Middle East, and Central Asia; Southern Hemisphere: Gabon, Mayotte, and Reunion; Pacific).

Inclusion criteria

We included confirmed cases of AFRTIs and influenza notified through the epidemiological surveillance of members of the French armed forces during the study period. Criteria for AFRTI notification were acute catarrh of the respiratory tract combined with fever ≥38.5°C and cough. Notification criteria for confirmed influenza were clinical influenza with biological confirmation (rapid test and/or PCR and/or culture) by nasopharyngeal specimens. Specific surveillance of influenza among members of the armed forces was stopped in 2013 as AFRTIs are an adequate indirect indicator to monitor influenza during seasonal outbreaks among soldiers [START_REF] Duron | The French Military influenza surveillance system (MISS): overview of epidemiological and virological results during four influenza seasons -2008-2012[END_REF]. Only the 2006-2013 data is therefore presented for influenza.

Data analysis

Data was analyzed using Excel ® . The incidence rates expressed as the number of cases notified per 1,000 persons and per year (PY for person-year) were calculated taking into account the mean number of soldiers deployed to each territory and the results expressed as the annual and weekly incidence. The mean incidence rates were calculated by area taking into account the duration of the French armed forces operations in that specific area.

RESULTS

The target population was all members of the French armed forces from 2006 to 2015, i.e. a mean annual population of 331,394 individuals.

Incidence and annual incidence rates

A total of 28,942 AFRTIs were notified as part of the epidemiological surveillance of members of the French armed forces from 2006 to 2015. A total of 22,818 cases were declared in metropolitan France, 3,211 in French overseas territories, 1,595 in the Navy (naval fleet in mission), and 1,318 in external operations (Table I).

From 2006 to 2013, 1,058 cases of influenza were declared including 934 cases in metropolitan France, 101 cases in French overseas territories, and 23 in external operations.

Incidence rate

The mean incidence rate of AFRTIs over the study period was 8.7 PY (95% CI [8.6-8.8]). The annual incidence rates of AFRTIs decreased over time, from 11. An exceptional peak was observed in 2009, with 8,793 cases and an incidence rate of AFRTIs of 25.9 PY (95% CI [25.4-26.4]). The mean incidence rate of influenza from 2006 to 2013 was 0.35 PY (95% CI [0.33-0.37]). A major increase in the incidence rate of confirmed cases of influenza was observed during the 2009 influenza pandemic, with an incidence rate of 1.9 PY versus 0.1 to 0.2 PY the other years.

Mean incidence rates by area AFRTI cases were declared in metropolitan France and in 24 other countries or areas where the French armed forces were deployed (Table I).

The mean incidence rate for the study period in metropolitan France was 8.2 PY [START_REF] Korzeniewski | Respiratory tract infections in the military environment[END_REF]818 incident cases). Countries of deployment with the highest number of incident cases, in absolute value, were New Caledonia (954 cases) and Afghanistan (700 cases). The highest incidence rates were observed in Tajikistan (82.3 PY), Afghanistan (45.1 PY), and the United Arab Emirates (42.3 PY).

Epidemic peaks were identified by assessing the incidence rate per year and per area (Figure 1). Thus, in the United Arab Emirates the incidence rate exceeded 100 PY in 2008 (181.2 PY) and in 2009 (378.5 PY). This incidence rate was also exceeded in New Caledonia in 2006 (117.9 PY), Tajikistan in 2006 (203.8 PY) and 2009 (437.8 PY), with the highest annual incidence rate reported in the present study. Incidences and mean incidence rates by country over the study period are presented in Figure 2. Low incidence rates were mainly observed in Sub-Saharan Africa (except for Uganda). Regions with high incidence rates were Central Asia, Middle East, and the Atlantic and Pacific islands.

Weekly incidence: the 2009 specific case A first peak in the incidence rates was observed on Week 33 of 2009 in the Pacific region, in Africa, and in the Indian Ocean (Figure 3). This was followed by another peak on Week 37 in the Atlantic (+4 weeks). The last peak was observed in Central Asia on Week 44 (+11 weeks). No increase in the incidence rates was observed in the Middle East. A first increase in the incidence rates was observed in Europe from Week 34 to Week 42 before the usual seasonal outbreak. The epidemic profile was similar for influenza cases over the same period, with a peak on Week 33 in the Pacific region, on Week 37 in the Atlantic, and with the highest peak observed in Central Asia on Week 45 (versus 44 for AFRTIs).

DISCUSSION

A military public health problem

The present study confirms the predominant role of AFRTIs among infectious episodes observed in members of the French armed forces. The epidemiological surveillance of these infections is therefore justified in this population. Performed over 10 years, i.e. for a longer duration than previously published studies [START_REF] Yoon | Characteristics of Adenovirus Pneumonia in Korean Military Personnel, 2012-2016[END_REF][START_REF] Koren | Typespecific clinical characteristics of adenovirus-associated influenza-like illness at five US military medical centers, 2009-2014[END_REF][START_REF] Gao | Safety and immunogenicity of 3 seasonal trivalent influenza vaccines in the Chinese military[END_REF][START_REF] Van Kerkhove | Risk factors for severe outcomes among members of the United States military hospitalized with pneumonia and influenza, 2000-2012[END_REF][START_REF] Kajon | Molecular Epidemiology of Adenovirus Type 21 Respiratory Strains Isolated From US Military Trainees (1996-2014)[END_REF][START_REF]Legionellosis in Military Health System beneficiaries, 1998-2013[END_REF][START_REF]Pneumonia and influenza among military members and other beneficiaries of the U.S. military health system, 2007-2012[END_REF][START_REF] Lee | Respiratory disease in Army recruits: surveillance program overview, 1995-2006[END_REF], the present study confirms the constant occurrence of these infections over the years among members of the French armed forces, irrespective of their missions. With a mean annual incidence rate of 8.7 PY, AFRTIs are among the leading infectious episodes in this population in metropolitan France, French overseas territories, and external operations, following diarrhea but ahead of vector-borne diseases [START_REF] Michel | Prevention of infectious diseases during military deployments: A review of the French armed forces strategy[END_REF]. This incidence rate is however probably underestimated because members of the armed forces do not seek medical advice for uncomplicated presentations or consult community physicians. Another reason may also be the underreporting of cases managed in military medical facilities or health facilities on the field. Underreporting is expected for the 2006-2013 period, during which influenza was specifically monitored, because of the obligation for microbiological confirmation. Comparing this incidence rate in young adults with data related to the French general population is difficult.

Excluding the 2009 epidemic peak, the study of the annual incidence rate highlights a progressive decline in the incidence of this syndrome over the 10 years under study. This trend may reflect poorer reporting and/or effective implementation of prevention measures. However, the comprehensive epidemiological surveillance of AFRTIs revealed outbreaks during external operations. The incidence rates observed in Tajikistan, Afghanistan, and the United Arab Emirates thus remain very high. The US army made the same observations with US soldiers based in Iraq and Afghanistan; this might be due to the continental climate and the significant drop in temperature, the winter season, and the constant exposure to dust, sand, and petrochemical fumes [START_REF] Barth | Prevalence of respiratory diseases among veterans of Operation Enduring Freedom and Operation Iraqi Freedom: results from the National Health Study for a New Generation of U.S. Veterans[END_REF].

As observed in the French general population, the incidence rates (AFRTIs and influenza) among the French armed forces varied as expected by seasons in metropolitan France, with an excessive incidence in autumn and winter. However, such seasonality was not observed overseas and in external operations, where sporadic cases were reported all year long, sometimes with localized outbreaks. We did not observe the expected six-month shift due to the reversal of seasons when comparing changes in the incidence rates among the armed forces between both hemispheres. This may be explained by the localizations of the missions, as they mainly occur in intertropical areas (Africa and French departments of the Americas).

Because of changes to the criteria for epidemiological surveillance in the armed forces (distinction between AFRTIs and confirmed influenza until 2013, and then AFRTIs only), we could not properly compare these two periods. However, it provides insight into the influenza burden among AFRTIs. The isolated study performed for the year 2009 (A H1N1 influenza pandemic) helps in revealing the impact of this outbreak on the armed forces. The influenza pandemic affected the armed forces in and outside metropolitan France, with an exceptional peak of confirmed cases (approximately 19 times higher than the mean rate observed in non-epidemic periods). This may be explained by the increased daily surveillance implemented at that time among the armed forces personnel [START_REF] Mayet | Novel influenza A(H1N1) outbreak among French armed forces in 2009: Results of Military Influenza Surveillance System[END_REF], but also by a more systematic biological confirmation of cases. An increased number of AFRTI cases was at the time observed in all areas of deployment, probably due to unconfirmed influenza cases or superinfections with other pathogens. As for the geographical distribution of influenza in 2009, we observed an initial spread of the infection in the Southern Hemisphere on week 33, followed by dissemination to the Northern Hemisphere in autumn. This distribution is typical of new influenza strains. Every year, the time interval is put to good use to synthesize influenza vaccines using the three predominant strains circulating in the Southern Hemisphere. Up to 2016, such time was not enough to have an adequate vaccine in time for members of the armed forces deployed in the Southern Hemisphere. In 2017 to immunize French soldiers deployed to the Southern Hemisphere, the army medical centers located in this region were authorized to locally buy vaccines ahead of this stage.

A daily challenge for the armed forces healthcare professionals

AFRTIs are a major medical problem in military operations because of the associated incidence rate, clinical impact, and the frequent need for care. AFRTIs are the leading infections in the US army. They require treatment initiation, are associated with poorer operational capacity for soldiers (up to 33.8%), and account for 25% to 30% of hospitalizations of military personnel [START_REF] Soltis | Self-Reported Incidence and Morbidity of Acute Respiratory Illness among Deployed U.S. Military in Iraq and Afghanistan[END_REF][START_REF] Korzeniewski | Respiratory tract infections in the military environment[END_REF]. The epidemic nature of these infections may also rapidly saturate the primary care system, especially in operational missions. This clinical situation is also associated with a diagnostic problem for the armed forces Although frequent, AFRTIs are rarely severe or complicated in members of the French armed forces according to internal data of the French army health service on medical evacuations and hospitalization of soldiers (unpublished data). This is probably due to the rapid access to care on the field, with early empirical antibiotic treatment when indicated. Members of the armed forces are also younger and healthier than the general population. By way of comparison, AFRTIs were the leading non-traumatic cause of medical evacuation in Afghanistan from 2010 to 2011 (all nationalities) [START_REF] Ziemba | Analysis of the causes of medical evacuation of injured and sick soldiers of the Polish Military Contingent in the Islamic State of Afghanistan taking part in International Security Assistance Force operations[END_REF]. This clinical situation may however lead to evacuation within the theater of operations or to metropolitan France, mainly because of the associated operational incapacity. Specific to the army, physicians deployed to external operations must consider a wider range of causative agents when confronted with severe pneumonia [START_REF] Dawood | Outbreak of Pneumonia in the Setting of Fatal Pneumococcal Meningitis among US Army Trainees: Potential Role of Chlamydia pneumoniae Infection[END_REF][START_REF] Balicer | Control of Streptococcus pneumoniae serotype 5 epidemic of severe pneumonia among young army recruits by mass antibiotic treatment and vaccination[END_REF][START_REF] Tate | Outbreak of severe respiratory disease associated with emergent human adenovirus serotype 14 at a US air force training facility in 2007[END_REF]. Alongside common pathogens (influenza virus, Streptococcus pneumoniae, etc.) that may be associated [START_REF] Van Kerkhove | Risk factors for severe outcomes among members of the United States military hospitalized with pneumonia and influenza, 2000-2012[END_REF][START_REF] Vento | Pneumonia in Military Trainees: A Comparison Study Based on Adenovirus Serotype 14 Infection[END_REF][START_REF] Sliman | Outbreak of Acute Respiratory Disease Caused by Mycoplasma pneumoniae on Board a Deployed U.S. Navy Ship[END_REF][START_REF] Klement | Identification of risk factors for infection in an outbreak of Mycoplasma pneumoniae respiratory tract disease[END_REF], specific endemic infections should be considered depending on the circumstances on the field: leptospirosis, melioidosis or schistosomiasis following exposure to fresh water, Q fever, or American histoplasmosis following a stay in the forest in Guyana, and anthrax in a context of bio-terrorism risk. Early detection and confirmation are essential for military public health and sometimes for national security.

Limitations of current prevention measures

With such a high incidence rate and a few outbreaks with operational impact, the incidence threshold of AFRTIs (influenza included) cannot be reduced despite current prevention measures. The same problem is observed for other community-acquired infections such as bacterial skin infections, diarrhea, and sexually transmitted infections [START_REF] Michel | Prevention of infectious diseases during military deployments: A review of the French armed forces strategy[END_REF]. Prevention of AFRTIs is currently based on operational readiness of members of the armed forces for each mission, hygiene measures, and vaccination against influenza.

The health of each soldier is monitored at medical visits every 24 months. Members of the armed forces must also attend a medical visit before each mission, where they receive targeted health education on the risks associated with the mission. If needed, they receive the required vaccines and are prescribed malaria prophylaxis. They are declared fit for duty if they do not present with any active disorder, even dental infections, and if their vaccinations are up to date. Health education on preventive measures against AFRTIs is given to soldiers on a regular basis as well as in cases of outbreaks: regular ventilation of living areas, strict hand hygiene following each contact with a sick person. Measures aimed at stopping secondary transmission from suspect cases are implemented as much as possible; physical means of prevention (masks, etc.) must therefore be available in all facilities. These measures have proved effective, but their implementation may be difficult because of the living conditions of operational missions; thus explaining some outbreaks observed in armed conflicts [START_REF]Military Strength Ranking The complete Global Firepower list for 2018 puts the military powers of the world into full perspective: 2018 France Military Strength Current military capabilities and available firepower for the nation of France[END_REF]. Since 1994 the French army health service has been implementing a prevention strategy against influenza through the vaccination of nonmedical armed forces personnel every three years to achieve collective immunity and maintain minimum operational capacity, and through annual vaccination of military health professionals. The effectiveness of the three-year strategy has yet to be proven [START_REF] Haus-Cheymol | groupe des médecins de services médicaux d'unité ayant participé à l'étude. [Effectiveness of triennial anti-influenza vaccination in French military during the 2003-2004 influenza season[END_REF].

Because of a lack of insight, we cannot assess the influence of antipneumococcal vaccination -mandatory for children since 2004 [START_REF]fixant les modalités de désignation et les missions des centres nationaux de référence pour la lutte contre les maladies transmissibles. 3 décembre[END_REF] -on the progressive decline in AFRTIs among the armed forces over the 2006-2015 period.

Besides, the French army has never documented any AFRTI outbreak caused by adenoviruses, unlike the US army. The US army has therefore implemented oral vaccination against adenoviruses in 2012, with a confirmed decrease in AFRTI cases over the initial training period of soldiers [START_REF] Clemmons | Acute Respiratory Disease in US Army Trainees 3 Years after Reintroduction of Adenovirus Vaccine1[END_REF][START_REF] Choudhry | Safety evaluation of adenovirus type 4 and type 7 vaccine live, oral in military recruits[END_REF][START_REF] Yun | Changes in Clinical Presentation and Epidemiology of Respiratory Pathogens Associated With Acute Respiratory Illness in Military Trainees After Reintroduction of Adenovirus Vaccine[END_REF][START_REF] Afhsc) | Incidence of acute respiratory illnesses among enlisted service members during their first year of military service: did the 2011 resumption of adenovirus vaccination of basic trainees have an effect?[END_REF]. Such vaccination is not considered by the French army.

We also could not assess whether antimalarial chemoprophylaxis with doxycycline (100 mg/day) in endemic areas had an impact on the incidence of AFRTIs in these areas.

Although doxycycline is active against most streptococci, some strains of Staphylococcus aureus, various Gram-negative bacilli, several anaerobes, Mycoplasma pneumoniae, Coxiella burnetii, and leptospirosis, no data suggests the reduced incidence of cosmopolitan or exotic bacterial pneumonia.

Limitations of the study

The case definition of AFRTI presented in the method complies with epidemiological surveillance criteria that do not make the distinction between lower and upper respiratory tract infections.

Non-biologically confirmed influenza cases are de facto included among AFRTIs. Confirmed cases of influenza are not included among AFRTIs, but they are included among influenza cases. However, as the influenza surveillance was stopped in 2013, cases are included among AFRTI cases reported in 2014 and 2015.

Exhaustiveness of confirmed cases (required in the army) by declaring units presented in the method does not seem possible.

CONCLUSION

The present study reports the high incidence of AFRTIs in military public health because of their morbidity and operational impact in cases of outbreaks. Although their severity is reduced because of an effective healthcare chain, prevention measures do not seem to significantly reduce the incidence threshold. This observation highlights several areas for improvement in the short-and mid-term: maximization of diagnosis on the field (e.g., rapid diagnostic test for influenza), oseltamivir on the field to limit the epidemic spread of influenza, study of the respiratory microbial ecology among members of the armed forces, assessment of the three-year vaccination strategy against influenza, strengthening of the response capacity to future emerging infections.

of interests

The authors report no conflict of interest.

  7 PY in 2006 to 4.3 PY in 2015 (2006=11.7 PY; 2007=11 PY; 2008=8 PY; 2009=25.9 PY; 2010=4.6 PY; 2011=5.4 PY; 2012=6.1 PY; 2013=6 PY; 2014=3 PY; 2015=4.3 PY).

  physicians, just like for any other community family physician. Poor specificity of the respiratory symptoms, limited access to microbiological diagnostic tools, and limited access to imaging techniques are major obstacles to clinical diagnosis. Physicians deployed with the armed forces to external operations or on small Navy vessels do not have access to rapid diagnostic tests for AFRTIs, and access to chest imaging techniques or routine microbiological tests is only possible in rear base medico-surgical facilities. Diagnostic means in metropolitan France are similar to those of family physicians. The French army health service is currently assessing several areas for improvement: maximization of biological diagnostic tools on the field, development of point-of-care facilities in areas with a high number of soldiers, and studies to determine the microbial ecology of AFRTIs.

Figure 1 . 1 .Figure 2 .Figure 2 . 1 WFigure 3 . 3 .

 1122133 Figure 1. Taux d'incidence moyens annuels de 2006 à 2015 (cas déclarés pour 1 000 personnes année) en fonction des territoires Figure 1. Mean annual incidence rates from 2006 to 2015 (declared cases per 1,000 personyears) by territory

Table I .

 I Mean incidence rates per 1,000 person-years of respiratory tract infections reported by territory over the 2006-2015 period

	Country	Number of cases	2006-2015 incidence rate as person year (PY)
	Tajikistan	196	82.25
	Afghanistan	700	45.11
	United Arab Emirates	175	42.33
	New Caledonia	954	30.26
	Uganda	2	29.85
	Cameroon	2	20
	French Polynesia	448	17.48
	Jordan	3	17.24
	Antilles	577	13.16
	Djibouti	329	13.14
	Lebanon	132	11.78
	Iraq	5	10.66
	Mayotte -Reunion	457	9.35
	Metropolitan France	22,818	7.94
	Senegal	71	7.97
	Niger	9	7.22
	Chad	84	6.63
	Kosovo	70	6.63
	Ivory Coast	80	5.17
	Guinea	1	4.22
	French Guyana	165	4.19
	Gabon	35	4.18
	Mali	19	2.78
	Central African Republic	14	2.34
	Qatar	1	1.2
	Naval fleet	1,595	13.29
	PY: person year;
	Naval fleet, offshore or onshore irrespective of the region


Quynh T T Nguyen

Nadhir Messai

Noureddine Manamanni

Sinuhé Martinez-Martinez

Fault estimation for networks of non-homogeneous agents with switching topologies

Keywords:

Introduction

Recently, a great interest has been drawn to the control and diagnosis of multi-agent systems. This is partly due to the trend of exploiting a network of simple agents to fulfill a complicated task instead of using a complex and powerful one (see,e.g., [START_REF] Djaidja | Stochastic consensus of leader-following multi-agent systems under additive measurement noises and time-delays[END_REF][START_REF] Frasca | Distributed randomized algorithms for opinion formation, centrality computation and power systems estimation: A tutorial overview[END_REF][START_REF] Pham | Consensus of Multi-Agent Systems in Clustered Networks[END_REF][START_REF] Oh | A survey of multi-agent formation control[END_REF][START_REF] Wei Ren | A survey of consensus problems in multi-agent coordination[END_REF][START_REF] Zhang | Distributed Cooperative Optimal Control for Multiagent Systems on Directed Graphs: An Inverse Optimal Approach[END_REF][START_REF] Zhang | Homogenous polynomially parameter-dependent HâĹđ filter designs of discrete-time fuzzy systems[END_REF][START_REF] Zhang | Leader-based optimal coordination control for the consensus problem of multiagent differential games via fuzzy adaptive dynamic programming[END_REF]). Since a typical specification of multi-agent systems is information sharing, if a fault occurs, its effect can propagate. This may lead to severe impacts on the overall performance of the system and make it crash to malfunctions quickly. Therefore, various approaches have been proposed to detect, isolate, and estimate faults in multi-agent systems (see, e.g., [START_REF] Chadli | Distributed state estimation, fault detection and isolation filter design for heterogeneous multiagent linear parameter-varying systems[END_REF][START_REF] Davoodi | Distributed Fault Detection and Isolation Filter Design for a Network of Heterogeneous Multiagent Systems[END_REF][START_REF] Davoodi | Simultaneous fault detection and control design for a network of multi-agent systems[END_REF][START_REF] Hashimoto | Distributed 1 -state-and-fault estimation for Multi-agent systems[END_REF][START_REF] Li | Cooperative fault estimation for linear multi-agent systems with undirected graphs[END_REF][START_REF] Li | Distributed Cooperative Fault Detection for Multi-agent Systems: A Mixed ∞ ∕ 2 Optimization Approach[END_REF][START_REF] Menon | Robust fault estimation using relative information in linear multi-agent networks[END_REF][START_REF] Nguyen | A distributed fault detection observer-based approach for a network of multi-agent systems with switching topologies[END_REF][START_REF] Sinafar | Distributed Fault Estimation for Multi-Agent Systems Using Relative Measurements-Distributed Adaptive Observer Approach[END_REF][START_REF] Xu | Distributed Fault Estimation Basing On A United Adaptive and Sliding-Mode Observer for Multi-Agent Systems[END_REF][START_REF] Zhang | Distributed fault estimation observer design for multi-agent systems with switching topologies[END_REF][START_REF] Zhang | Adjustable Parameter-Based Distributed Fault Estimation Observer Design for Multiagent Systems With Directed Graphs[END_REF]).

Different architectures of fault diagnosis problem in multi-agent systems have been proposed from centralized (see, e.g. [START_REF] Huang | Fault estimation of multi-agent systems with transient performance improvement[END_REF][START_REF] Meskin | Fault Detection and Isolation of discrete-time Markovian jump linear systems with application to a network of multi-agent systems having imperfect communication channels[END_REF]), decentralized (see, e.g. [START_REF] Arrichiello | Observer-Based Decentralized Fault Detection and Isolation Strategy for Networked Multirobot Systems[END_REF][START_REF] Boem | Decentralized fault diagnosis for heterogeneous multi-agent systems[END_REF]), to distributed (see, e.g., [START_REF] Chadli | Distributed state estimation, fault detection and isolation filter design for heterogeneous multiagent linear parameter-varying systems[END_REF][START_REF] Han | Robust fault estimation in the finite-frequency domain for multi-agent systems[END_REF][START_REF] Nguyen | A distributed fault detection observer-based approach for a network of multi-agent systems with switching topologies[END_REF]). In the earlier results, typical fault diagnosis schemes known as centralized schemes that require all the information related to the system available to a single agent executing the fault diagnosis algorithm. In comparison to the other architectures, the centralized scheme is less attractive in practical implementations due to the stringent use of computational resources, communication bandwidth limitations, and the poor scalability of the overall system. Solutions for these limitations have been given in later results, they are non-centralized schemes known as decentralized/distributed ones where the fault diagnosis task on an agent is fulfilled based on only the local information. Since all measurements may not be available in every agent, the decentralized/distributed approaches are seemed to have more potential applications (see, e.g. [START_REF] Martínez-Villegas | Review of two non-centralized observer-based diagnosis schemes for interconnected systems âĄŐ[END_REF] and the references therein). This is one of the reasons that motivate us to investigate the distributed approach.

Otherwise, the information graph is crucial for the behavior of a multi-agent system. Predominantly, there are two categories of information graph in the context of multi-agent systems: the fixed topology and the switching ones. Several interesting works (see, e.g., [START_REF] Menon | Robust fault estimation using relative information in linear multi-agent networks[END_REF][START_REF] Li | Cooperative fault estimation for linear multi-agent systems with undirected graphs[END_REF][START_REF] Davoodi | Simultaneous fault detection and control design for a network of multi-agent systems[END_REF][START_REF] Chadli | Distributed state estimation, fault detection and isolation filter design for heterogeneous multiagent linear parameter-varying systems[END_REF][START_REF] Li | Distributed Cooperative Fault Detection for Multi-agent Systems: A Mixed ∞ ∕ 2 Optimization Approach[END_REF][START_REF] Han | Robust fault estimation in the finite-frequency domain for multi-agent systems[END_REF][START_REF] Zhang | Adaptive technique-based distributed fault estimation observer design for multi-Agent systems with directed graphs[END_REF]), have been done dealing with the problem of fault diagnosis in multi-agent systems with fixed topology when the communication graph does not change through all the progress of the system operation. Switching topologies is the case that the communication graph can switch from one topology to another [START_REF] Olfati-Saber | Consensus problems in networks of agents with switching topology and time-delays[END_REF]. Because of the practical factor such as environment changes or energy constraints, the problem of switching topologies is explicitly more realistic and more challenging to cope with in comparison to the issue of fixed topology. Up to now, just a few papers focus on the problem of fault detection and isolation (FDI) and fault estimation (FE) with switching topologies, such as [START_REF] Zhang | Distributed fault estimation observer design for multi-agent systems with switching topologies[END_REF] and [START_REF] Nguyen | A distributed fault detection observer-based approach for a network of multi-agent systems with switching topologies[END_REF]. In [START_REF] Zhang | Distributed fault estimation observer design for multi-agent systems with switching topologies[END_REF], an observer-based distributed fault estimation approach is proposed from an overall perspective, while the local fault estimation observers are presented for each agent and based on the relative output estimation error. In [START_REF] Nguyen | A distributed fault detection observer-based approach for a network of multi-agent systems with switching topologies[END_REF], we have proposed a distributed observer-based FD approach, in which, an agent can detect not only its own fault but also its neighbors' ones in spite of switching specifications. Actually, it is worth to note that in some works, to implement the scheme, a bank of observers whose dimension is equal to the dimension of the entire system is required at each node, which brings a computational burden on the node (see, e.g., [START_REF] Li | Cooperative fault estimation for linear multi-agent systems with undirected graphs[END_REF]). This burden becomes heavier when the number of agents in the network increases. We have avoided this problem in [START_REF] Nguyen | A distributed fault detection observer-based approach for a network of multi-agent systems with switching topologies[END_REF], where the orders of detection observers are only equal to the sum of the order of each agent and its neighbors. In current work, we also aim to cope with the switching topologies issue because of its prospective practical implementation.

Generally, fault detection and isolation and fault estimation are effective solutions for enhancing system safety and reliability (see, e.g., [START_REF] Ding | Model-based Fault Diagnosis Techniques[END_REF]). The role of fault diagnosis in multi-agent systems becomes essential because the problem of FTC in this kind of system is even more complex and suffers from more difficult challenges in comparison to the one in classical single systems [START_REF] Patton | A generic strategy for fault-tolerance in control systems distributed over a network[END_REF]. The strongly and rapidly development of exploitation of multi-agent systems in practice urges researchers more intensely focusing on the problem of FDI/FE in multi-agent systems. The problem of FDI/FE is also significant when it is implemented in Fault-Tolerant Control (FTC) problems, where the more exact information of fault is known, the higher the control performance quality will be (see, e.g., [START_REF] Gao | A Survey of Fault Diagnosis and Fault -Tolerant Techniques Part I : Fault Diagnosis with Knowledge -Based and Signal-Based Approaches[END_REF][START_REF] Liu | Hierarchical-Structure-Based Fault Estimation and Fault-Tolerant Control for Multiagent Systems[END_REF]). In this work, FE will be pursued. While FDI is to set up alarms whenever there is a fault (fault detection) and after that indicate where the fault occurs (fault isolation), FE reconstructs the magnitude as well as the shape of the fault. With no doubt, FE is more useful and powerful.

In two past decades, FD for multi-agent systems has attracted attention from the community of researchers and become a very hot topic [START_REF] Qin | A survey of fault diagnosis for swarm systems[END_REF]. Compared with the research on FD, the published literature on FE for multi-agent systems is sparser, and most of the existing research is observer-based approaches. In classical single systems, faults can be reconstructed from the estimated states, estimated system outputs, or the difference between the estimated and actual system outputs which can be obtained by some known techniques (e.g., [START_REF] Ding | Model-based Fault Diagnosis Techniques[END_REF][START_REF] Bejarano | Switched observers for switched linear systems with unknown inputs[END_REF][START_REF] Messai | Fault Estimation for for a Class of of Switched Systems[END_REF][START_REF] Shi | Some novel approaches on state estimation of delayed neural networks[END_REF]). In addition, in multi-agent systems, estimate faults are also reproduced by using relative measured output, which should be available for the cooperation among agents. A robust FE method is given in [START_REF] Menon | Robust fault estimation using relative information in linear multi-agent networks[END_REF], which is based on sliding mode observers and uses only relative output. The authors proposed transformations and scalings to create observable subsystems from the unobservable initial systems using relative output. The FE observer design in [START_REF] Li | Cooperative fault estimation for linear multi-agent systems with undirected graphs[END_REF] and [START_REF] Zhang | Distributed fault estimation observer design for multi-agent systems with switching topologies[END_REF] are also based on the relative measured output rather than the absolute one. In [START_REF] Li | Cooperative fault estimation for linear multi-agent systems with undirected graphs[END_REF], the authors use an integral action in the fault estimator to enhance the estimation performance. In [START_REF] Zhang | Distributed fault estimation observer design for multi-agent systems with switching topologies[END_REF], a distributed FE observer is proposed for homogeneous multi-agent systems subject to switching topologies. Nevertheless, these mentioned design still rely on defining global vectors, and such constraint requires a high capacity of resources which may not be affordable by all agents. In [START_REF] Liu | Hierarchical-Structure-Based Fault Estimation and Fault-Tolerant Control for Multiagent Systems[END_REF], a decentralized FE algorithm is proposed, in which, the estimated fault is obtained by using only the information from the corresponding agent. It should mention to two recently interesting research on FE for multi-agent systems, [START_REF] Huang | Fault estimation of multi-agent systems with transient performance improvement[END_REF], and [START_REF] Han | Robust fault estimation in the finite-frequency domain for multi-agent systems[END_REF]. However, they are out of our purpose in this work. While [START_REF] Huang | Fault estimation of multi-agent systems with transient performance improvement[END_REF] focuses on the transient performance improvement with a centralized fault reconstruction, [START_REF] Han | Robust fault estimation in the finite-frequency domain for multi-agent systems[END_REF] solves the problem in the finite-frequency domain.

But most of these works only deal with the problem of FE for homogeneous multi-agent systems with fixed topology. As a result, they are no longer applicable in many practical scenarios where the task is fulfilled by non-homogeneous agents and/or the topology of the communication network is changed due to various factors such as external interference and hardware limitations. So, it can be said that the problem of FE for non-homogeneous/ heterogeneous multi-agent systems with switching topologies is still very challenging. To the best of our knowledge, there is no existing publication that deals with this problem.

Motivated by the desire to overcome the specified limitations mentioned above, our objective is to cope with the problem of distributed fault estimation for non-homogeneous multi-agent systems with switching topologies. An average dwell-time approach is employed to guarantee the stability of the distributed FE observers. It can be said that our approach is distributed in both design and implementation because the observer design for each agent does not need the information of the entire system. A model based on both, absolute and relative outputs, is associated with each agent with respect to its neighbors. The so-called virtual model is constructed with the assumption that the network has to switch among some known topologies. Each topology is considered as an operating mode of the system. Furthermore, the relation between the agents' states before and after the switching is taken into account by a transition matrix. As such, we consider the multi-agent system with switching topologies as an impulsive switched system. Model-based fault estimation observers are synthesized taking into account each agent's virtual model and the switching behavior. Then, a set of optimal gains are found for each known topology. When the system switches from one topology to another, these gains are updated by the appropriate pre-calculated optimal values.

The main contributions of this paper are:

(i) A new decoupling method that allows to decouple the fault of the agent and its neighbors under the switching topologies is proposed. This method is developed based on the previous works of our team [START_REF] Messai | Fault Estimation for for a Class of of Switched Systems[END_REF] for a class of switched linear systems.

(ii) A design of a distributed observer-based fault estimation for a class of non-homogeneous multi-agent system with switching topologies that allows each agent to estimate its faults and those of its neighbors is introduced. The proposed approach is distributed in both design and implementation because it does not need the information of the entire system, and therefore, this approach may help to reduce the burden of computations in comparison to the other research on FE in multi-agent system with switching topologies, for example [START_REF] Zhang | Distributed fault estimation observer design for multi-agent systems with switching topologies[END_REF].

Moreover, the proposed approach allows to estimate the faults that can occur in a large-scale multi-agent systems by using only a few number of agents. In other words, the diagnosis task is shared among a number of diagnosis agents [START_REF] Martínez-Villegas | Review of two non-centralized observer-based diagnosis schemes for interconnected systems âĄŐ[END_REF]. This scheme have prospective applications in the realistic world, where a group of agents may evolve into several subgroups (see,e.g., [START_REF] Pham | Consensus of Multi-Agent Systems in Clustered Networks[END_REF][START_REF] Peng | A novel optimal bipartite consensus control scheme for unknown multi-agent systems via model-free reinforcement learning[END_REF]). Then, a remarkable amount of computation is reduced when the fault diagnosis modules are set up on some specific agents in each cluster. Another advantage is the ability of our approach to handle nonhomogeneous multi-agent systems with switching topologies without needing the synthesis of estimators that have the same dimension of the whole system (i.e. the dimension of our estimator is related only to the number of neighbors of the monitoring agents)

The rest of this paper is organized as follows. After the introduction in Section 1, notations, preliminaries, and problem formulation are presented in Section 2 and Section 3. The main results are addressed in Section 4 and Section 5, in which three propositions, one theorem, their proofs, and remarks are presented. The proposition of fault estimator is given in Section 6. Before the conclusion in Section 8, the simulation results that supports and illustrates our proposed theoretical development is provided in Section 7.

Notation and preliminaries

The communication topology among the agents is represented by an undirected graph  = ( , ), consisting of the node set = {1, 2, ..., } and the edge set ⊆ × . As we consider switching topology, the communication topology is time-variant, or  = ( ). Let us denote the set of all topologies = {1, 2, ..., } and a right continuous function ( ) ∶ [ , ∞) → as the switching rule.

We adopt the following notations throughout this paper: ℝ denotes the set of all dimensional real column vectors; for a matrix , denotes its transpose; and denote the identity and zero matrices with appropriate dimensions, respectively;

denotes the ( × 1) column vector whose elements are all ones; the Kronecker product of matrices is denoted by ⊗; | | denotes the cardinality of a given set ; (.) + , (.) -denotes the right-limit and the left-limit operator.

For given matrices ( ) , 1 ( ) , ...,

| | ( )
, let us define the following augmented matrices:

( ) = ( ) ( ) (1)
and

̄ ( ) = | |+1 ⊗ ( ) - ( ) , (2)
where

( ) = 1 ( ) , ..., | |
( ) . Moreover, since the practical implementation of our approach needs a slow switching between the topologies ( i.e., this is needed in order to estimate the evolution of the possible faults), average dwell time constraints are considered to deal with the observer synthesis [START_REF] Liberzon | Basic problems in stability and design of switched systems[END_REF], [START_REF] Hespanha | Stability of switched systems with average dwell-time[END_REF], [START_REF] Mahmoudi | On observer design for a class of impulsive switched systems[END_REF], and [START_REF] Lin | Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results[END_REF]. The definition of Average dwell time is recalled as follows:

Definition 1. (Average dwell-time) Let
( , 0 ) be the number of discontinuities of the switching signal ( ) on a given interval [ 0 , ). If there exist two positive numbers 0 and then ( ) has average dwell-time such that

( , 0 ) ≤ 0 + ( -0 )∕ , 0 ≤ 0 ≤ , ( 3
)
where 0 is the chatter bound, 0 ≥ 0.

The switching signal is constrained by the following average dwell time given in:

> = ( )∕(-2 ), (4)
where < 0 and > 1 are given constants.

Before presenting the problem formulation, the following lemmas [START_REF] Chilali | ∞ design with pole placement constraints: an LMI approach[END_REF] and [START_REF] Bernstein | Matrix mathematics[END_REF], which will be used in this sequel, are recalled.

Lemma 1. (LMI region) Let  be a subregion of the complex left-haft plane. The eigenvalues of a given matrix

∈ ℝ × belong to , is then called  -if and only if it exists a symmetric positive definite matrix ∈ ℝ × such that:

 ( , ) < 0, (5)
where the block matrix characterizes for the LMI region , and

 = ⊗ + ⊗ ( ) + ⊗ ( ) < 0, (6)
where the symmetric matrix ∈ ℝ × and the matrix ∈ ℝ × are defined corresponding the shape of .

Lemma 2. (Intersected LMI regions)

Given two LMI regions  1 and  2 , a matrix is both  1 -and  2if and only if there exists a symmetric positive definite matrix satifying Lemma 1 and:

 1 ∩ 2 =  1 0 0  2 < 0. ( 7
)
Lemma 3. (Matrix inversion) Consider the matrices ∈ × , ∈ × and ∈ × with and are nonsingular matrices. Then:

-1 = -1 --1 -1 -1 (8)

Problem formulation

System description

The multi-agent system studied here is considered as the interconnection of different dynamical systems that we call sub-systems or agents. Such agents, communicate together through a network, are supposed to be non-homogeneous in the meaning that they have different dynamics but the same number of states and outputs. From an agent viewpoint, each one of the agents with which it communicates is called a neighbor. Let us consider a group of agents described by the following state-space model [START_REF] Menon | Robust fault estimation using relative information in linear multi-agent networks[END_REF][START_REF] Zhang | Adjustable Parameter-Based Distributed Fault Estimation Observer Design for Multiagent Systems With Directed Graphs[END_REF][START_REF] Li | Cooperative fault estimation for linear multi-agent systems with undirected graphs[END_REF][START_REF] Xu | Distributed Fault Estimation Basing On A United Adaptive and Sliding-Mode Observer for Multi-Agent Systems[END_REF]:

̇ ( ) = ( ) + ( ) + ( ) ( ) = ( ) (9)
= 1, 2, ..., , ( ) ∈ ℝ , ( ) ∈ ℝ and ( ) ∈ ℝ are the state, the measured output and the control input of the system, respectively. ( ) ∈ ℝ represents an unknown time varying fault in the ℎ agent with > , i.e., ( ) = 0 there no fault in the ℎ agent, and ( ) ≠ 0 otherwise. , , , are constant matrices with appropriate dimensions.

We define the relative measured output between the ℎ agent and the ℎ neighbor as

( ) = ( ) -( ) for = 1, 2, ..., where ∈ = 1 , 2 , ..., | | ⊆ [1,
] denotes the set of agents that agent can sense. The synthesis of the estimator is based on the impulsive switched virtual model, introduced in the next section, for which some assumptions are necessary.

Assumption 1: The number of neighbors with respect to agent remains constant when the topology changes. Assumption 2: Each agent has access to the relative output of its neighbors.

The impulsive switched virtual model

Our objective in this work is to propose a distributed design of fault estimators for multi-agent systems with switching topologies, and moreover, the fault estimators can estimate both the faults affecting the agent for which it is designed and its neighbors. To deploy the idea, each one of the agents uses what is called a virtual model, which takes into account the number of its neighbors as well as their dynamics. Then, such system modeling is the basis for constructing the proposed fault estimator. Otherwise, as the multi-agent system's topology switches among different topologies, one can model such a multi-agent system as an impulsive switched system with modes of operation corresponding to the topologies. Consequently, taking into account both, virtual model and switching topologies, the following impulsive switched virtual model with respect to agent ℎ can be established as:

⎧ ⎪ ⎨ ⎪ ⎩ ̇ ( ) = ( ) ( ) + ( ) ( ) + ( ) ( ) ( ) = ̄ ( ) ( ) ( + ) = ( + ), ( -) ( -) (10)
where

( ) = [( ) , ( 1 ) , ( 2 ) , ..., ( | | ) ] ∈ ℝ ̃ with ̃ = (| | + 1) × ,
includes states of agent and its neighbors. Notice that, by assumption 1, ( ) has a constant dimension when the topology switches. All the information that the ℎ agent can sense is collected in the vector state ( ) = [( ) , ( 1 ) , ( 2 ) , ...,

( | | ) ] , ( ) = [( ) , ( 1 ) , ( 2 ) , ..., ( | | ) ]
is the control input supposed to be known. When the topology switches, the ℎ agent's neighbors change, so the states of the system (10) may have a jump at the switching instant. Let ( + ), ( -) denotes the state transition matrix which shows relation between states before and after the switching instant. In the case of fixed topology ( + ), ( -) = . Notice that matrices, ( ) , ( ) , and ( ) , are constructed as defined in (1), while the matrices ̄ ( ) follow definition [START_REF] Bejarano | Unknown Input and State Estimation for Unobservable Systems[END_REF]. The upper index | | means the ℎ neighbor of the ℎ agent and the lower index denotes the active mode. Hereafter, let ( + ) = and ( -) = .

In other words, is the successor mode of mode .

For the sake of clarity, the superscript index and the time dependence will be omitted when there is no ambiguity. Readers should keep in mind that all the following developments are for agent and its neighbors.

Fault reconstruction methodology

This section describes the proposed fault estimation methodology which is depicted by figure 1. The process to obtain the fault estimation in our proposed approach can be divided into three main step. Step 1 includes two sub-steps, model representation and coordinate transformations. In the former, a virtual model of each agent is created by taking into account its neighbors, which is done at subsection 3.2. In the latter, some coordinate transformations are introduced in order to decouple a subset of state from the fault. Next, in step 2, a distributed observer is synthesized to estimate the subset ̃ 1 , which is decoupled from the fault. Once, the estimation ̂ 1 is obtained, the estimation of the subset which is affected by the fault, ̃ 2 , is calculated and they are merged together in order to obtain an estimate of the whole virtual model ̂ = ̂ 1 , ̂ 2 . To feasible step 3, a differentiator is used to calculate the derivative of the transformed output of the virtual model ̃ . Finally, the estimation of the fault ̂ is achieved by using the estimated state ̂ , the input of the system , as well as the trasnsformed output ̃ , and its derivative.

Coordinate transformations

The objective of this section is to fulfill the step of coordinate transformations in the fault reconstruction architecture described in Figure 1. This means that some coordinate transformations are introduced in order to obtain the vector state subset ̃ 1 , which is decoupled from faults and the state vector ̃ 2 which is affected by the faults. First, it will be assumed that the following properties are held:

Assumption 3:

( ̄ ) = (| | + 1) × = ̃ Assumption 4: ( ) = (| | + 1) × = ̃ Assumption 5: ( ̄ ) = (| | + 1) × = ̃ Assumption 6:
All the invariant zeros of the matrix triplets ( , ̄ , ) have negative real part. Assumption 3, 4 and 5 are necessary for the feasbiblity of the transformations presented below. Assumption 6 is refered to as the strong detectability condition ( [START_REF] Molinari | A Strong Controllability and Observatbility in Linear Multivariable Control[END_REF], [START_REF] Hautus | Strong detectability and observers[END_REF]), which is a necessary and sufficient requirement for asymptotic estimation of the state vector [START_REF] Bejarano | Unknown Input and State Estimation for Unobservable Systems[END_REF]. These assumptions are quite general and commonly used in the literature (see, e.g., [START_REF] Menon | Robust fault estimation using relative information in linear multi-agent networks[END_REF][START_REF] Laboudi | Simultaneous Estimation of the State and the Unknown Inputs for a Class of Switched Linear Systems[END_REF][START_REF] Li | Cooperative fault estimation for linear multi-agent systems with undirected graphs[END_REF][START_REF] Xu | Distributed Fault Estimation Basing On A United Adaptive and Sliding-Mode Observer for Multi-Agent Systems[END_REF]).

The main results of this section are as follows.

Proposition 1. Due to assumption 4, one can find the non singular matrix [START_REF] Djaidja | Stochastic consensus of leader-following multi-agent systems under additive measurement noises and time-delays[END_REF], then system [START_REF] Ding | Model-based Fault Diagnosis Techniques[END_REF] can be rewritten as (12a), (12b), and (12c).

= ( , 1 , ..., | | ) ∈ ℝ ̃ × ̃ (11) ̇̃ ( ) = ̃ ̃ ( ) + ̃ ( ) + ̃ ( ) (12a)
( ) = ̃ ̃ ( ) (12b)
̃ ( + ) = ( ) -1 , ̃ ( -) (12c
̃ = ( ) -1 = ( ̃ 1 ) , ( ̃ 2 ) , ( ̃ 1 1 ) , ( ̃ 1 2 ) , ..., ( ̃ | | 1 ) , ( ̃ | | 2 ) , ( 13
)
with ̃ 1 ∈ ℝ -, ̃ 2 ∈ ℝ . The matrices ̃ , ̃ , ̃ , and ̃ are given as follows

̃ = ⎛ ⎜ ⎜ ⎝ ̃ ,11 ̃ ,12 ̃ ,21 ̃ ,22 , ̃ 1 ,11 ̃ 1 ,12 ̃ 1 ,21 ̃ 1 ,22
, ...,

⎡ ⎢ ⎢ ⎣ ̃ | | ,11 ̃ | | ,12 ̃ | | ,21 ̃ | | ,22 ⎤ ⎥ ⎥ ⎦ ⎞ ⎟ ⎟ ⎠ , ̃ = ⎛ ⎜ ⎜ ⎝ ̃ ,1 ̃ ,2 , ̃ 1 ,1 ̃ 1 ,2 , … , ⎡ ⎢ ⎢ ⎣ ̃ | | ,1 ̃ | | ,2 ⎤ ⎥ ⎥ ⎦ ⎞ ⎟ ⎟ ⎠ , ̃ = | |+1 ⊗ ̃ , ( 14
)
̃ = | |+1 ⊗ - 1 1 1 1 , … , | | | | | | | | , where ̃ ,11 , … , ̃ | | ,11 ∈ ℝ ( -)×( -) , ̃ ,12 , … , ̃ | | ,12 ∈ ℝ ( -)× , ̃ ,21 , … , ̃ | | ,21 ∈ ℝ ×( -) , ̃ ,22 , … , ̃ | | ,22 ∈ ℝ × , ̃ ,1 , … , ̃ | | ,1 ∈ ℝ ( -)× , and ̃ ,2 , … , ̃ | | ,2 ∈ ℝ × .
PROOF. The proof of this proposition is obvious by pre-multiplying the first equation of ( 10) by ( ) -1 and using ̃ = ( ) -1 . Then, the matrices of (12a) and (12b) are given by

̃ = ( ) -1 , ̃ = ( ) -1 ̃ = ( ) -1 , ̃ = with ( ) -1 = ̃ ,11 ̃ ,12 ̃ ,21 ̃ ,22 , ( ) -1 = ̃ ,1 ̃ ,2
, ( ) -1 = 0 , = , and these other elements in ( 14) are obtained similarly.

Remark 1:

It should be noted that after the transformation in Propostion 1, the sub-states decoupled from faults

̃ 1 , ̃ 1 1 , ̃ 2 1 , … , ̃ | | 1
, and the rest denoted by

̃ 2 , ̃ 1 2 , ̃ 2 2 , … , ̃ | | 2
are interleaved, see equation [START_REF] Gao | A Survey of Fault Diagnosis and Fault -Tolerant Techniques Part I : Fault Diagnosis with Knowledge -Based and Signal-Based Approaches[END_REF]. The following transformation allows us to gather all the states decoupled from faults into one vector state

̃ 1 = [( ̃ 1 ) , ( ̃ 1 1 ) , ( ̃ 2 1 ) , ..., ( ̃ | | 1
) ] , and the rest into another

̃ 2 = [( ̃ 2 ) , ( ̃ 1 2 ) ,( ̃ 2 2 ) ,...,( ̃ | | 2 ) ] .
Proposition 2. The transformation matrix [START_REF] Hashimoto | Distributed 1 -state-and-fault estimation for Multi-agent systems[END_REF] permits to rearrange the component of the decoupled virtual model (12a), (12b), and (12c) and to rewrite them in the form (16a), (16b), and (16c).

= ⎡ ⎢ ⎢ ⎣ | |+1 ⊗ - ( -)× | |+1 ⊗ ×( -) ⎤ ⎥ ⎥ ⎦ (15
)

̇̃ 1 = ̃ ,11 ̃ 1 + ̃ ,12 ̃ 2 + ̃ ,1 (16a) ̇̃ 2 = ̃ ,21 ̃ 1 + ̃ ,22 ̃ 2 + ̃ ,2 + (16b) ̃ 1 ( + ) = , ̃ ( -) (16c)
where

̃ = ̃ with ̃ 1 = [( ̃ 1 ) , ( ̃ 1 1 ) , ( ̃ 2 1 ) , ..., ( ̃ | | 1
) ] and

̃ 2 = [( ̃ 2 ) , ( ̃ 1 2 ) , (
= ( ̃ ,1 , ̃ 1 ,1 , … , ̃ | | ,1 ) and ̃ ,2 = ( ̃ ,2 , ̃ 1 ,2 , … , ̃ | | ,2 ).
PROOF. The proof of (16a) and (16b) in the proposition is obvious by pre-multiplying both sides of (12a) by .

In addition, this coordinate transformation also makes the jump of system states change. From ̃ ( ) = ̃ ( ) and (12c), it implies to ̃ ( + ) = ( ) -1 , -1 ̃ ( -). This is equivalent to:

̃ 1 ( + ) ̃ 2 ( + ) = ̃ -̃ ̃ ( ) -1 , -1 ̃ ( -) By setting , = ̃ -̃ ( ) -1 , -1 = ,1 , ,2 , , (16c)
is obtained.

Remark 2:

In comparison to the impulsive switched system before the coordinate transformations [START_REF] Ding | Model-based Fault Diagnosis Techniques[END_REF], the jump of states at the switching instant in (12c) changes due to the coordinate transformation.

Remark 3: The sub-state ̃ 1 in (16a) is decoupled from the faults , but its dynamics still have the presence of ̃ 2 . To solve this problem, the following proposition is introduced. Proposition 3. Under assumption 5, the non-singular matrix [START_REF] Hespanha | Stability of switched systems with average dwell-time[END_REF] decouples the substate ̃ 1 from the substate ̃ 2 , and the decoupled system [START_REF] Huang | Fault estimation of multi-agent systems with transient performance improvement[END_REF] is obtained.

= | |+1 ⊗ - 1 1 1 , … , | | | | | | (17)
where , … , | | ∈ ℝ ×( -) are arbitrary matrices. Moreover, the decoupled system is given by:

⎧ ⎪ ⎨ ⎪ ⎩ ̇̃ 1 = ̃ ̃ 1 + ̃ ,1 + ̃ ,12 ̃ ,1 ̃ ̃ ,2 ̃ = ̃ ,2 ̃ 1 ̃ 1 ( + ) = , ̃ ( -) (18)
where

̃ ,1 = ( ,1 , 1 ,1 , … , | | ,1 ), ̃ ,2 = ( ,2 , 1 ,2 , … , | | ,2 ), ̃ ,1 = ( 1 , 1 1 1 1 , … , | | 1 | | | | ), (19)
̃ ,2 = ( 2 , 1 2 1 1 , … , | | 2 | | | | ), (̃ ) = [ , -1 , … , -| | ],
and

̃ = ̃ ,11 -̃ ,12 ̃ ,1 .
PROOF. Let us define

-1 = ,1 ,2 , ..., | | | | | | -1 = ⎡ ⎢ ⎢ ⎣ | | ,1 | | ,2 ⎤ ⎥ ⎥ ⎦ .
Then, by applying Lemma 3 (for more detail, see the Appendix 1), it yields the inverse matrix of in [START_REF] Hespanha | Stability of switched systems with average dwell-time[END_REF] as below:

( ) -1 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ,1 … ,2 … 1 ,1 -1 ,1 … 1 ,2 -1 ,2 … ⋮ ⋮ … ⋮ | | ,1 … - | | ,1 | | ,2 … - | | ,2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (20)
From ( 17) and ( 20), one obtains

( ) -1 = ⎛ ⎜ ⎜ ⎝ ,1 ,1 ,2 ,2 , 1 ,1 1 1 1 ,1 1 1
,2

1 1 1 ,2 1 , ..., ⎡ ⎢ ⎢ ⎣ | | ,1 | | | | | | ,1 | | | | ,2 | | | | | | ,2 | | ⎤ ⎥ ⎥ ⎦ ⎞ ⎟ ⎟ ⎠ (21)
We also certaintly have ( ) -1 = . This equality can be rewritten as

( ) -1 = ̃ ⊗ - .
Consequently, it straightforwardly implies that ,1

= ... = | | ,1 | | | | = (22)
and ,2

= ... = | | ,2 | | | | = (23)
Let pre-multiply both sides of (12b) by ( ) -1 , then we have

( ) -1 = ,1 ,1 ,2 ,2 , 1 ,1 1 1 1 ,1 1 1 1 ,2 1 1 1 ,2 1 1 , .. ..., ⎡ ⎢ ⎢ ⎣ | | ,1 | | | | | | ,1 | | | | | | ,2 | | | | | | ,2 | | | | ⎤ ⎥ ⎥ ⎦ ⎞ ⎟ ⎟ ⎠ ̃ ( 24
)
with = [( ) , ( 1 ) , ( 2 ) , ...,

( | | ) ] and ̃ = ( ̃ 1 ) , ( ̃ 2 ) , ( ̃ 1 1 ) , ( ̃ 1 2 ) , ..., ( ̃ | | 1 ) , ( ̃ | |
2 ) . By substituting [START_REF] Li | Distributed Cooperative Fault Detection for Multi-agent Systems: A Mixed ∞ ∕ 2 Optimization Approach[END_REF] and ( 23) into [START_REF] Lin | Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results[END_REF], it yields

( ) -1 = ⎛ ⎜ ⎜ ⎝ ,1 ,2 , 1 ,1 1 1 1 ,2 1 1 , ..., ⎡ ⎢ ⎢ ⎣ | | ,1 | | | | | | ,2 | | | | ⎤ ⎥ ⎥ ⎦ ⎞ ⎟ ⎟ ⎠ ̃ ( 25
)
) ] . This means that the Step 1 introduced in subsection 3.3 is completed, and the two next steps are carried out in the next section: an observer is designed for the agent in each mode in order to obtain the sub-state estimation ̂ 1 , calculate the entire state estimation ̂ , and then estimates the faults.

Switched observer design

To estimate the sub-state ̃ 1 we propose to design a switched observer by taking into account LMI region. Afterthat, the sub-state ̃ 2 is calculated. Then, a fault estimator based on the state estimation ̂ = [ ̂ 1 ; ̂ 2 ] and the sliding mode technique is introduced.

Let us consider the observer as follow:

⎧ ⎪ ⎨ ⎪ ⎩ ̇̂ 1 = ̃ ̂ 1 + ̃ ,1 + ̃ ,12 ̃ ,1 ̃ + ( ̃ ,2 ̃ -̃ ,2 ̂ 1 ) ̂ 1 = ̃ ,2 ̂ 1 ̂ 1 ( + ) = , ̂ ( -) (28)
where denote the observer gains. By setting

̂ = ̃ - ̃ ,2 and using , = ,1 , ,2 , , (28) becomes:
⎧ ⎪ ⎨ ⎪ ⎩ ̇̂ 1 = ̂ ̂ 1 + ( ̃ ,12 ̃ ,1 + ̃ ,2 )̃ + ̃ ,1 ̂ 1 = ̃ ,2 ̂ 1 ̂ 1 ( + ) = ,1 , ̂ 1 ( -) + ,2 , ̂ 2 ( -) (29)
Once the estimation ̂ 1 is obtained and based on [START_REF] Meskin | Fault Detection and Isolation of discrete-time Markovian jump linear systems with application to a network of multi-agent systems having imperfect communication channels[END_REF], the estimated value of the sub-state ̃ 2 is calculated by [START_REF] Messai | Fault Estimation for for a Class of of Switched Systems[END_REF].

̂ 2 = ̃ ,1 ̃ -̃ ,1 ̂ 1 , ( 30
)
and

̂ = [ ̂ 1 ; ̂ 2 ]
is the entire state estimation.

Let the estimation error for the sub-state ̂ 1 be defined as

1 = ̃ 1 -̂ 1 ( 31
)
where ̃ 1 and ̂ 1 are recalled from the dynamics ( 18) and ( 28) as follows:

̇̃ 1 = ̃ ̃ 1 + ̃ ,1 + ̃ ,12 ̃ ,1 ̃ ̇̂ 1 = ̃ ̂ 1 + ̃ ,1 + ̃ ,12 ̃ ,1 ̃ + ̃ ,2 ̃ -̃ ,2 ̂ 1
Then, the derivative of the estimation error is taken as:

̇ 1 = ̇̃ 1 -̇̂ 1 = ̃ 1 - ̃ ,2 ̃ -̃ ,2 ̂ 1 (32)
From (27b), we have ̃ ,2 ̃ = ̃ ,2 ̃ 1 , and then, let us substitude it into the above equation, one obtains:

̇ 1 = ̃ 1 - ̃ ,2 ̃ 1 -̃ ,2 ̂ 1 = ̃ - ̃ ,2 ̃ 1 -̂ 1 ( 33
)
Finally, by using the notation ̂ = ̃ -̃ ,2 , we obtain the estimation error dynamic as follows:

̇ 1 = ̂ 1 1 ( + ) = ̂ , 1 ( -) (31)
where

̂ , = ,1 , -,2 , ̃ ,1
. The problem now is to find the matrices in order to guarantee that the estimation error [START_REF] Molinari | A Strong Controllability and Observatbility in Linear Multivariable Control[END_REF] is stable. In the following theorem, the sufficient conditions for the design of a switched state estimation observer under the switching constraint satisfying average dwell time are given. Moreover, in order to enhance the performance of the observer, one can implement the pole locations approach mentioned in Lemma 1, in which the eigenvalues of the matrix ̂ should be in an LMI region. One can propose the region corresponding to the desired performance given in Figure 2. This region is an intersection of two regions 1 and 2 . 1 is defined as a vertical band with the center at = and 2 width. 2 is a horizontal band with the center at = 0 and 2 width. Then, one can find:

1 = -2 -2 -2 -2 , 1 = 0 1 1 0 , 2 = -2 0 0 -2 , 2 = 0 1 -1 0 . ( 32
)
With the above description of the LMI region = 1 ∩ 2 , the following theorem is proposed.

Theorem 4. Consider the switched model [START_REF] Huang | Fault estimation of multi-agent systems with transient performance improvement[END_REF], the switched observer [START_REF] Meskin | Fault Detection and Isolation of discrete-time Markovian jump linear systems with application to a network of multi-agent systems having imperfect communication channels[END_REF] and let the three parameters > 0, > 0, and < 0 be chosen. If there exist symmetric matrices > 0, matrices , such that:

( ̃ ) + ̃ -( ̃ ,2 ) -( ) ̃ ,2 -2 ≤ 0 (33) ̂ , ⋆ ≥ 0 ( 34
)
⎡ ⎢ ⎢ ⎢ ⎢ ⎣ -2 Φ ,1 0 0 ( * ) -2 0 0 0 0 -2 Φ ,2 0 0 -Φ ,2 -2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ < 0 (35)
, ∈ {1, 2, ..., }, ≠ where

Φ ,1 = -2 + ( ̃ ) + ̃ -( ̃ ,2 ) -( ) ̃ ,2 Φ ,2 = ( ̃ ) -̃ -( ̃ ,2 ) + ( ) ̃ ,2
Then, estimation error dynamic ( 31) is asymptotically stable with any switching signal verifying [START_REF] Bernstein | Matrix mathematics[END_REF]. Moreover, the observer gain is given by = ( ) -1 ( ) and the eigenvalues of the observer are inside the defined LMI regions .

PROOF. To study the convergence of the estimation error, let us define the following multiple Lyapunov function for any two successive modes " " and " ":

1 ( ) = ( 1 ( )) 1 ( ) ( 36
) 1 ( ) = ( 1 ( )) 1 ( ) (37)
To guarantee that the estimation error (31) converges exponentially to zero and satisfies the constraint of average dwell time (4) with given constants < 0 and > 1, the following conditions are given [START_REF] Mahmoudi | On observer design for a class of impulsive switched systems[END_REF]:

̇ 1 ( ) ≤ 2 1 ( ) (38)
1 ( + ) ≤ 1 ( -) (39)
The time derivative of the Lyapunov candidate function ( 36) is:

̇ ( 1 ( )) = ( ̇ 1 ( )) 1 ( ) + ( 1 ( )) ̇ 1 ( ) (40)
Then, ( 38) is equivalent to:

( ̂ ) + ̂ -2 ≤ 0 (41)
By setting = ( ) , one can obtain [START_REF] Oh | A survey of multi-agent formation control[END_REF]. Next, let us consider [START_REF] Shi | Some novel approaches on state estimation of delayed neural networks[END_REF], it leads to

( 1 ( + )) 1 ( + ) ≤ ( 1 ( -)) 1 ( -) (42)
or

̂ , ̂ , - ≤ 0 (43)
by using Schur complement, (34) is proved.

From Lemma 1, the matrix ̂ isif and only if there exists a symmetric matrix > 0 and a matrix such that:

( ̂ , ) < 0 (44)
Implement the Lemma 2 with:

1 = 1 ⊗ + 1 ⊗ ̂ , + 1 ⊗ ̂ , < 0 (45)
2 = 2 ⊗ + 2 ⊗ ̂ , + 2 ⊗ ̂ , < 0 (46)
where 1 , 2 , 1 , and 2 are introduced in (32). After some manipulations, one can obtain the inequality [START_REF] Patton | A generic strategy for fault-tolerance in control systems distributed over a network[END_REF]. This ends the proof.

Remark 5: Condition ( 33) and ( 34) are used to guarantee exponential convergence of the global error dynamics (31) which also satifies the constraint for average dwell time.

Remark 6: Regional pole placement condition [START_REF] Patton | A generic strategy for fault-tolerance in control systems distributed over a network[END_REF] can enhance the stability performance of the switched system [START_REF] Molinari | A Strong Controllability and Observatbility in Linear Multivariable Control[END_REF]. The region is given in the left-half complex plane, this means that the parameters chosen must be > 0, > 0, and ( + ) < 0.

Remark 7: For each agent, matrices and can be obtained, if conditions of Theorem (4) are satisfied. Then the observer gain is given by = ( ) -1 ( ) .

Fault estimator

According to the fault reconstruction methodology described in subsection 3.3, until now, step 1 and step 2 have been already completed. The state estimation is obtained, but the problem of faults estimation has not been dealt with yet. From (16a) and (16b), it is straightforward to see that the estimation of the faults can be attained from the state estimation ̂ 1 and ̂ 2 . This can be formulated through the following proposition:

Proposition 4. Consider the system (12a), the reduced system [START_REF] Huang | Fault estimation of multi-agent systems with transient performance improvement[END_REF], and the observer gains obtained from theorem 4, then the following estimator ensures the asymptotic reconstruction of the faults:

̂ = ̃ ,1 ̇̃ + ,1 ̂ 1 + ,2 ̃ + ,3 (47)
where

,1 = -̃ ,1 ̃ - ̃ ,2 -̃ ,21 -̃ ,22 ̃ ,1 , ,2 = -̃ ,1 ̃ ,12 ̃ ,1 + ̃ ,2 -̃ ,22 ̃ ,1 , ,3 = -̃ ,1 ̃ ,1 -̃ ,2 .
PROOF. Under the conditions ( 33), ( 34) and ( 35) of Theorem 4, and from (16b), the faults ( ) can be estimated from the values of ̂ 1 ( ) and ̂ 2 ( )as follow:

̂ = ̇̂ 2 -̃ ,21 ̂ 1 -̃ ,22 ̂ 2 -̃ ,2 (48)
By using [START_REF] Messai | Fault Estimation for for a Class of of Switched Systems[END_REF] and its derivative, after some manipulations, ( 47) is proved.

To obtain the estimation of faults in [START_REF] Zhang | Distributed fault estimation observer design for multi-agent systems with switching topologies[END_REF], it is required to calculate the derivative of the output ̃ . This problem is handled by using a high-order robust exact differentiator based on sliding mode techniques [START_REF] Levant | Higher-order sliding modes, differentiation and output-feedback control[END_REF]. Let us introduce the following ℎ -order differentiator of which the signal ̃ is the input and the recoverd derivatives , = 0, 1, ..., are the outputs.

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ̇ 0 = 0 0 = -0 | | | 0 -̃ | | | ∕( +1) ( 1 -̃ ) + 1 1 = 1 1 = -1 | | | 1 -0 | | | ∕( +1) ( 1 -0 ) + 2 . . . ̇ -1 = -1 -1 = --1 | | | -1 --2 | | | 1∕2 ( ̇ -1 --2 ) + ̇ = - ( --1 ) (49)
where are chosen differentiator parameters. The estimation ( 49) is initialized such that: 0 (0) = ̃ (0), (0) = -1 (0) = 0. For more detail about the used differentiator, the reader is invited to see [START_REF] Levant | Higher-order sliding modes, differentiation and output-feedback control[END_REF].

Numerical example

In this subsection, some simulation results are provided to verify the theoretical proposed approach.

Example 1

Let us consider a team of four aircraft dynamical systems borrowed from [START_REF] Zhang | Adaptive technique-based distributed fault estimation observer design for multi-Agent systems with directed graphs[END_REF] and [START_REF] Zhang | Observer-based fault estimation and accomodation for dynamic systems[END_REF]. These dynamics satisfy the assumptions 1 and 2. Without loss of generality, let us consider that agent 1 and agent 2 have the same dynamics, so do the agent 3 and agent 4. These linearized models are given by:

1 = ⎡ ⎢ ⎢ ⎢ ⎣
-0.0366 0.0271 0.0188 -0.4555 0.0480 -1.0100 0.0024 -4.0208 0.1002 0.3681 -0.7070 1.4200 0.0000 0.0000 1.0000 0.0000

⎤ ⎥ ⎥ ⎥ ⎦ , 2 = 1 ,
⎤ ⎥ ⎥ ⎥ ⎦ , 2 = 1 3 = ⎡ ⎢ ⎢ ⎢ ⎣
-1.5539 0.0154 0.0000 1.3284 -0.0398 -0.0007 0.0000 0.0000

⎤ ⎥ ⎥ ⎥ ⎦ , 4 = 3 , 1 = ⎡ ⎢ ⎢ ⎣ 1 0 0 0 0 1 0 0 0 0 1 0 ⎤ ⎥ ⎥ ⎦ , 2 = 3 = 4 = 1 , 1 = ⎡ ⎢ ⎢ ⎣ 0 0 1 0 0 0 ⎤ ⎥ ⎥ ⎦ , 2 = 3 = 4 = 1 .
The system switches among three topologies according to the sequence depicted in Figure 4. Note here that the chosen communication topologies correspond to the case when an UAV keeps the connection with a neighbor that is still in its communication range and in the same time interacts with a new agent in its vicinity. So, for each agent, its virtual model has three transition matrices, and their virtual models satisfy the assumptions 3 to 6. For the simulation task, because the role of all agents in the system is equal, let consider the effectiveness of the fault estimation observer design on the agent 1. Assume that the transition matrices for the virtual model of agent 1 at the switching instants are as follows: With the given = 1.6 and = -0.17, this switching signal satisfies the constraint of average dwell time > = 1.6∕(2.0.17) = 1.384( ). The simulated switching signal respecting this constraint of average dwell-time is given in Figure 3. The meaning of Figure 3 is that when its values are 1, 2, and 3, the corresponding respectively information graph are topology I, II, and III in Figure 4. To illustrate the performance of the proposed methodology, multiple faults 1 ( ) and 4 ( ) ∈ ℝ 2 are assumed to occur in agent 1 and 4, respectively. These faults are shown in Figure 5, where the multi-shape signal is the second element of fault 1 ( ), and the signal 4 1 ( ) = 3 10 is the first element of fault occuring on agent 4.

Under the region with = 4, = -5, and = 8, by solving the LMI in Theorem 4, it yields the observer gains as follow:

1 1 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ -34
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .
From the output ̃ , a five order differentiator is employed. The following parameters are used: { } 5 =0 = 30, 25, 20, 15, 10, 5.

The simulation results are shown in Fig. 6 -Fig. 8. The time evolution of the estimation error 1 1 = ̃ 1 1 -̂ 1 1 is presented in Fig. 6. Simulation results of fault estimation are shown in Fig. 7 and Fig. 8, where 1 ( ), ̂ 1 ( ) with = 1, 2, 3 are the faults affect on the system in the point of view of agent 1 and the faults estimated by agent 1, respectively. Even though there exist some chattering phenomena at the switching instants due to the change of the dynamics and transition matrices at these instants, Fig. 7 shows that the estimated fault converges to the actual ones in agent 1. Fig. 8 indicates that the fault on agent 4 is able to be estimated by the observer on agent 1 only when they are neighbors (mode 2 and mode 3).

The following example is presented o evaluate the effectiveness of our proposed approach in the scenario that the fault estimators are set up on several agents in a multi-agent system.

Example 2

We consider a leader-follower multi-agent system which is modified from the example used in [START_REF] Peng | A novel optimal bipartite consensus control scheme for unknown multi-agent systems via model-free reinforcement learning[END_REF] to illustrate to their proposed of bipartite consensus scheme. This muli-agent system has 1 leader and 7 followers and its communica-

= -2 0 0 -4 , 2 = 5 = 7 = 1 , 4 = -5 0 0 -3 , 3 = 6 = 4 , 1 = 1 0 , 4 = 6 = 7 = 1 , 2 = 0.75 0.1 , 3 = 5 = 2 , 1 = 1 0 0.1 0.1 , 2 = 3 = 4 = 5 = 6 = 7 = 1 , 1 = 0 1 , 2 = 5 = 7 = 1 , 3 = 1 0.5 , 5 = 1 0 .
In this example, we consider the scenario that the followers are divided into two groups. Agent 1 and agent 4 are the monitor agents in their group. In order to estimate the faults occurring on the agents in each group, the fault estimation modules are set up on the monitor agents. In addition, there always exists one connection between the monitor agent of a group and an agent of the other group. The system alternately switches between two topologies and , which are depicted in Fig. 10. For each agent, it virtual models satisfy assumption 3 -6 and the transition matrices are chosen as follows: The other simulation parameters are 1 = 1.8, 1 = 0.15, 4 = 1.5, 4 = -0.12, LMI regions 1 , , = 4 , , = -5, 4, 8. The faults are assumed to occur in agent 1, agent 2, and agent 6 as depicted in Fig. 11 . Then, the observer gains obtained by solving the LMI in Theorem 4 are: 12 and the simulation results of fault estimation are epicted in Fig. 13 and Fig. 13. It is observered that the effectiveness of our proposed approach is still achieved in this example when the estimation errors converge to zero and the estimate faults converge to the actual fault.

1 12 = ⎡ ⎢ ⎢ ⎣ 0 0 0 1.05 0 0 0 0.97 ⎤ ⎥ ⎥ ⎦ , 1 21 = ⎡ ⎢ ⎢ ⎣ 0 0 0 1.02 0 0 0 0.95 ⎤ ⎥ ⎥ ⎦ , 0 2

Conclusions and Perspectives

This paper addresses the problem of fault estimation for a class of network of a non-homogeneous multi-agent system subject to switching topologies. By using the "virtual" model to represent the model of each agent, the information of their neighbors is included. Through some coordinate transformations and with some classical hypothesis, a sub-state of the system is decoupled from faults. An impulsive switched observer has been proposed to estimate the state and reconstruct the fault. Sufficient condition for the solvability of the observer design is obtained based on the LMI feasibility conditions. The average dwell-time-based design is introduced to guarantee the stability of the proposed observer approach.

Moreover, the LMI region approach was exploited to enhance the observer performances. Then, the fault estimations are obtained by simultaneously using state estimations and a robust exact differentiator. Finally, a numerical example has been presented to verify the efficiency of the proposed approach. It illustrates that the proposed method can be used successfully for the fault estimation problem in a network of a non-homogeneous multi-agent system.

This proposed approach can be prospectively implemented in multi-robot systems, a network of sensors, or a group of unmanned vehicles such as UAVs given in the example. In these implementations, the network switches among some known topologies and an agent can estimate not only its faults but also its neighbors' ones. This information can be useful for the problem of consensus with the presence of faults, which we will consider in the future.

A few challenges which restrict the implementation of our proposed approach in some cases in practice are also recognized. Therefore, in the future, we aim to relax some constraints which are expressed as the assumptions in this paper. For example, the problem of dimension changing which is corresponding to the case of changing the number of each agent's neighbors when the communication graph switches from topology to the others. We also desire to find the actual transition matrices which are pre-assigned in this paper, even though the chosen parameters of these matrices affect only the transient response of the system after each switching instant. Moreover, solving the problem of FDI and FE with heterogeneous multi-agent systems subjects to modeling error and disturbances are also the subject of our study.

Appendix 1

The matrix U in (17) can be rewritten as follow:

U = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ … -1 1 -1 … ⋮ ⋮ ⋮ ⋮ … ⋮ ⋮ … - | | | | - | | ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (A1.1)
In order to obtain (U ) -1 , let us use the following notations: = , 1 = -1 1 -1 ,..., and

= - | | | | - | |
. Then, the matrix U becomes:

U = ⎡ ⎢ ⎢ ⎢ ⎣ … 1 … ⋮ ⋮ … ⋮ … ⎤ ⎥ ⎥ ⎥ ⎦ (A1.2)
By using Lemma 3, we can have:

(U ) -1 = ⎡ ⎢ ⎢ ⎢ ⎣ -1 … --1 1 -1 -1 1 … ⋮ ⋮ … ⋮ --1 -1 -1 … -1 ⎤ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎣ -1 … --1 1 -1 1 … ⋮ ⋮ … ⋮ --1 … -1 ⎤ ⎥ ⎥ ⎥ ⎦ (A1.3) With -1 = ,1 ,2
, -1 1 = -

Figure 1 :

 1 Figure 1: Fault reconstruction architecture

∈

  ℝ × , and , ..., | | are arbitrary matrices such that is non singular,

Figure 2 :

 2 Figure 2: The LMI region

AmplitudeFigure 3 :

 3 Figure 3: The switching signal

Figure 4 :

 4 Figure 4: The information graphs

Figure 5 :Figure 6 :

 56 Figure 5: The faults effect on agent 1 (upper) and agent 4 (lower)

Figure 7 :

 7 Figure 7: The actual fault 1 1 2 ( ) and the estimated fault ̂ 1 1 2 ( )

Figure 8 :

 8 Figure 8: The actual fault 1 3 1 ( ) and the estimated fault ̂ 1 3 1 ( )

Figure 9 :

 9 Figure 9: The information graph

1

 1

AmplitudeFigure 10 : 2 Figure 11 :

 10211 Figure 10: The switching signal

Figure 12 :Figure 13 :Fig. 12 -

 121312 Figure 12: The estimation errors 1 1 = ̃ 1 1 -̂ 1 1 and 4 1 = ̃ 4 1 -̂ 4 1

Figure 14 :

 14 Figure 14: The actual fault 4 ( ) and the estimated fault ̂ 4 ( )

  retain (U ) -1 in[START_REF] Levant | Higher-order sliding modes, differentiation and output-feedback control[END_REF].

which is equivalent to:

By using the notations [START_REF] Laboudi | Simultaneous Estimation of the State and the Unknown Inputs for a Class of Switched Linear Systems[END_REF], the equation ( 26) can be represented in its compact form as follow:

Then, by substituting (27a) into (16a) and from (27b), (16c), the dynamics of sub-state ̃ 1 in ( 18) is obtained.

Remark 4: Finally, there is no more the presence of the sub-state ̃ 2 in the dynamics of the sub-state ̃ 1 , which is decoupled from faults. In the other words, the vector state = [( ) , ( 1 ) , ( 2 ) , ..., ( | | ) ] is decomposed into two transformed sub-states ̃ 1 = [( ̃ 1 ) , ( ̃ 1 1 ) , ( ̃ 2 1 ) , ..., ( ̃