A Collective Biography of an Emerging Professional Group Chemists Funded by the Caisse Nationale des Sciences (National Fund for Sciences) in the 1930s

Martine Sonnet1

In France, since 1901, the Caisse des Recherches Scientifiques (Fund for Scientific Research) funds, with a very limited budget, laboratory equipment and scientific publications, but does As early as 1931, the CNS set up four levels of renewable grants, which could last for one, three or five years. These could be full grants for scientists without any other paid employment, or partial grants (a half or a quarter), awarded according to personal employment positions. Definitions and stipends of the three highest levels represent the academic ranks of Head of Works, Lecturers and Professors. Nevertheless, the CNS only offered precarious positions with no social protection. The CNS grade scale includes:

• Fellowships (bourses): to give an opportunity to beginners in research who are not yet graduated;

• Grants for Researchers (Chargés de Recherche): for researchers who had successfully demonstrated professional ability and skills, and were expected to later devote their careers to science;

• Grants for Heads of Research (Maîtres de Recherche): for distinguished scholars with authoritative publications;

• Grants for Senior Researchers (Directeurs de Recherche): for the same profile as the previous, but even more distinguished and authoritative -usually older.

Scientists awarded grants were required to submit annual activity reports to the disciplinary scientific committees, which decided on the renewals and examine new candidates. Field committees were set up for Mathematics, Physics, Chemistry, Biology, and To fully assess the place of Chemists, their number must be related with the status they had at the time of recruitment. When joining the CNS, 88% of the male and 92% of the female Chemists received only a fellowship grant. In comparison, those number fall to 73% male and 90% female scholars in the other areas combined. The figures fall to 56% for male Mathematicians. Chemists seem to be the most disadvantaged starters, and it is unsurprisingly worse for women. Only three women appear among the 21 Chemists who were initially granted with more than a fellowship grant. Of the 18 men, 12 were Researchers, two were Head of Research and four were seniors 9 . 77% of the male and 73% of the female Chemists received a reduced grant versus 71% and 69% in other areas 10 . Regardless of the area, women more often received full grants, but this small relative advantage also shows that women face more difficulties when trying to get employment outside of the CNS.

Table 2 compares the ages of recruitment for each level 11 in order to determine if the Chemists' higher rate of fellowships could reflect their youth. As can be seen, fellowships in Chemistry are presented one or two years earlier than those of their colleagues 12 . Conversely, male Chemists recruited at a higher level were older than other scholars at the same stage of their career. This observation suggests a specific career path in Chemistry. Furthermore, 9 Joseph Bougault, Robert Lespieau, Paul Sabatier and Jean-Baptiste Senderens. 10 Based on the available amount of grants for 142 of the male and 33 of the female Chemists, and 518 of the men and 89 of the women from other areas. 11 The years of birth, like all the biographical information, have been mainly collected from archives listed in the notes above 5, 6, 7. General, scientific and academic biographical dictionaries and yearbooks, catalogs of libraries, reviews tables and indexes, and "Google ocean", have provided complementary identification (especially genealogical websites). Birth years are known for 37/37 of the female and 150/155 of the male Chemists, and for 95/97 women and 527/565 men from other areas. Median ages were not calculated for groups that had less than 10 scholars.

12 Only in Mathematics new fellowship beneficiaries are younger, with an average age of 27 for men; the only two female mathematicians arrived at 28 (Marie-Louise Dubreil-Jacotin) and 29 (Marie Charpentier).

Table 2 shows that, overall, female careers were delayed, and even more so when considering that men spent at least one year fulfilling their military obligations. Even though a PhD, even in progress, was not compulsory, 79% of the female and 61%

of the male Chemists advocated getting one before or during the year that follows their recruitment. In the long term, 89% of the women and 88 % of the men got one. In other areas, 85% of the women, the same rate as men 14 , would obtain this degree at some point, but only 59% among the women and 62% among the men would get it as early as the Chemists. The relative youth of Chemists does not mean that they had a lower degree than their colleagues, and regarding women the opposite was actually true.

Apart from having a doctorate, or instead of it, 44% of the male and 16% of the female CNS Chemists were Chemical engineers; 13% of the men and 11% of the women were Pharmacists; and 14% of the men and 5% of the women had the secondary school agrégation before being recruited. The agrégation was more frequent in other areas: 32% male and 9% female who were scholars from other areas had it. Among the men, at least, the presence of alumni from the Ecole normale supérieure was lower in Chemistry (7%) than in other areas (17%). This figure, considered along with the relatively strong presence of engineers among them, places the Chemists' cohort on the fringe of the most prestigious academic path, thus confirming their specific career path.

Once recruited, Chemists tended to leave their position in the CNS earlier and more frequently than their colleagues from other areas (Table 3) 15 . 13 The unique case of Marcelle de Heredia Lapicque. 14 Based on all CNS female non-Chemists (98) and on a representative sample of 105 male non Chemists. Considering the length of service and both sexes, Chemists remained proportionally less than their colleagues. This trend is particularly strong after the regular four years in the CNS, which was the normal duration required to become a Researcher. Nevertheless, the high departure rate among Chemists at this stage cannot be explained by a lack of promotions.

After 5 years as fellowship beneficiaries, the five female Chemists represented got a

Researcher grant, as well as 33 of the 38 men; in other areas the rate of promotion is lower for women -23/29 -and slightly lower for men: 81/97. Consequently, the final gap between Chemists' and other scholars' length of service in the CNS seems mainly due to a labour market that offers many opportunities in the chemical industry. For non-Chemists, another remarkable point in Table 3 is a constant relative female over-representation: this trend confirms that scientific graduate women, under-recruited in universities, were especially available for long-term CNS grants.

Robot-portraits of typical CNS Chemists

After assessing the Chemists' place in the CNS, I shall briefly summarize the main aspects of their profiles and the balance between their professional/personal lives through robotportraits, based upon the cohort's prevailing statistical features. The female Chemist shall be named Suzanne and her male colleague Pierre, the most frequently encountered first names among them. We already know that both Suzanne and Pierre have joined CNS by age 30 in order to receive half-fellowship grants, having already defended their PhD or being very close to doing so, and that they will stay there for four years.

Suzanne and Pierre are born in mainland France, like 78% of female and 88% of male CNS Chemists and, respectively, 79% and 81% of their colleagues 16 . Among all scholars, male Chemists represent the lowest rate of scholars originating from Colonial and Overseas France (2%) and from foreign countries (10%). Like 38% of the female Chemists born in mainland France, Suzanne is born in either Paris or its suburbs and Pierre, as 38% of the male Chemists, in a small town 17 or village -as only 28% of them were born in Paris or its suburbs. Being a Parisian seems to be an asset for women. Regardless of their place of birth, of the known personal addresses, 30/39 of the women and 63/88 of the men are located there.

The proximity between their home and the laboratory makes it easier to balance family and work, particularly when some experiments need constant monitoring.

This issue of balance is all the more important since Pierre -like three-quarters of the male CNS Chemists, and Suzanne -like half of the female ones, will get married at least once in their lives 21 . Pierre is already married or will be married during the first year of his grant (based on 53 of 83 men with a documented wedding year) but it's not so obvious for Suzanne: only 9 out of18 of the female Chemists experience marriage so early. Among the married CNS Chemists, at least three-quarters of the men and half of the women already have or will have children 22 . Therefore, Pierre and Suzanne can be both considered parents, but with significantly different family sizes. Two children at most for Suzanne, but three, four or more for Pierre; in both cases, the first one is born within the first year of marriage.

The smaller size of Suzanne's family is linked with the fact that she keeps working after childbirth, whereas Pierre's wife could be a housewife. The marriage pool for Pierre includes unemployed women, while Suzanne's sustained commitment to a scientific career leads her to choose her husband from a pool of scientists, researchers or teachers. This is the case for 11 out of 15 of the married CNS female Chemists whose husbands' occupation is documented. Among these 11, five of them are even married to a CNS colleague. For the 117 married men in the cohort, the occupation of only 29 of their wives is known, which suggests a low rate of employment for the wives. However, as shown from many of the male scholar obituaries, unemployment does not necessarily mean a lack of unpaid scientific cooperation.

The 29 wives with identified occupations reflect a strong professional endogamy: if Pierre's wife happens to have a paid job, she is likely to work in chemical research (public rather than industrial) or teaching, and they might have met while they were students.

Endogamy prevails in the small world of CNS scholars, but Chemistry seems to be a little more open from the point of view of its social recruitment23 . If Suzanne and Pierre, like 15%

of CNS Chemists, have a father -and a mother if ever she had a profession -leading a teaching career (regardless of level), it corresponds to the case of about 30% of their colleagues in other areas. Half among Chemists also reduce the second largest group of paternal professions -Physicians/Pharmacists/Veterinarians: 10% versus 20%. The social origin of the Chemists is consequently more diverse, particularly with parents coming from some professions such as engineers, or jobs connected with commercial and manufacturing activities, and also from lower classes (farmers or craftsmen, for example). For the CNS, this confirms observations about academics in science faculties, where Chemists seem to be « new men »24 when compared with scholars from other scientific disciplines -particularly Mathematics and Physics, the most socially elitist. What is more, in the CNS, 1/5th of these new men are women!

  not directly fund researchers through grants. The Caisse Nationale des Sciences (National Fund for Sciences, hereafter referred to as CNS), created in 1930, reflected a new political will of the State to provide human resources for public research outside of the academic system. The CNS rewarded the efforts of prestigious scientists 2 , encouraging the French Government to align itself with the German Kaiser-Wilhelm-Gesellschaft and the Belgian Fonds national de la recherche scientifique. In 1935, the Caisse des Recherches Scientifiques merged with the CNS into the Caisse Nationale des Recherches Scientifiques. The first leftwing French Government of the Front Populaire then established an Under-State Secretariat for Scientific Research 3 , which would provide a final decisive impulse before the existent CNRS (Centre National de la Recherche Scientifique) was created, in September 1939. This article shall focus on the two Caisses, from 1931/32 to 1938/39.

  Suzanne and Pierre get their doctorates in Paris, like 85% of the female and 74% of the male CNS Doctors in Chemistry. Chemical Engineering Degrees are less centralized: only 59% of male Engineers are graduates of the Parisian Chemical School or Institute 18 , the others coming mainly from Strasbourg, Lille, Mulhouse and Nancy; some of them later defend a PhD in Paris. Suzanne and Pierre work in a Parisian laboratory, like 81% of the female and 69% of the male CNS Chemists -women again being the most Parisian. Suzanne is based in the Science Faculty, like 17/28 of Parisian female Chemists, compared with only 28/103 of her male colleagues. Pierre gets other opportunities, at the Collège de France (13/103) or at the Pharmacy Faculty (8/103) 19 for example. Receiving partial grants, Suzanne and Pierre complement their income, most likely as Assistants (30 of the male and 8 of the female Chemists) rather than as Préparateurs (20 and 3) or Chefs de Travaux (15 and 5). Only male CNS Chemists are also secondary teachers (6) or higher education professors, active (4) or retired (4). Working in Paris, Suzanne and Pierre live on the left bank of the Seine, where the Science Faculty and major scientific institutes are located. Almost three-quarters of Parisian CNS Chemists live in the 5th arrondissement and its boundaries with the 6th, 13th and 14th 20 :

place for Chemists in the CNS?

  Natural Sciences 4 . In 1935, new rules established two new specific committees, one for Mechanics, Statistics and Astronomy, and another for Medicine.

	What This text is based on the documentation produced each year by the CNS 5 , (lists of
	beneficiaries and activity reports), completed for 1938/1939 by the census of scientific
	personnel, drawn up to prepare scientific mobilisation in a pre-war context 6 , and on the
	CNRS 7 archives. It was possible to reconstitute the scientific population funded by the CNS
	between 1931/1932 and 1938/1939: 135 women and 715 men in the area of the Exact
	Sciences 8 , which included 37 female and 155 male Chemists. Scholars who belonged to the
	area of Chemistry are referred to as Chemists, with no regard for the specific nature of their
	works. This article shall focus on the individual researchers, their living conditions and career
	development.
	Inside the CNS's Exact Sciences cohort, Chemists account for the largest group, and
	second largest in terms of the number of women rate (Table 1). The 16% women average rate
	in the Exact Sciences -with a gap between Mathematics and Biology or Chemistry -, is
	higher than in the Humanities and Social Sciences, where it only reaches 12%.

Table 1 .

 1 Scholars funded by the CNS between 1931/1932 and 1938/1939: Distribution of disciplines and number of women rates.

	Areas	Total number of scholars		Women	Men
			Number	Women rates	Number
	Chemistry	192	37	19%	155
	Physics	179	17	9%	162
	Natural Sciences	166	28	17%	138
	Biology	150	40	27%	110
	Mathematics	71	2	3%	69
	Medicine	51	8	16%	43
	Mechanics, Statistics and	41	3	7%	38
	Astronomy
	Total in the Exact Sciences	850	135	16%	715

Table 2 .

 2 Average and median ages of scholars at recruitment by the CNS

			Chemists			Non Chemists
		Women	Men		Women	Men
		Average	Median	Average	Median	Average	Median	Average	Median
	Fellowships	30,5	29	29,5	28	31,5	30	31,5	30
	Researchers	42,5	-	43	40	44	-	40,5	37
	Heads of R.			56	-	65 13		48	47
	Senior Res.			73,5				66,5	71

Table 3 .

 3 The length of service of Chemists and other scholars in the CNS.

	Time at the CNS	Chemists		Other scholars
		Women	Men	Women	Men
	At least 3 years	83%	78%	85%	81%
	At least 4 years	67%	65%	73%	67%
	At least 5 years	39%	47%	71%	55%
	At the CNS for at least 6 years	32%	29%	67%	51%

  Préfecture nor a Sous-Préfecture. 18 Ecole Supérieure de Physique et de Chimie Industrielles de Paris (ESPCIP) or Institut de Chimie de Paris (ICP). 19 A few female or male CNS Chemists are also hosted by the Institut de Biologie Physico-Chimique, the Ecole des Hautes Etudes, Medicine and Pharmacy Faculties, the Conservatoire National des Arts et Métiers or the Muséum National d'Histoire Naturelle.

17 Neither a

20

Including the Montagne Sainte-Geneviève and the Quartier Latin, where housing is financially impossible for young scholars today.

21

Marriages are confirmed for 117/155 of the male and 18/37 of the female CNS Chemists. Meanwhile almost 1/3 of them remain definitively single (13/37).

IHMC CNRS/ENS/PARIS I

Mainly Jean Perrin, André Mayer and Marie Curie.

The Front Populaire would promote both scientific research and women's role in it, appointing Irene Joliot-Curie as the first Under-State Secretary (5 June -28 September 1936).

Humanities and Social Sciences included only two committees, Historical and Philological Sciences, Philosophical and Social Sciences, which were later subdivided.

Mainly Archives Nationales, F17 13331 and F17 17458-17461.

Archives Nationales, 1980284/24-27.

Mainly personal files, Archives Nationales 20070296.

In addition 29 women and 225 men in Humanities and Social Sciences, a relatively undeveloped field inside CNS.

Based on 37/37 identified women's birthplaces and 139/155 for men in Chemistry; for the other areas, 90/97 and 484/565, respectively.

The male rate is probably undervalued because, at this stage of the study, biographical information of men is less comprehensive than of women.

Based on the paternal professions (and very few maternal ones) that are known for 58 of the male and 14 of the female CNS Chemists and, unfortunately, for a very small sample of 26 fellowships belonging to other areas. Such a small group of women represented in this topic makes us consider women and men together, which needs to be considered with a great deal of caution.

Christophe Charle, La République des universitaires,1870-1940 (Paris: Seuil, 1994), 155-156.

Conclusion: long careers and long lives for CNS Chemists

What happens to Suzanne and Pierre after their years in the CNS? Among the 37 female Chemists, six unfortunately died prematurely, before the age of 60, 10 were disappear from the record before that age, 14 can be followed until retirement, and 7 others are still working at the age of 60 -and are not followed up to their retirement. 25 Altogether 21 women had long careers, attested until at least the age of 60. 26 We can thus presume that Suzanne shall be one of them. Among the 155 male Chemists, 11 died before the age of 60, and 27 disappeared from the record before that age. Among the other 117, 68 were followed until their retirement (at least 57 of them) or their death while still active after 60 (11 of them), and 47 remain alive and working after 60, and are not followed until their retirement. We can consider Pierre to belong to the group of 115 27 male workers with long careers. Referring to the initial cohort and with the exception of premature deaths, an active career carried on until the age of 60 and over represents at least 68% of the women and 80% of the men. This relatively low difference confirms a strong and sustained commitment to public research or higher education, in the case of women devoting themselves to scientific careers before the Second World War. CNS Chemists that were followed until the end of their career retired mainly from the CNRS or science faculties. Suzanne shall retire from the CNRS, where she would become a Senior Researcher, and Pierre from a Science Faculty as a Professor. The faculties remain reluctant to allow female access to higher positions. The industrial activities identified concern men only, at least 20/74 among those who disappeared from the record before retirement, and one among those whose entire career is known.

Referring to the initial cohort of young Chemistry fellowship beneficiaries who began a long career with the support of the CNS, only one, Alfred Kastler, would win a Nobel Prize.

Suzanne and Pierre, who did not achieve this fame have, last but not least, a long retirement ahead of them: the average age of death for female CNS Chemists is 86, 81 for their male colleagues, and the cohort's deans reach 104 in both sexes! 25 The legal age for retirement in these careers for this generation may reach 70. 26 Among the 21 females with the longest careers, 10 are married and nine are single, the two others remained single at the ages of 41 and 44, but their final marital status is unknown. 27 The unknown year of birth for two of the men explain the difference between 117 and 115.


Lucas Brivadis
email: lucas.brivadis@gmail.com

Vincent Andrieu
email: vincent.andrieu@gmail.com

Ulysse Serres
email: ulysse.serres@gmail.com

Luenberger observers for discrete-time nonlinear systems

published or not. The documents may come

Introduction 1.Context

The design of observers for nonlinear discrete-time systems remains a challenging and open problem despite a burgeoning literature. Since no universal method exists, several approaches have been developed. Most of them have first been developed for continuous-time systems, and then extended to the discrete case. Some of them, such as the well-known extended Kalman filter ( [START_REF] Boutayeb | A strong tracking extended kalman observer for nonlinear discrete-time systems[END_REF][START_REF] Reif | The extended kalman filter as an exponential observer for nonlinear systems[END_REF]), provide only a local convergence of the observer, and are based on a linearization of the system. Others (as [START_REF] Califano | On the observer design in discrete-time[END_REF] or [START_REF] Huijberts | On existence of extended observers for nonlinear discrete-time systems[END_REF]) consist in applying an invertible change of coordinates that transforms the original system in an other form for which it is much more easier to design an observer. Still others deal with Lipschitz nonlinear systems ( [START_REF] Zemouche | Observers design for discrete-time lipschitz nonlinear systems. state of the art and new results[END_REF][START_REF] Zemouche | Observer design for lipschitz nonlinear systems: The discrete-time case[END_REF], among others), that occur frequently in practice, and are based on linear matrix inequalities that provide Lyapunov functions for the error system.

A completely different idea is to try to reproduce the Luenberger's initial methodology originally developed for linear continuous-time system in [START_REF] Luenberger | Observing the state of a linear system[END_REF], which differs from what is now usually called Luenberger observer. This path has been mapped in the case of discrete-time systems by N. Kazantzis and C. Kravaris in [START_REF] Kazantzis | Discrete-time nonlinear observer design using functional equations[END_REF]. It consists to estimate first a function of the state, thanks to a linear stable system fed by the output, and then to inverse this mapping. However, strong assumptions such as analyticity of the system and observablity of the linearized system are required, and the invertibility of the function is obtained only locally.

In the following, we relax those assumptions following the strategy developed in the continuous case in [START_REF] Andrieu | On the existence of a kazantzis-kravaris/luenberger observer[END_REF] and later in [START_REF] Andrieu | Convergence speed of nonlinear luenberger observers[END_REF] and [START_REF] Bernard | Luenberger observers for nonautonomous nonlinear systems[END_REF]. We require the system to be time reversible, and replace the observability hypothesis of the linearized system by a backward distinguishability hypothesis on the nonlinear system itself. In so doing, we obtain the existence and the injectivity (not only locally) of such a function of the state.

This paper is organized as follows. In the next part of the introduction (Section 1.2), we state our problem in a more precise way and introduce some notations and definitions. We also prove a first result that guarantees the existence of an observer as soon as there exists a continuous uniformly injective map satisfying some functional equation. Our main results can be found in Section 2. We state sufficient conditions for the existence, injectivity and also unicity of such a map. We provide in Section 3 some examples and applications of those results. we examine linear systems with polynomial output and also discrete-time systems that approximate continuous-time systems.

Throughout the paper, we denote by | • | the usual Euclidean norm and by • the induced matrix norm.

Problem statement

We consider the discrete-time system

x k+1 = f (x k ), y k = h(x k ), (1)
with state x ∈ R n , output y ∈ R p and suitable functions f and h. In this paper, we deal with the problem of existence of an observer for system [START_REF] Andrieu | Convergence speed of nonlinear luenberger observers[END_REF]. We denote X k (x 0 ) = f k (x 0 ) the value at time k of the unique solution of system (1) initialized at x 0 ∈ R n , and

Y k (x 0 ) = h(X k (x 0 )) the corresponding output. Let X 0 ⊂ X ⊂ R n such that for all initial condition x 0 ∈ X 0 and all k ∈ N ∪ {0}, X k (x 0 ) ∈ X .
Definition 1. Let m be a positive integer, ϕ : R n × R p → R n and ψ : R m → R n . The discrete-time dynamical system given by

ξ k+1 = ϕ(ξ k , y k ), xk = ψ(ξ k ), (2)
is called an observer for (1) if and only if, for all (x 0 , ξ 0 ) ∈ X 0 ×R m , the solution of the coupled system (1)-( 2), denoted by (X k (x 0 ), Xk (x 0 , ξ 0 )) k 0 , satisfies

lim k→+∞ X k (x 0 ) -Xk (x 0 , ξ 0 ) = 0. (3)
Note that, even if Xk seems to depend directly of x 0 , it is actually not the case. As (2) says, Xk depends only of the measurements Y 0 (x 0 ), Y 1 (x 0 ), . . . , Y k-1 (x 0 ) through the dynamic of (ξ k ) k 0 .

We follow the Luenberger-like methodology in order to design an observer for system [START_REF] Andrieu | Convergence speed of nonlinear luenberger observers[END_REF]. Let m be a positive integer. First, we try to transform (1) into

ξ k+1 = Aξ k + By k . ( 4
)
with A ∈ R m×m a matrix with spectral radius ρ(A) < 1 and B ∈ R m×p . In order to do this, we look for a continuous map T : X → R m such that, for any x 0 ∈ X 0 and any k ∈ N ∪ {0},

T (X k+1 (x 0 )) = AT (X k (x 0 )) + BY k (x 0 ). ( 5
)
Let Ξ k (x 0 , ξ 0 ) denote the value at time k of the unique solution of system (4) with initial condition ξ 0 ∈ R m and measurements y k = Y k (x 0 ). Note that, for any (x 0 , ξ 0 )

∈ X 0 × R m , Ξ k+1 (x 0 , ξ 0 ) -T (X k+1 (x 0 )) = A(Ξ k (x 0 , ξ 0 ) -T (X k (x 0 ))) (6)
and since ρ(A) < 1, Ξ k (x 0 , ξ 0 ) -T (X k (x 0 )) converges geometrically towards zero. Hence, implementing system (4), one can deduce an approximation of T (x k ) as k goes to infinity. Then, if T is injective, one can estimate the state of system (1). More precisely, we have the following theorem.

Theorem 1. Let m be a positive integer, A ∈ R m×m such that ρ(A) < 1 and B ∈ R m×p . Let T : X → R m be a continuous map. Assume the following:

1. For all x ∈ X , T satisfies

T (f (x)) = AT (x) + Bh(x). (7)
2. T is uniformly injective, that is, there exists α a class K ∞ function such that for all

(x 1 , x 2 ) ∈ X 2 , |x 1 -x 2 | α(|T (x 1 ) -T (x 2 )|. ( 8
)
Then there exists a map T * : R m → R n such that ( Xk ) k 0 defined by Xk (x 0 , ξ 0 ) = T * (Ξ k (x 0 , ξ 0 )) for all (x 0 , ξ 0 ) ∈ X 0 × R m is the solution of an observer for (1).

Proof. Clearly, [START_REF] Huijberts | On existence of extended observers for nonlinear discrete-time systems[END_REF] implies that ( 5) is satisfied for all x 0 ∈ X 0 and all k ∈ N ∪ {0}.

Let (x 0 , ξ 0 ) ∈ X 0 × R m . Since ρ(A) < 1, it follows from (6) that lim k→+∞ Ξ k (x 0 , ξ 0 ) -T (X k (ξ 0 )) = 0. ( 9
)
From the uniform injectivity of T , there exists a pseudo-inverse

T -1 : T (X ) → R n such that for all x in X T -1 (T (x)) = x and for all (ξ 1 , ξ 2 ) ∈ T (X ) 2 , |T -1 (ξ 1 ) -T -1 (ξ 2 )| α(|ξ 1 -ξ 2 |). (10)
According to [START_REF] Mcshane | Extension of range of functions[END_REF]Theorem 2], there exists a function T * : R m → R n , that is an extension to R m of T -1 , satisfying [START_REF] Mcshane | Extension of range of functions[END_REF] for all (ξ 1 , ξ 2 ) ∈ (R m ) 2 . Hence,

|T * (ξ) -x| α(|ξ -T (x)|) ∀ξ ∈ R m , ∀x ∈ X . (11)
Thus |T * (Ξ k (x 0 , ξ 0 )) -X k (ξ 0 )| → 0 as k goes to infinity. Setting ϕ : (ξ, y) ∈ R n × R p → Aξ + By and ψ = T * , it follows from the Definition 1 that ( Xk ) k 0 defined by Xk (x 0 , ξ 0 ) = T * (Ξ k (x 0 , ξ 0 )) is the solution of an observer for [START_REF] Andrieu | Convergence speed of nonlinear luenberger observers[END_REF].

Then it is sufficient to prove the existence of a uniformly injective continuous map T : X → R m satisfying [START_REF] Huijberts | On existence of extended observers for nonlinear discrete-time systems[END_REF] for some positive integer m in order to design an observer for (1). In the next section, we state sufficient conditions for the existence, injectivity, and also unicity of a continuous map T solution of [START_REF] Huijberts | On existence of extended observers for nonlinear discrete-time systems[END_REF].

Remark 1. Note that if X is a compact subset of R n , then every continuous injective map T : X → R m is also uniformly injective in the sense of (8). In the following, we are interested in the injectivity of T . If uniform injectivity is required (for example to apply Theorem 1), then one must either assume X compact or prove the uniform injectivity by other means.

Results and comments 2.1 Existence of the transformation

First, we are interested in the existence of a map T satisfying [START_REF] Huijberts | On existence of extended observers for nonlinear discrete-time systems[END_REF]. In [START_REF] Andrieu | On the existence of a kazantzis-kravaris/luenberger observer[END_REF], V. Andrieu and L. Praly have proved the existence of a so-called Kazantzis-Kravaris/Luenberger observer for continuous-time systems of the form

ẋ = f (x), y = h(x). (12)
We follow the same methodology and adapt it in the discrete case. We need to make some assumptions on the system.

Assumption 1. f is invertible and f -1 and h are continuous.

Assumption 2. There exist four non-negative constants

C 1 , C 2 , C ′ 1 and C ′ 2 such that, for all x ∈ R n , |x| C 1 + C 2 |f (x)|, |h(x)| C ′ 1 + C ′ 2 |x|. ( 13
)
Remark 2. Note that Assumptions 1 and 2 are satisfied in particular if f is invertible and both f -1 and h are globally Lipschitz. We will use this remark in the next section about the injectivity of T .

For all non-negative integer i, we denote • the composition operator and

f i = f • f • • • • • f i times , f -i = (f -1 ) i .
Theorem 2. Let m be a positive integer, A ∈ R m×m a normal matrix such that ρ(A) < min {1, 1/C 2 } and B ∈ R m×p . Assume that Assumptions 1 and 2 are satisfied. For all x ∈ X , set

T (x) = +∞ i=0 A i Bh(f -(i+1) (x)). ( 14
)
Then T : X → R m is well defined, continuous, and satisfies [START_REF] Huijberts | On existence of extended observers for nonlinear discrete-time systems[END_REF].

Proof. For all x ∈ X and all non-negative integer i, let a i (x) = A i Bh(f -(i+1) (x)). According to Assumption 1, each a i is continuous on X . Note that, since A is normal, ρ(A) = A . Then, according to Assumption 2, we have for all x ∈ X

|a i (x)| ρ(A) i B C ′ 1 + C ′ 2 C i+1 2 |x| + C 1 i j=0 C j 2 . ( 15
)
Since ρ(A) < 1 and ρ(A)C 2 < 1 the Lebesgue dominated convergence theorem applied on any compact set implies that (14) defines a continuous function. Moreover, for any x ∈ X ,

T (f (x)) = +∞ i=0 A i Bh(f -(i+1) (f (x))) = A +∞ i=0 A i-1 Bh(f -i (x)) = A +∞ i=0 A i Bh(f -(i+1) (x)) + Bh(x) = AT (x) + Bh(x),
which shows that T satisfies (7).

Injectivity with backward distinguishability

In order to obtain that T defined by ( 14) is injective, we introduce the following backward distinguishability assumption on the system.

Assumption 3. For all (x 1 , x 2 ) ∈ X 2 , if x 1 = x 2 , then there exists a positive integer i such that h(f -i (x 1 )) = h(f -i (x 2 )).
We also need stronger hypothesis on the system than in the previous section.

Assumption 4. f is invertible and f -1 and h are of class C 1 and globally Lipschitz.

According to the Remark 2, if Assumption 4 holds, then Assumptions 1 and 2 are satisfied. We denote by I k the identity k×k matrix, by ⊗ the Kronecker product and by A * the conjugate transpose matrix of A.

⊗ I p ∈ C m×p . Let C 2 = sup{|(f -1 ) ′ (x)|, x ∈ X } and D be the open disc of C of radius min {1, 1/C 2 }.
Then there exists a subset R ⊂ D n+1 of zero Lebesgue measure in C n+1 such that, for any

(λ 1 , . . . λ n+1 ) ∈ D n+1 \ R, the matrix A = diag(λ 1 , . . . , λ n+1 ) ⊗ I p ∈ C m×m is such that the map T : X → C m defined by (14) is well-defined, of class C 1 and one-to-one. Proof. Let (λ 1 , . . . λ n+1 ) ∈ D n+1 and A = diag(λ 1 , . . . , λ n+1 ) ⊗ I p ∈ C m×m . Let T : X → C m be defined as in (14). For all λ ∈ D, let T λ (x) = +∞ i=0 λ i h(f -(i+1) (x)), ∀x ∈ X . (16)
Let a i (x) = λ i h(f -(i+1) (x)) for all x ∈ X . Then each a i is of class C 1 on X by Assumption 4, and we have the following domination:

a ′ i (x) λ i C ′ 2 C i+1 2 with C ′ 2 = sup{|h ′ (x)|, x ∈ X }.
Moreover, λC 2 < 1. So the Lebesgue dominated convergence theorem implies that for each λ ∈ D, T λ : X → C p is well-defined and of class C 1 . Considering the structure of A and B, remark that up to a permutation of coordinates we have

T (x) = T λ 1 (x), . . . , T λ n+1 (x) * It is sufficient to prove that T : X → C p is one-to-one for almost all (λ 1 , . . . , λ n+1 ) ∈ D n+1 .
In order to do this, we need the following lemma, established by L. Praly and V. Andrieu in [2, Lemma 1], which is a modified version of [6, Lemma 3.2] due to J.-M. Coron. We apply this lemma to Γ = {(x 1 , x 2 ) ∈ X 2 | x 1 = x 2 } and g = ∆T defined as follows:

∆T : (x 1 , x 2 , λ) ∈ X 2 × D → T λ (x 1 ) -T λ (x 2 ) (18)
Clearly, ∆T (x 1 , x 2 , •) is holomorphic on D for each (x 1 , x 2 ) ∈ X 2 and ∆T (•, λ) is of class C 1 on X 2 for each λ ∈ D. Fix (x 1 , x 2 ) ∈ Γ. Now, we prove that ∆T (x 1 , x 2 , •) is not identically zero on D. Assume the contrary. By unicity of the power series expansion, we get that for all positive integer i, h(f

-i (x 1 )) = h(f -i (x 2 )) (19)
According to the backward distinguishability Assumption 3, it implies that x 1 = x 2 which is contradictory with the fact that (x 1 , x 2 ) ∈ Γ. Hence, ∆T (x 1 , x 2 , •) is not identically zero on D.

Since D is a convex subset of C and ∆T (x 1 , x 2 , •) is holomorphic, its zero are isolated and with finite multiplicity. Hence the hypotheses of Lemma 1 are satisfied. Thus, R ⊂ D n+1 has zero Lebesgue measure and for all (λ 1 , . . . , λ n+1 ) ∈ D n+1 \ R, T is injective by definition of ∆T .

Remark 3. The function T and the matrices A and B defined Theorem 3 take complex values while previous Theorems 1 and 2 remain in the real frame. However, one can choose two different ways to bridge this gap.

• State Theorems 1 and 2 in the complex frame. The proofs remain identical. One should simply change the domains and codomains of f and h.

• Instead of considering A = diag(λ 1 , . . . , λ n+1 ) ⊗ I p ∈ C m×m and B = (1, . . . , 1) * ⊗ I p ∈ C m×p , one should either consider à = diag(Λ 1 , . . . , Λ n+1 ) ⊗ I p ∈ R 2m×2m and B = (I, . . . , I) * ⊗ I p ∈ R 2m×p
, where

Λ i = ℜ(λ i ) -ℑ(λ i ) ℑ(λ i ) ℜ(λ i ) , I = 1 0 .
Then for all real sequence of measurements (y k ) k 0 the solutions of ξk+1 = Ãξ k + By k contain the real and imaginary parts of the solutions of ξ k+1 = Aξ k + By k .

Unicity

One can also wonder in which cases does the unicity of T satisfying [START_REF] Huijberts | On existence of extended observers for nonlinear discrete-time systems[END_REF] holds. More than a theoretical question, this fact may be useful in practice in order to obtain the injectivity of T . Most of the time, the function T given by ( 14) is difficult to compute. Since the matrix A has spectral radius strictly inferior to 1, an approximation of T is given by

T N (x) = N i=0 A i Bh(f -(i+1) (x)), ∀x ∈ X . ( 20
)
for all N 0. Then |T (x) -T n (x)| → 0 as N → +∞. However, if f and h have more properties (for example if f is linear and h is polynomial, see Section 3.1), there may exist another solution T of (7) much more easier to compute than T . Then, the question of the injectivity of that new T remains open a priori. But if [START_REF] Huijberts | On existence of extended observers for nonlinear discrete-time systems[END_REF] has a unique solution for A and B complex matrices chosen has in Theorem 3, then T = T and hence T is injective. Now, we state our unicity theorem. Theorem 4. Let m be a positive integer, A ∈ R m×m such that ρ(A) < 1 and B ∈ R m×p . Let Assumption 1 hold and make the following backward stability hypothesis on X :

∀x ∈ X , ∀i 1, f -i (x) ∈ X . (21)
Assume also that X is compact. Then there exists one and only one continuous function T : X → R m that satisfy (7) for all x ∈ X .

Proof. First, we prove that the continuous solution of ( 7) is unique. Let T 1 , T 2 : X → R m be two continuous solutions of [START_REF] Huijberts | On existence of extended observers for nonlinear discrete-time systems[END_REF]. Let x ∈ X . Then for all i ∈ N ∪ {0},

T 1 (x) -T 2 (x) = (T 1 -T 2 )(f i (f -i (x))) = A i (T 1 -T 2 )(f -i (x)).
(from ( 7))

Since X is compact, satisfy (21) and T 1 and T 2 are continuous, there exists a constant

K > 0 such that |(T 1 -T 2 )(f -i (x))| K for all i ∈ N ∪ {0}. Since moreover ρ(A) < 1, A i (T 1 - T 2 )(f -i (x)) → 0 as i → +∞. Thus T 1 (x) -T 2 (x) = 0.
The existence of a continuous T satisfying ( 7) follows from the Theorem 2 and from the fact that Assumption 2 can be replaced in its proof by the fact that X is compact and backward stable1 . Indeed, the series (14) still defines a continuous function since the domination

|a i (x)| ρ(A) i ||B|| sup x∈X h(x) (22)
holds for all x ∈ X and can replace (15). Then one may apply the Lebesgue dominated convergence on X .

To conclude this section, recall that we have now at our disposal three theorems that ensures under different conditions on (1) the existence, unicity and injectivity of a continuous map T satisfying [START_REF] Huijberts | On existence of extended observers for nonlinear discrete-time systems[END_REF]. In the next section, we illustrate on examples how to use those tools. In particular, we study systems with linear dynamics and polynomial output, and emphasize the link between the Luenberger observers developed in [START_REF] Andrieu | On the existence of a kazantzis-kravaris/luenberger observer[END_REF] for continuous-time systems and the discrete-time observers developed in this paper for theirs first-order approximations.

Examples

Linear dynamics with polynomial output

We consider first the system with linear dynamic and polynomial output of degree d

x k+1 = F x k , y k = HP d (x) ( 23
)
with P d : R n → R k d a vector containing the k d possible monomials with degree less or equal than d, F ∈ R n×n and H ∈ R p×k d . Then we have the following proposition.

Proposition 1. Let m be a positive integer and B ∈ R m×p . There exists a subset S of zero Lebesgue measure in R m×m such that for all A ∈ R m×m \S, there exists a function T : R n → R m of the form

T (x) = M P d (x), ∀x ∈ R n (24)
for some M ∈ R m×k d , that satisfies (7) for any x ∈ R n .

Proof. First, note that since P d (F x) is a vector containing polynomials of x with degree inferior to d, there exists a matrix D ∈ R k d ×k d such that

P d (F x) = DP d (x), ∀x ∈ R n . (25)
Since the set of eigenvalues of D is finite, the spectra of D and -A are disjoint for almost all A ∈ R m×m i.e. there exists a subset S ⊂ R m×m of zero Lebesgue measure such that the spectra of D and -A are disjoint for all A ∈ R m×m \ S. For such matrices A the Sylvester equation

M D = AM + BH (26)
has a unique solution M ∈ R m×k d . Set T as in (24). It remains to check that ( 7) is satisfied for f = F and h = HP d . For all x ∈ R n ,

T (F x) = M P d (F x) (from (24)) = M DP d (x) (from (25)) = AM P d (x) + BHP d (x) (from (26)) = AT (x) + BHP d (x).
Remark 4. Note that the result is still true if A and B are complex matrices. Then T takes complex values. The proof remains identical.

Remark 5. Choose a set X 0 ⊂ R n of initial condition and let X be as usual such that X k (x 0 ) ∈ X for all x 0 ∈ X 0 and all k ∈ N ∪ {0}. Note that if F is invertible and if X is compact and backward stable, then the assumptions of the Theorem 4 hold. Assume also that Assumptions 3 and 4 hold and apply Theorem 3 with m = (n+1)p. Then, for almost all (λ 1 , . . . , λ n+1 ) ∈ C n+1 , and for complex matrices A and B as in Theorem 3, we have

T (x) = M P d (x) = +∞ i=0 A i Bh(f -(i+1) (x)) (27)
for all x ∈ X . In particular, T defined by (24) is injective.

Link with the continuous Luenberger observer

In this section, we are interested in the link between the continuous Luenberger observer developed in [START_REF] Andrieu | On the existence of a kazantzis-kravaris/luenberger observer[END_REF] for system [START_REF] Zemouche | Observer design for lipschitz nonlinear systems: The discrete-time case[END_REF] and the discrete observer developed in the previous sections for a discrete-time version of (12).

Continuous-time system

We consider the following example with linear dynamic and polynomial output:

ẋ1 = x 2 ẋ2 = -x 1 , y = x 2 1 -x 2 2 + x 1 + x 2 . (28)
It can be shown that this system is weakly differentially observable 2 of order 4 on R 2 in the sense of [START_REF] Bernard | Luenberger observers for nonautonomous nonlinear systems[END_REF]Definition 1]. Following [START_REF] Bernard | Luenberger observers for nonautonomous nonlinear systems[END_REF], we seek

T λ : R → R n such that d dt T λ (x) = λT λ (x) + y (29)
2 First, the map (x for some λ < 0. Since (28) has linear dynamic and polynomial output of degree 2, one can look for T of the form

T λ (x) = x * a c/2 c/2 b x + d e x (30)
for some (a, b, c, d, e) ∈ R 5 . Then (29) holds if and only if

-c = λa + 1, c = λb -1, 2(a -b) = λc, -e = λd + 1, d = λe + 1. ( 31
)
The only solution of this equation is

a = - λ 4 + λ 2 , b = λ 4 + λ 2 , c = - 4 4 + λ 2 , d = 1 -λ 1 + λ 2 , e = - 1 + λ 1 + λ 2 . ( 32
)
Since T λ is stationary, one could believe that this function provide an observer that could be efficient even for a numerical approximation of (28). However, as we will see in the following, it is not the case: for a given discrete approximation of (28), it is better to design an observer based on the discrete-time system rather than to use the one given by T λ .

Associated first-order discrete-time system

For some discretization parameter dt > 0, the associated first-order approximation3 of ( 28) is

     x 1 (k + 1) = x 1 (k) + dtx 2 (k) x 2 (k + 1) = x 2 (k) -dtx 1 (k) y k = x 1 (k) 2 -x 2 (k) 2 + x 1 (k) + x 2 (k) . (33)
We seek a function T d λ : R → R n satisfying a first-order approximation of (29) given by the Euler explicit method:

T d λ (x(k + 1)) = (1 + λdt)T d λ (x(k)) + dty k . (34)
Since λ < 0, it is sufficient to choose λdt > -2 to have -1 < 1 + λdt < 1. Now, we seek T d λ of the form

T d λ (x) = x * a ′ c ′ /2 c ′ /2 b ′ x + d ′ e ′ x (35)
for some (a ′ , b ′ , c ′ , d ′ , e ′ ) ∈ R 5 . Then (34) holds if and only if (d ′ , e ′ ) satisfy the same equation that (d, e) in ( 31) and (a ′ , b ′ , c ′ ) satisfy

     -c ′ + b ′ dt = λa ′ + 1, c ′ + a ′ dt = λb ′ -1, 2(a ′ -b ′ ) -c ′ dt = λc ′ . (36)
Remark that this equation is the same than (32) when dt = 0. This is coherent with the fact that (33) is a discretization of (28). Then, the only solution of (36) is such that (d ′ , e ′ ) = (d, e) for all dt > 0 and (a ′ , b ′ , c ′ ) converges to (a, b, c) as dt goes to 0:

               a ′ = - λ + dt 4 + (λ + dt) 2 , b ′ = λ + dt 4 + (λ + dt) 2 , c ′ = - 4 4 + (λ + dt) 2 .
(37) For dt > 0, the discrete observer given by T d λ is therefore different from the continuous observer given by T λ , even if their difference goes to 0 as dt goes to 0.

Comparison of the observers

Consider a numerical simulation of the continuous-time system (28) obtained by the Euler explicit first-order method, which corresponds to the discrete-time system (33). Then the map T d λ given by ( 34) is much more adapted to the design of a numerically efficient observer than the function T λ given by (29) that has been designed for (28). More generally, in order to implement an observer for a continuous-time varying system, it is better to develop a discrete-time observer based on the numerical approximation of the system, rather than a continuous-time observer based on the original system itself.

In order to highlight numerically this fact, we simulate the system (28) thanks to (33) and compare the accuracy of two observers: one based on functions of the form T d λ , and another based on functions of the form T λ . To obtain the observers, we fix dt > 0 and three arbitrary values λ i < 0 satisfying λ i dt > -2 and use the fact that 

   1 0 1 1 a 1 c 1 d 1 e 1 a 2 c 2 d 2 e 2 a 3 c 3 d 3 e 3         x 2 1 -x 2 2 x 1 x 2 x 1 x 2     =     y T 1 (x) T 2 (x) T 3 (x)     . (38)
where (a i , c i , d i , e i ) is given by (32) (resp. (37)) with λ = λ i and T i = T λ i (resp. T i = T d λ i ). Fix the following parameters and initial conditions:

dt = 0.01, x(0) = (1, 0), λ i = -10 × i, ξ i (0) = 0. ( 39
)
Then the 4×4 matrix defined in (38) is invertible. Hence one can reconstruct an approximation (x 1 , x2 ) of the state (x 1 , x 2 ) from the measurement y and approximations of T i (x) given by the dynamic ξ i k+1 = (1 + λ i dt)ξ i k + dty k .

On Fig. 1, we plot on a semi-log scale the evolution of the absolute error ε k = |x k -xk | between the state and its observer for k ∈ {0, . . . , 500} (i.e. t ∈ [0, 5]) for the observer based on functions T λ i designed for the original continuous-time system. Similarly, we make on Fig. 2 the same plot but for the observer based on functions T d λ i designed for the discrete-time system. We clearly see that the observer based on T d λ i is much more efficient than the one based on T λ i . On one hand, using T d λ i , the error go to zero until it achieve 10 -12 , which is close to the machine epsilon (≈ 10 -16 ). Moreover, the state observer seems to converge exponentially to the state, with a rate r ≈ -4.58 (estimation based on a linear regression made on [0.5, 3]). On the other hand, with T λ i , the observer does not converge to the state: it keeps an absolute error oscillating around 10 -2 . This phenomenon is due to the fact that the trajectory of (33) is not invariant for this observer: even if it is well initialized (i.e. x(0) = x(0)), the observer will oscillate around the state.

Conclusion

We have shown how the initial Luenberger methodology can be applied to nonlinear discretetime systems. It is based on the existence of a map satisfying some functional equation linked to the system, that transform the original system into a linear asymptotically stable one fed by the output. As soon as this map is uniformly injective, it allows us to estimate the state of the nonlinear system by simulating an autonomous system fed by the output and inverting this map. We stated sufficient conditions for the existence of such a map. In particular, we need the system to be reversible in time. Under a backward distinguishability hypothesis, we also proved that this map is injective.

Theorem 3 .

 3 Let Assumptions 3 and 4 hold. Let m = (n + 1)p and B = (1, . . . , 1) *

Lemma 1 .

 1 Let D and Γ be open subsets of C and R 2n , respectively. Let g : Γ × D → C p be a function which is holomorphic in λ for each x ∈ Γ and C 1 in x for each λ ∈ D. If for each x ∈ Γ, the function λ ∈ D → g(x, λ) is not constantly zero, then the set R = x∈Γ (λ 1 , . . . , λ n+1 ) ∈ D n+1 ∀i ∈ {1, . . . , n + 1}, g(x, λ i ) = 0 (17) has zero Lebesgue measure in C n+1 .

Figure 1 :Figure 2 :

 12 Figure 1: Evolution of the error between the state and the observer based on T λ i in semi-log scale

Similarly, using the same trick, one can easily show that the hypothesis of globally Lipschitz in Assumption 4 can be replaced in the proof of Theorem 3 by the fact that X is compact and backward stable.

Since (28) is weakly differentially observable, it can be shown that (33) is backward distinguishable as soon as dt is small enough.