Marianne Huchard

Towards a catalog of modeling patterns for knowledge discovery with relational concept analysis

Formal Concept Analysis

Formal concepts are "a natural feature of information representation which is as fundamental to hierarchies and object/attribute structures as set theory or relational algebra are for relational databases".

Uta Priss. 40th anniv. vol. of Annual Review of Inf. Sc. and Tech., 2006 Simple but powerful basics Dissemination 1982-2022

• Conceptualization, philosophical developments

• Data analysis, data mining, clustering

• Knowledge representation (ontology construction)

• Classification, indexation (information retrieval)

• Unsupervised learning

• Supervised learning (adding classes in description)

• Tools From:

Person, the common generalizations between the attributes of the associations (see [22]) are not sufficient to sugges the generalization of the Lease and Acquisition associations to the Housing Transaction association: associations ar first and foremost defined by the classes they relate. This is another instance of the more general problem of "circular" classification, often encountered in knowledg representation languages [13]: classification is expressed in terms of equivalence of logical expressions whose trut values can be established through classification. The consequences of such "circularity" go from undecidability t unpredictability. In this case, if the lattice generation method happens to generalize Tenant and Landlord first, the Lease and Acquisition will be properly generalized. A more serious problem occurs if we could not generaliz Tenant and Landlord without generalizing the corresponding associations 9 . We had to amend the algorithm for con ceptual graphs by "revisiting" some nodes of the lattices as new generalizations are found. In this case, we would hav To: • Premières thèses de doctorat

• Cyril Roume, Univ. Montpellier ( 2004)

• Mohamed Amine Rouane-Hacène, UQAM (2007)
• Foundational papers • Set of Formal contexts (object-attribute)

• Represent classes/instances

• Set of Relational contexts (object-object) • Industry (Wajnberg+) Modeling for RCA

• Naive view: A model is provided with data which can be used • Practice is more complex, e.g.

• Associations: bidirectional or N-ary • Particular relations: E.g. is-a, instance-of, contains

• The analyst may want to focus on specific parts of the data • Data may be provided without a model (e.g. textual data)

⇒ The RCF has to be built by considering a transformation of the initial model, or even created from scratch

The spirit is similar to scaling of FCA that transforms non-Boolean attributes into Boolean ones.

Here the needed modeling addresses models, instances (objects) and values.

FCA RCA DPs Conclusion

Rationalizing RCF design

Focus on Separate Viewpoints: an application

Assist Feature location in Software Products Lines (Hlad, Lemoine, Huchard, Seriai, GPCE 2021)

• Separately group features and artefacts to extract:

• Feature variability (variability model)

• Artefact variability (potentially reusable code pieces)

• Traceability between propositional logic formula on features vs. code artefacts

• Identification of apparition context of artefacts in terms of positive vs. negative features followed by formula reduction • Generation of code annotations #if formula-on-features

Focus on Separate Viewpoints: an application

For more details • Workflow

Azmeh+ (web services workflow classification)

• Query

Azmeh+ (web service replacement in a workflow)

• Temporal

Le Ber+ (hydrological data)

Conclusion

• FCA as a symbolic/white-box/structured-data KD method

• RCA, a variant for multi-relational datasets

• Practicing RCA demands a quite complex data preparation

• Recurring situations to be capitalized in Design Patterns

•

  Systematization of the idea of Godin&Mili OOPSLA'93 • Projects • MACAO (LIRMM, Orange Labs) 2001-2003/2006-2009 • PICS CNRS "CAdOE" (LORIA, LIRMM, UQAM) 2011-2013

•

  have all their pizzas in concept pizzas that have all their ingredients in concept vegan ingredients eataly and lafelicita share:∃∀ serves (∃∀contains (Vegan) ) r (o) ⊆ Ext(c) Universal strict ∀∃ r .c r (o) ⊆ Ext(c) and r (o) = ∅ Universal-percent ∀∃_≥ n% r .c |r (o) ∩ Extent(C )| ≥ n|r (o)|/100) Covers ⊇ r .c r (o) ⊇ Ext(c) Covers-percent ⊇ _≥ n% r .c |r (o) ∩ Extent(C )| ≥ n|Extent(C )|/100) Existential (wide) ∃ r .c r (o) ∩ Ext(c) = ∅ Universal strict ∀∃ r .c r(o) ⊆ Ext(c) and r (o) = ∅ Qualif. card. ≥ n r .c r (o) ⊆ Ext(c) and |r (o)| ≥ n restriction Card. restriction ≥ n r . L |r (o)| ≥ n Environment (Le Ber+, Martin+, Miralles+) • Health (Wajnberg+) • Ontologies (Rouane-Hacène+), Link key extraction (Atencia+) • Software engineering • UML class model, use case model normalization (Huchard+, Dolques+) • Design defects refactoring (Moha+) • Model transformation pattern extraction (Dolques+) • Web service, component classification (Azmeh+, Urtado/Vauttier+) • Multiple variability modelling (Galasso+, Waffo Kouhoué+), Feature mining or location (Al-Msie'deen, Eyal-Salman, Seriai+, Hlad+),

  to derive a feature model: J. Galasso-Carbonnel et al. Modelling equivalence classes of feature models with concept lattices to assist their extraction from product descriptions. J. Syst.Softw. 152: 1-23 (2019) May be used to suggest reusable code pieces: E.g. methods fly, takeoff, land can be grouped in a component AirVehicle

  if (AIR || GROUND) && ! ( AIR && WATER ) public void ride_(){...} //#endif ... } Extent(C_Prod._30)={Twingo, Squba, TringaT650, A380} Positive part from C_Art._20: (AIR || GROUND) Extent(C_Prod._30) ∩ Extent(C_Prod._27) = ∅ Extent(C_Prod._27 = {Akoya} Negative part from C_Prod._27 = ! ( AIR && WATER )

•

  Pursue the formalization of DPs • Develop the corresponding interpretation tools • Analyze effects on extracted knowledge: • conceptual structures • logical constraints: implication and association rules, mutual exclusion, OR groups, XOR groups • Develop a methodology integrating DPs usage • Build bridges with Graph-FCA and Polyadic Concept Analysis Supported by the ANR SmartFCA project Grant ANR-21-CE23-0023 of the French National Research Agency This reflexion could not have happened without ... 58/58

  O is a finite set of objects, A is a finite set of attributes R ⊆ O × A is a binary relation (o, a) ∈ R means that object o owns attribute a.

	Intro Intro Intro Intro	FCA FCA FCA FCA FCA FCA FCA	RCA RCA RCA RCA RCA RCA RCA	DPs DPs DPs DPs DPs DPs DPs	Conclusion Conclusion Conclusion Conclusion Conclusion Conclusion Conclusion
			Formal Context = Triple (O, A, R) Formal Concept = Pair (Extent, Intent) Concept Lattice Another view of concept building Another view of concept building Another view of concept building
		Set of all Extent = Maximal set of owner objects Concept_Ingredient_9
		Intent = Maximal set of their shared attributes concepts Links between objects and attributes Links between objects and attributes vege / vegan Concept of vegan ingredients
		Ingredient goatcheese × v e g e Ingredient v e g e provided goatcheese × with extent ↑ or intent ↓ inclusion Concept_Ingredient_13 Concept_Ingredient_14 spring scallop scallop burrata goatcheese scallop burrata goatcheese scallop burrata goatcheese burrata × scallop tomato × shallot × mushroom × eggplant × eggplant × eggplant tomato burrata × scallop C_I_11 burrata shallot shallot shallot tomato × shallot × mushroom × vege mushroom mushroom mushroom vegan shallot mushroom tomato eggplant tomato eggplant eggplant tomato	v e g a n v e g a n Concept_Ingredient_10 s p r i n g s u m m e r s p r i n g s u m m e r × vege Concept_Ingredient_11 Concept_Ingredient_16 a u t u m n a u t u m n spring summer spring spring summer summer × × × × × × × × × × × × Concept_Ingredient_12 × × vegan summer goatcheese autumn autumn autumn × × × × × × × Concept_Ingredient_15 Concept_Ingredient_17 vege vege vege tomato eggplant autumn shallot mushroom vegan vegan vegan
					8/58 10/58 12/58 14/58

• Formal Context • Galois connection • Concept Lattice • Implications and associations rules

  (Ganter et Wille, Diday, Polaillon, ...) • Fuzzy (Belohlavek et al., Cabrera, Cordero, Enciso, Mora, Lòpez-Ròd., Ojeda-Aciego et al., Cornejo, Medina et al., Yahia et al., Dubois Prade ...) • Value taxonomies (Godin et al., Carpineto et Romano, ...) • Logical description (Chaudron et al., Ferré et al., ...) • Graphs (Ganter and Kuznetsov, Liquière, Prediger et Wille, Kötters et

	Intro Intro	FCA FCA FCA	RCA RCA RCA	DPs DPs DPs	Conclusion Conclusion Conclusion
		Dissemination 1982-2022 Complex data in FCA Roots: Class model refactorization [Godin&Mili 1993]
		• Multi-valued attributes : integers, double, terms, structures, " A formal method that organizes a set of class interfaces into a
		Introduction Formal Concept Analysis Relational Concept Analysis symbolic objects, etc. al., Graph-FCA Ferré et al....) lattice structure " -Associations in Sect. 3.2.3
		• Multi-relational, RCA (Priss, Rouane-Hacène et al., ...);
		RCA+Fuzzy (Boffa et al.) • Polyadic (Sacarea, Tronca et al.) Design patterns for RCA
		• Sequences (Boukhetta, Demko, Bertet et al., Buzmakov et al.) • Applications • Temporal data (Wolff et al., Nica, Braud, Dolques, Le Ber et al., Conclusion
		ICCS Boukhetta, Demko, Bertet et al.) ICFCA CLA	FCA4AI	IPMU
		• Pattern Structures (Ganter et al., Kuznetsov et al., Napoli et al.,
		Buzmakov et al.)
					16/58 18/58

Credits Uta Priss https://upriss.github.io/fca/fca.html

Thus, prior to establishing that Tenant and Landlord generaliz to Person, we cannot say that Lease and Acquisition generalize to Housing Transaction. TENANT name address LANDLORD name address ACQUISITION

		Transaction		Effective
	LEASE	Value		Starting
	Month. rate Start date	HOUSE
		Monthly rate	Purchase Price	Starting date	Date of Acquisition
	Purch. price
	Date acqui
		Figure 12. A case of "circular" generalization.

  one RC i connecting an object of FC to its corresponding view object in an FC i

	Intro Intro Intro	FCA FCA FCA FCA FCA FCA	RCA RCA RCA RCA RCA RCA	DPs DPs DPs DPs DPs DPs	Conclusion Conclusion Conclusion Conclusion Conclusion Conclusion
		Design Pattern Separate/Gather Viewpoints Focus on Separate/Gather Viewpoints RCF multi-views model Focus on Separate Viewpoints: an application Focus on Separate Viewpoints: Vehicles
		Problem Solution • Formal contexts (FC) Assist Feature location in Software Products Lines (Hlad, Lemoine, Huchard, Seriai, GPCE 2021) Output Input Annotated SPL code Features and code FC1 RC1 of similar products public class Vehicle{ • Migrate a similar software family into a software product line //#if MOVE & SECU & PROGCONT
		• Problem • Recurring solution • Typical example • Variants and discussion • Feature location: find the software artefacts (E.g. code) [MOVE, SECU,PROGCONT,WATER] public void move_(){...} • Objects are described by Boolean attributes that can be put into different categories; or initial multi-valued attributes have been transformed into Boolean attributes • Each attribute category is a view or a concern FC2 RC2 public class Vehicle{ public void passControl_(){...} corresponding to a feature public void move_(){..} //#endif • Principle: if a group of artefacts and a group of features are public void passControl_(){..} //#if WATER FC ....... shared by products, this suggests that these features may be public void float_(){..} public void float_(){...} ....... coded through these artefacts } //Oceanis30 //#endif • It is relevant to analyze objects through the perpective of: • a single view //#if AIR • Challenges: [MOVE, SECU,PROGCONT,AIR] public void fly_(){...} • several views (including all) FCn 1. have a sufficient product number to discriminate and associate public class Vehicle{ //#endif RCn a single feature to a group of artefacts public void move_(){..} }
		public void passControl_(){..} 2. some pieces of code do not correspond to a feature but suggest an interaction between features, or missing features in the product description public void fly_(){..} Products derived according } //A380 selected features
					36/58 38/58 40/58

• Recurring situations

• Capitalizing them would accelerate future applications • Design Pattern approach (C. Alexander, GOF: E. Gamma, J.

Vlissides, R. Johnson, R. Helm)

• One FC for the initial objects

• FC objects are the initial objects, FC attributes are initial object identifiers, or other description, or none

• One FC i for each view i, 1 ≤ i ≤ n, one per attribute category

• FC i objects are views on the initial objects, FC i attributes are initial Boolean attributes of one view

• Relational contexts (RC)

• For each i, 1 ≤ i ≤ n,

  Univ. Montpellier 2022, ISIA • B. Lemoine, thèse de Master, Localisation de caractéristiques dans le cadre des lignes de produits logiciels, Univ. Montpellier 2021, prix IESF-CODIGE • N. Hlad et al. Leveraging relational concept analysis for automated feature location in software product lines. GPCE 2021: 170-183 Analyzing Visual Accessibility Options in Operating Systems • Objects are operating systems (OS); Views are three visual accessibility options categories (contrast, text, zoom) • Allows to analyze constraints on these options e.g.: implications, mutual exclusion • Objectives: recommendations for a new system version (OS developer), or user assistance in changing versions • A. Waffo Kouhoué et al. Exploring Variability of Visual Accessibility Options in Operating Systems.

	Intro	FCA FCA FCA	RCA RCA RCA	DPs DPs DPs	Conclusion Conclusion Conclusion
		Focus on Separate Viewpoints: an application Focus on Separate Viewpoints Towards a catalog of DPs
		Opportunities for other design patterns
		Another developed application • Collapse Specialization For more details (Java interfaces, UML), Martin+ (plants) Godin/Mili (classes), Huchard/Miralles+ • Identification of artefacts at the instruction-level • N-ary relation reification /split Martin+ (plants) • Reduction of annotation formulas (e.g. by factorization, using implications from the AOC-poset, etc.) • Separate topics Martin+ (plants), Miralles+ (UML)
		• Tools: CLEF (J. Galasso), IsiSPL (N. Hlad, B. Lemoine) • Instances2Model Dolques+ (Model transformation pattern
		• Experimentations extraction), B.Seriai+ (RDF)
		• FakePrinterSPL, DrawSPL, ArgoUML, NotePad, GameOfLife,
		Elevator
		• evaluation: correct product re-generation ; formulas reduction Future Internet
		in number (∼-30%) and size (∼-60% of features) 13(9): 230 (2021)
					52/58

• Readings

• N. Hlad, thèse de doctorat, IsiSPL : un processus automatisé pour faciliter l'ingénierie des lignes de produits logiciels selon une stratégie d'adoption industrielle réactive ou extractive,

•


Giuseppe Fedele

Francesco Chiaravalloti

Cédric Join

AN ALGEBRAIC DERIVATIVE-BASED APPROACH FOR THE ZERO-CROSSINGS ESTIMATION

A new approach to the design of a zero-crossings estimation algorithm is proposed. The approach uses elementary differential algebraic operations in the frequency domain for accurate derivative estimation. Such estimates are composed of iterated integrals of noisy observed signal. A detector-signal, which is exactly equal to zero when there is no intersection between the observed signal and the real axis and is greater than zero when a zero-crossing occurs, is obtained. To justify the theoretical analysis and to investigate the performances of the developed method, simulated experiments are performed.

INTRODUCTION

Given a piecewise continuous signal, the purpose of this paper is to detect its zero-crossings and estimate their locations. The problem is especially challenging for applications requiring on-line detection: the main difficulties stem from corrupting noises which blur the zero-crossings, and the combined need of fast calculations for real-time implementation and of reliable detection. A large amount of literature is devoted to these questions in fields such as signal processing [START_REF] Friedman | A Zero Crossing Algorithm for the Estimation of the Frequency of a Single Sinusoid in White Noise[END_REF] [2] [START_REF] Sadler | Frequency estimation via sparse zero crossings[END_REF], industrial electronics [START_REF] Vainio | Adaptive Filtering Using Multiplicative General Parameters for Zero-Crossing Detection[END_REF] [START_REF] Weidenbr Üg | New Synchronization Method for Thyristor Power Converters to weak AC-Systems[END_REF], fluid mechanics, speech processing [START_REF] Niederjohn | A Mathematical Formulation and Comparison of Zero-Crossing Analysis Techniques which have been Applied to Automatic Speech Recognition[END_REF] [START_REF] Scarr | Zero Crossings as a Means of Obtaining Spectral Information in Speech Analysis[END_REF], biomedical engineering, optics, neurophysiology, structural dynamics, communications, image processing, to name just a few. The use of the zero-crossing detection and calculation of the number of cycles that occur in a predetermined time interval is a simple and well-known methodology, especially in signal processing [START_REF] Friedman | A Zero Crossing Algorithm for the Estimation of the Frequency of a Single Sinusoid in White Noise[END_REF]. The importance of the zero-crossings is well documented in [START_REF] Kedem | Spectral Analysis and Discrimination by Zero-Crossings[END_REF] where the fruitful connection between zerocrossing counts and time-invariant linear filtering is investigated. This connection leads to interesting properties for the fast analysis of random signals. Moreover, the zero-crossing counts in random signals and their filtered versions essentially constitute a domain which is equivalent to the spectral domain. In this paper a new method to estimate the zerocrossings of a generic signal is proposed. The main idea is to create a function with no defined first order derivative and the related discontinuity in the second order derivative in the zero-crossing points. These discontinuities can be effectively detected by using reliable techniques of parameters estimation [START_REF] Fliess | An algebraic framework for linear identification[END_REF]. The paper is organized as follows. Section 2 describes the proposed approach for zero-crossings estimation; in Section 3 the main aspects of the algebraic identification method and its application to the derivatives estimation are discussed; Section 4 contains some considerations about the implementation and the robustness of the proposed approach; Section 5 illustrates the performances obtained by simulated experiments and it is followed by final conclusions.

PROPOSED APPROACH

The logic scheme of the proposed zero-crossings detector is depicted in Fig. 1. The piecewise smooth signal y(t), which has a zero-crossing at t = t s , is the input signal. Based on the observation y n (t) = y(t) + n(t), where n(t) is an additive noise corruption, it is desirable to estimate the zero-crossing location. Clearly both y(t) and |y(t)| have the same zerocrossing, moreover in t s the first order derivative of |y(t)| is not defined as shown in Fig. 2. Even though the first order derivative of |y(t)| is not defined in t = t s , from a numerical point of view, the second order derivative of the function |y(t)| presents a positive peak around the zero-crossing t s (see Fig. 3). The proposed scheme is capable to obtain a detectorsignal, namely d(t), which is exactly equal to zero when there is no intersection with real axis and is greater than zero when there is a zero-crossing. The idea is to compute the second order derivative of the signals

y 1 (t) = |y n (t)| + y n (t) 2 , (1)
and

y 2 (t) = |y n (t)| -y n (t) 2 , (2)
d 1 (t) = 0, t ∈ [0,t s ), d 2 y(t) dt 2 , t > t s , d 2 (t) = -d 2 y(t) dt 2 , t ∈ [0,t s ), 0, t > t s ,
and the detector signal defined as

d(t) = d 1 (t)d 2 (t) (3)
has a peak in the instant t = t s and is zero for t = t s .

NUMERICAL DIFFERENTIATION: A SHORT SUMMARY

Derivative estimation of noisy time signals is a longstanding difficult ill-posed problem. Here the numerical differentiation problem is dealt with the algebraic parameter estimation initially presented in [START_REF] Fliess | An algebraic framework for linear identification[END_REF], [START_REF] Mboup | Parameter estimation for signals described by differential equations[END_REF]. Given a smooth signal, a key point of this approach is to consider its second order derivative in t = τ, for each fixed τ ≥ 0, as a single parameter to be estimated from a noisy observation of the signal. A pointwise derivative estimation therefore follows by varying τ. The main aspect of the algebraic parameter estimation is to operate in the operational calculus domain [START_REF] Mikusiǹski | Operational Calculus[END_REF], [START_REF] Mikusiǹski | Operational Calculus[END_REF], [START_REF] Yosida | Operational Calculus -A Theory of Hyperfunctions[END_REF], where an extensive use of differential elimination and a series of algebraic manipulations yield, back in the time domain, an explicit expression for the estimate of the second order derivative in t = τ as an integral operator of the noisy observation within a short time interval [τ, τ +T ]. Let us consider the estimation of x (2) (t), the second order derivative of a smooth signal x(t) defined on an interval

I ∈ R + = [0, +∞).
Assume that x(t) is analytic on I so that it is possible to consider the approximation of the signal x(t) with a second order polynomial

x(t) ≈ a 0 + a 1 t + a 2 t 2 . ( 4
)
By considering classic operational calculus operators, x(t) can be rewritten as

X(s) = a 0 s + a 1 s 2 + 2a 2 s 3 . (5)
Multiply both sides of ( 5) by s 3 , the following expression holds:

s 3 X(s) = a 0 s 2 + a 1 s + 2a 2 . ( 6
)
Taking the derivative of both sides of ( 6) with respect to s, one and two times respectively, expressions ( 7) and ( 8) are obtained

3s 2 X(s) + s 3 dX(s) ds = 2a 0 s + a 1 , (7) 6sX
(s) + 6s 2 dX(s) ds + s 3 d 2 X(s) ds 2 = 2a 0 . (8)
The coefficients a 0 , a 1 and a 2 are obtained via the triangular system of Eqs. ( 6)- [START_REF] Kedem | Spectral Analysis and Discrimination by Zero-Crossings[END_REF]. Derivative operations in time domain are avoided by multiplying both sides of Eqs. ( 6)-( 8) by s -n , n ≥ 3. To express such equations back in the time domain, let us recall that for a given signal u(t), and a positive integer α, the time domain analog of

V (s) = 1 s α d β ds β U(s)
is the iterated integral, of order α, of (-1) β t β u(t). Using the Cauchy formula, this leads to a single integral

v(t) = 1 (α -1)! t 0 (t -ξ ) α-1 (-1) β ξ β u(ξ )dξ . (9)
The corresponding iterated time integrals are low pass filters which attenuate the corrupting noises. A quite short time window is sufficient for obtaining accurate values of a 0 , a 1 and a 2 The extension to polynomial functions of higher degree is straightforward. For the second order derivative estimation, assume that the second order polynomial be the truncated Taylor expansion around a given time instant τ, and apply the previous computations to obtain the coefficient a 2 . Resetting and utilizing sliding time window, an estimation of the derivative at any sampled time instant can be performed. Note also that unstructured noises, which can be considered as high frequency perturbations, are attenuated by the iterated integrals, which are simple examples of low-pass filters. Of course it is possible to estimate all the coefficients a i , i = 0, 1, 2, simultaneously. However, not only the coefficients a i , i = 0, 1 are not necessary for the estimation of x (2) (t), but also simultaneous estimation is more sensitive to noise and numerical computation errors. In the proposed approach, all the terms a i , i = 0, 1 are consequently considered as undesired perturbations to annihilate. To this aim it suffices to find a differential operator, i.e.

Π = ∑ finite ρ l (s) d l ds l , ρ l (s) ∈ C(s), (10)
satisfying

ΠX(s) = ρ(s)a 2 , (11)
for some rational function ρ(s) ∈ C(s). Such a linear differential operator, called an annihilator for a 2 , obviously exists and is not unique. It is also clear that to each annihilator Π, there is a unique ρ(s) ∈ C(s) such that [START_REF] Mikusiǹski | Operational Calculus[END_REF] holds. In this case Π and ρ(s) are said associated.

Lemma 1

The linear differential operator Π = Π 1 Π 0 , where

Π k = d ds s k+1 , k = 0, 1, is an annihilator of a 2 associated to the function ρ(s) = 4 s 2 . Proof. Applying the operator Π 0 to (5) it is easy to obtain that Π 0 X(s) = - a 1 s 2 -4 a 2 s 3 . (12)
Finally

Π 1 Π 0 Y (s) = d ds s 2 Π 0 Y (s) = 4a 2 s 2 , (13)
from which the proof follows.

According to Lemma 1 it is easy to verify that the following relation holds:

2sX(s) + 4s 2 dX(s) ds + s 3 d 2 X(s) ds 2 = 4 a 2 s 2 . ( 14
)
To eliminate the time derivations, which can amplify the noise effects on the signal x(t), Eq. ( 14) is divided by s 4 thus introducing at least an integral effect on each term which contains the signal x(t):

2 X(s) s 3 + 4 dX(s) ds s 2 + d 2 X(s) ds 2 s = 4 a 2 s 6 . (15)
Therefore, Eq. ( 15) can be expressed in time domain as:

a 2 = 30 T 0 (T -ξ ) 2 -4(T -ξ )ξ + ξ 2 x(ξ ) T 5 dξ , ( 16
)
where T denotes the time.

In the same way, considering the terms a i , i = 0, 2 as undesired perturbations to annihilate, the following lemma gives an annihilator of a 1 .

Lemma 2

The linear differential operator Π = Π 1 Π 0 , where Π k = d ds s 2k+1 , k = 0, 1, is an annihilator of a 1 associated to the function ρ(s) = -1.

By considering Lemma 2 it follows that

a 1 = - 24 T 0 3 2 (T -ξ ) 2 -5(T -ξ )ξ + ξ 2 x(ξ )dξ T 4 . (17)
Since the noisy signal x n (t) = x(t) + n(t) is available then, after a change of variable to reduce the estimation interval from [0, T ] to [0, 1], the estimations of a 1 and a 2 , namely â1 and â2 respectively, are computed as â1 = -12

1 0 15ξ 2 -16ξ + 3 x n (T ξ ) T dξ , (18)
â2 = 30 1 0 6ξ 2 -6ξ + 1 x n (T ξ ) T 2 dξ . (19)

DETECTOR SYNTHESIS AND IMPLEMENTATION

To implement the detector signal d(t), a moving window of length T is used. The samples of the current window are then used to compute the value of d 1 and d 2 referred to its end-point. In this section it is shown that, when a peak is detected in d(t), the corresponding t being the mid-point of the analyzing window is declared to be the zero-crossing location. Let us consider, for example, the signal y 1 (t). The same approach can be used for the signal y 2 (t).

According to the previous assumptions, y 1 (t) is equal to zero from 0 to t s , if t s is the position of the zero-crossing. Suppose also that in the interval [t s , T ], with T sufficiently small, such a signal can be approximated with a second order polynomial which, obviously, must have the same zerocrossing in t s

y 1 (t) ≈ a(t -t s ) 2 + b(t -t s ). (20)
Note that, by Eq. ( 4), a 2 = a and a 1 = b -2at s .

Let op[y 1 (t)] be the operator consisting in the computation of a 2 for the signal y 1 (t), i.e.

op[y 1 (t)] =

(Tt s ) 2 15bt 2 s + a(Tt s )(T 2 + 3T t s + 6t 2 s ) T 5 .

(21) Eq. ( 21) represents the value of the operator in t = T as function of t s . To find the position of the maximum of op[y 1 (t)], corresponding to the peak which is desirable to detect, the stationary points of op[y 1 (t)] are needed. Such points can be computed as solutions of the following equation

∂ op[y 1 (t)] ∂t s = 0 → t s (T -t s ) (T -t s )t s - b a (T -2t s ) = 0.
(22) Therefore two stationary points are in t = t s and t = T . The others are solutions of

(T -t s )t s + K(T -2t s ) = 0, ( 23
)
where K = -b a . As the value of K increases, which means that a decreases, one roots tends to t = T 2 while the other one is outside the interval [0, T ], as it can be easily observed by the root-locus in Fig. 4 [START_REF] Ogata | Modern Control Engineering[END_REF].

Since

∂ 2 op[y 1 (t)] ∂t 2 s = - 30 T 5 2at s (T 2 -3T t s + 2t 2 s ) -b(T 2 -6T t s + 6t 2 s ) , (24)
t s →0 ∂ 2 op[y 1 (t)] ∂t 2 s = 30b T 3 , lim t s →T ∂ 2 op[y 1 (t)] ∂t 2 s = 30b T 3 , lim t s → T 2 ∂ 2 op[y 1 (t)] ∂t 2 s = - 15b T 3 .
and the function op[y 1 (t)] presents a positive peak in t = T of amplitude

A peak = lim t s → T 2 op(ρ) = a 2 + 15b 16T (25)
when the function has a zero-crossing for t = T 2 , i.e. there is a delay in the identification of the zero-crossing of about T 2 seconds.

Remark 1

The analysis based on a local quadratic approximation of the signal indicates the possibility of a bias (i.e. t s = T /2) if the quadratic coefficient is not small enough respect to the linear one. This drawback can be overcome by an appropriate choice of the window length, i.e. T should be sufficiently small to have a linear approximation of the signal.

In the case of noisy signals, as it will be shown by numerical experiments in the next section, the method gives satisfactory results, i.e. the zero-crossing is effectively detected with a delay of T /2. A demonstration approach of this behavior is reported below. Suppose that

• a b, i.e. the window length T is opportunely chosen in order to have an acceptable linear approximation of the signal;

• Prob {y 1 (t) = 0, t < t s } < ε.
With such hypothesis

y 1 (t) = b(t -t s ) + n(t), t ≥ t s , 0, t < t s . (26)
To compute op[n(t)] starting from samples of white noise n, let us consider the continuous signal

n(t) = n-1 ∑ k=0 n(kT c )e - (t-kTc) 2 2σ 2 , (27)
with T c = T /(n -1) and σ → 0. In this case by using the above approach

∂ ∂t s op[n(t)] = 30 T 5 lim σ →0 ∂ ∂t s T t s (T -ξ ) 2 -4(T -ξ )ξ + ξ 2 n(ξ )dξ = (28) - 30 T 5 (T -t s ) 2 -4(T -t s )t s + t 2 s n(t s ). (29)
Therefore the stationary points of op[y 1 (t)] are the solutions of the following equation

t s (T -t s )(T -2t s ) + K t s - 3 + √ 3 6 T t s - 3 - √ 3 6 T = 0, (30)
with K = -n(t s ) b .

T/2 T K<0 0.21T 0.79T T/2 T K>0 0.21T 0.79T

Figure 5: Roots-locus of (30).

Fig. 5 shows the root-locus of Eq. (30) varying K. As it can be observed, for small values of K, i.e. n b, the zero-crossing remains localized in T 2 . Moreover for reasonable SNR, the error in the location of the zero-crossing is approximatively bounded by T 2 √ 3 .

NUMERICAL RESULTS

This section includes some numerical results that highlight and point out the advantages and the strengths of the proposed method, in particular it will be devoted to simulated experiments on several signals having a zero-crossing in the interval [0,[START_REF] Vainio | Adaptive Filtering Using Multiplicative General Parameters for Zero-Crossing Detection[END_REF]. For each signal one hundred experiments have been performed adding a zero-mean gaussian white-noise. In all the experiments the SNR, measured in decibels as the logarithm of the average power of the signal's samples and the noise's samples, over the time of the experiment, is equal to 40dB; the sampling time is T s = 4 × 10 -4 . The goodness of the proposed method will be measured in terms of the mean value and the variance of the error ê between the true zero-crossing and the estimated one. The proposed method, namely (FCJ), is compared with a zero-crossing detection method by interpolation [START_REF] Friedman | A Zero Crossing Algorithm for the Estimation of the Frequency of a Single Sinusoid in White Noise[END_REF], [START_REF] Wall | Simple Methods for Detecting zero Crossing[END_REF], [START_REF] Mog | Zero Crossing Determination by Linear Interpolation of Sampled Sinusoidal Signals[END_REF], namely (ZCI). The ZCI implementation identifies two points of the signal: the first just before the positive going zero-crossing and the second just after the same zero-crossing. The hypothesis is that the shape of the signal is very close the straight line near the zero-crossing. The true zero-crossing is then computed by linear interpolation between these two points. A severe test for the zero-crossing methods is when the signal is tangent to the real axis. In this case noise affecting the signal could create false zero-crossings. For example the signal y(t) = sin(2t) cos(t) is tangent to the real axis in t = π/2 and has a zero-crossing in t = π. Interpolation based methods fail to estimate zero-crossings while the proposed one is able to correctly detect zero-crossing as it is shown in Fig. 6.

CONCLUSIONS

A method for estimating the zero-crossings instants of a sampled signal using algebraic derivative approach in the frequency domain was presented. This method provides a detector signal which is equal to zero when the signal has no intersection with real axis and has a peak when a zero-crossing occurs. The main idea was to numerically compute the second order derivative of two signals, built on the available one, which have a discontinuity in the zero-crossing instant. According to a local polynomial model of the signal, the second order derivative was estimated by the iterated integrals of the signal, which behave as low pass filter and mitigate the noise effects. The analysis of the estimator in terms of error on the zero-crossing was discussed. Numerical comparisons with other techniques show a quite satisfactory performance of the proposed method. It seems to be a promising technique for the detection of zero-crossing rates and locations since it is also able to discriminate false zero-crossings, for example in case of signal tangent to the real axis, and true ones.

Figure 1 :Figure 2 :

 12 Figure 1: Block diagram of the zero-crossing system detector.

Figure 3 :

 3 Figure 3: Second order derivative of |y(t)|.

Figure 4 :

 4 Figure 4: Roots-locus of (23).

Figure 6 :

 6 Figure 6: Test function y(t) = sin(2t) cos(t) and the detector signal.

Table 1 :

 1 Table 1 resumes the results obtained in the performed experiments. Mean and variance of the index ê over 100 tests.

	Test function sin(tπ/3 + π/7) -2.24 × 10 -4 1.62 × 10 -5 FCJ / ZCI µ( ê) σ ( ê) 9.70 × 10 -3 1.16 × 10 -5 5 -√ t 3 + 5 1.10 × 10 -3 2.37 × 10 -5 1.43 × 10 -2 1.76 × 10 -5 1 -t + sin(3t) 1.49 × 10 -4 5.50 × 10 -6 5.90 × 10 -3 6.77 × 10 -6


Fabien Theron

Yannick Bidel

Emily Dieu

Nassim Zahzam

Malo Cadoret

Alexandre Bresson

Frequency doubled telecom fiber laser for a cold atom interferometer using optical lattices

A compact and robust laser system, based on a frequency-doubled telecom laser, providing all the lasers needed for a rubidium cold atom interferometer using optical lattices is presented. Thanks to an optical switch at 1.5 µm and a dual-wavelength second harmonic generation system, only one laser amplifier is needed for all the laser system. Our system delivers at 780 nm a power of 900 mW with a detuning of 110 GHz for the optical lattice and a power of 650 mW with an adjustable detuning between 0 and -1 GHz for the laser cooling, the detection and the Raman transitions.

Introduction

Cold atom interferometers [START_REF] Tino | Atom Interferometry[END_REF] have demonstrated to be very accurate and sensitive sensors to measure gravity acceleration [START_REF] Peters | [END_REF]3,4], gravity gradient [5,6] and rotation [7,8]. Now, more and more experiments are using optical lattices [9] to improve the performance of these sensors. Indeed, optical lattices allow to control in an extremely precise way the momentum of the atoms and thus its movements. It is therefore possible to levitate atoms with a perfectly controlled force and thus to build compact gravimeter with levitating atoms [10,11]. It is also possible to increase the momentum transfer in an atomic splitter and thus to increase the sensitivity of the sensor [12,13,14]. Optical lattices can also be used to impart precisely a well controlled velocity to the atoms which is very important for gyroscope or large atomic fountain experiment [15]. Finally optical lattices can be used to separate and transport a cold atom cloud in order to build a sensor with spatially separated clouds of cold atoms and thus to measure gravity gradient [16]. The disadvantage of using optical lattices in atom interferometry sensors is that one needs another laser generally detuned around 100 GHz from the atomic transition with a power of at least few hundred mW. Until now, the optical lattices were implemented with an independent laser different from the one used for the atom interferometer experiment. The whole laser system for the sensor is thus more complicated, larger and less robust. This is a real inconvenient if one builds a compact and robust sensor [17] for geophysics, navigation or space application. Many developments [18,19,20] have been done over the last few years to achieve compact and robust laser systems for atom interferometers but none of them were addressing optical lattices.

In this article, we present a laser system which gen-1 erates the lasers for a cold atom interferometer and optical lattices with only one laser amplifier which is usually the less reliable part. Our laser system is based on a frequency doubled telecom fiber bench which addresses rubidium atoms (λ = 780 nm) and is derived from the laser system described in the reference [10] in which two independent laser systems composed of a laser source at 1.5 µm, an EDFA (Erbium Dopped Fiber Amplifier) and a second harmonic generation bench were implemented: one for the cold atom interferometer and one for the optical lattices. The laser system presented in this article combines the two laser sources at 1.5 µm thanks to an optical switch and thus allows to use only one EDFA. This trick is possible because the optical lattice laser and the cold atom interferometer laser are not needed at the same time. In the first part of this article, we will present the general architecture of our laser system then we will describe the dual wavelength second harmonic generation and finally we will present the lock of the optical switch.

General description of the laser system

Our system generating the lasers for the cold atom interferometer and the optical lattices is described on Fig. 1. The atom interferometer laser source (AI laser) which provides the frequencies for laser cooling and trapping, detection and Raman transition is an erbium fiber DFB laser at 1.5 µm (output power: 20 mW, linewidth 2 kHz). This laser source is locked relative to the rubidium transitions with a detuning which can be adjusted from 0 to -1 GHz. The repumper and the second Raman frequency are generated thanks to a phase modulator fed by a radio frequency at 6.5 GHz or 6.8 GHz. This part of the laser is described in details in reference [21]. The optical lattice laser source (OL laser) is a DFB laser diode at 1.5 µm (Avanex DFB, output power: 10 mW, linewidth 1 MHz). The OL laser is detuned by 55 GHz at 1.5 µm relative to the AI laser which gives a detuning of 110 GHz at 780 nm. The detuning of the OL laser is controlled thanks to a beat-note at 1.5 µm between the two lasers. In order to have a beat-note at lower frequency easily measurable, a phase modulator fed by a radio frequency at 6.7 GHz is put on the optical lattice path and generates sidebands separated by 6.7 GHz. Then, a beat note at 3.1 GHz is measured between the AI laser and the 7 th sideband of the modulated OL laser which allows to control the frequency of the OL laser.

The AI laser and the OL laser are combined at 1.5 µm thanks to an electro-optical modulator which acts like a continuous optical switch between each laser. This component is a wave-guided Mach-Zehnder interferometer (MZ) from EO Space with an extinction of 30 dB and a period of 5 V. Because the path difference in the MZ is close to zero, interference signals of OL laser and AI laser are in phase opposition and the output of the MZ can be described by Eq. 1. Thus changing the MZ voltage command V allows to select the AI laser or the OL laser at the output.

I out = I AI × cos 2 (αV ) + I OL × sin 2 (αV ), α ∈ ℜ (1)
Then, the laser goes through a 5 W Erbium Doped Fiber Amplifier (EDFA). In order to avoid damaging the EDFA, one must make sure to have a constant power at its input. The power of both laser sources have therefore been adjusted to have the same power at the output of the MZ regardless of the MZ command. Finally, the laser is sent to the dualwavelength second harmonic generation bench.

Dual-wavelength frequency second harmonic generation

The dual-wavelength second harmonic generation configuration that we have implemented is similar to the one described in reference [22]. It consists of two bulk PPLN crystals, in double pass configuration (Fig. 1). At the output of the EDFA fiber, the laser is focused in a first PPLN crystal thanks to a 75 mm focal length lens (L1), and the double pass configuration is realized thanks to a 10 cm curvature radius gold mirror (CM). At the center of the crystal, the waist of the laser is 23 µm. This first PPLN is phase matched for OL frequencies (T OL = 133.45 o C) and allows to get an OL laser power of 900 mW at 780 nm with an input power at 1.5 µm of 4.3 W. The measurement of the power at 780 nm versus the power at 1.5 µm is given on Fig. 2. The experimental data are fitted by Eq. 2 which takes into account the pump depletion [23].

P 780 = P 1.5 tanh 2 ( ηP 1.5 ) (2)
The fit gives a conversion efficiency of η = 5.4 %/W . The dichroic mirror (D) transmits the AI laser at 1.5 µm to the second PPLN, and reflects the OL laser at 780 nm. The AI laser at 1.5 µm is focused in the second PPLN crystal thanks to a 60 mm focal length lens (L2) in 2f-2f design, in order to get the same waist for both crystals. The double pass configuration in the crystal is realized thanks to a 10 cm curvature radius gold mirror. The second PPLN is phase matched for AI frequencies (T AI = 134 o C) and allows to get an AI laser power of 650 mW at 780 nm with an input power at 1.5 µm of 3.5 W. For this crystal, the fit of the power at 780 nm versus the power at 1.5 µm gives a conversion efficiency of η = 5.1 %/W (see Fig. 2).

Effect of the finite extinction of the optical switch

Due to the finite extinction of the optical switch, the OL and the AI laser beams will be contaminated by emission rate. We measure an extinction of the optical switch equal to η MZ = 1 × 10 -3 . By measuring the power at 780 nm versus the frequency of the laser for the OL crystal (see Fig. 3), one finds that the ratio between the conversion efficiency at the AI frequency and the conversion efficiency at OL frequency is equal to η conv = 1.5×10 -2 . The relative power at AI frequency in the OL laser beams is equal to ǫ AI = η 2 MZ × η conv = 1.5 × 10 -8 . During the optical lattice stage, the AI can be detuned by ∆ AI = 1 compared to the atomic resonance. Thus, the AI parasite laser line causes a spontaneous emission rate ǫ AI × ∆ 2 OL ∆ 2 AI = 1.8 × 10 -4 less than the spontaneous emission rate caused at the OL laser line. We can conclude that the effect of AI parasite frequency in the OL beam will be completely negligible.

Lock of the Mach-Zehnder optical switch

The MZ switch presents a long term drift on hours time scales which results in a drift of the command voltages needed to select AI and OL frequencies. A continuous correction of the MZ voltage command has thus been implemented by modulating the volt-age of the MZ at 50 kHz and by detecting this modulation on the power at 780 nm of the OL beam (Fig. 1). For that purpose, a part of the OL beam is sent to a photodiode. Then the signal is amplified, demodulated at 50 kHz and low pass filtered. The error signal obtained at this point is integrated, summed to the electric command which switch from AI to OL and sent to the MZ input. Because the error signal is only available when the OL frequency is selected, an electronic switch (LS) has been inserted before the integrator in order to activate the lock only during the optical lattice stage. During the AI stage, the output of the integrator will thus stay at a constant value. This non correction of the drift of the MZ during the AI stage is totally acceptable because the duration of this step is very short (∼ 1 s) compared to the time scale of the drift of the MZ (few hours).

Conclusion

We have developed a compact and robust laser system providing all the lasers needed for a cold atom interferometry sensor using optical lattices. The association of a fibered Mach-Zehnder electro-optical switch and a dual-wavelength second harmonic generation unit has allowed us to use only one laser amplifier and thus lead to a laser system which is more reliable, more compact and with a lower electrical consumption. We have checked that the finite extinction of the switch does not cause additional spontaneous emission in the optical lattice. A power of 650 mW for the cold atom interferometer and 900 mW for the optical lattice have been obtained. These laser powers can be increased by using a 10 W laser amplifier which is a component commercially available. By taking parameters of the dual-wavelength architecture and using Eq. 2, one obtains an OL laser power of 3 W, and an AI laser power of 2 W. The laser system presented in this article is addressing transportable and onboard atomic sensor with enhanced performance thanks to optical lattice technology and can have applications in space technology, geophysics or navigation.

Figure 1 :

 1 Figure 1: Diagram of the laser system and the Mach-Zehnder lock. AI: Atom Interferometry, OL: Optical Lattice, C: coupler, PM: Phase Modulator, M-Z: Mach-Zehnder, EDFA: Erbium Doped Fiber Amplifier, L: Lens, PPLN: Periodically-Poled Lithium Niobate, D: Dichroic mirror, CM: gold Concave Mirror, S : Shutter, PD: PhotoDiode, LPF: Low Pass Filter, LS: Lock Switch.

Figure 2 : 3 :

 23 Figure 2: Second harmonic generation at 780 nm of the laser. On top (a), doubling efficiency of the OL laser, with the OL PPLN, depending pump power. On bottom (b), doubling efficiency of the AI laser, with the AI PPLN, depending pump power.

ACKNOWLEDGEMENTS We thank F. Nez, from the Laboratoire Kastler Brossel (LKB), for his help on the project. We acknowledge funding support from the Direction Scientifique Générale of ONERA and the Direction Générale de l'Armement (DGA).


Damien Lefebvre

Philippe Chevrel

Sébastien Richard

Control analysis tools for active attenuation of vehicule longitudinal oscillations

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

This paper presents a set of control analysis to evaluate the quality of a given controller, in the context of driveline control. Aft er having introduced the simplified mechanical model from which the analysis is perfonned, different indicators are presented. These indicators are chosen in order to give concise but essential infonnation on the control system. In addition to the classical robustness margins, they evaluate some aspects of robust perfonnance and the influence of the backlashes.

Automotive systems, Robust control, Vehicles and transportation systems, µ analysis

I. INTRODUCTION
Embedded processors in actual vehicles allow to implement some "advanced" control strategy in order to improve vehicles behavior. However the lack of methodology restrains the use of such strategy and few paper demonstrates their interest in the context of vehicle dynamics control [START_REF] Hrovat | Bond graph modeling of automotive power trains[END_REF]. This study takes place in a more Aerodynamics forces and the road profile are not considered.

Each parameter of this model is related to vehicle characteristics:

� Ieng and Iveq (kg.m2) correspond respectively to the inertia of the engine flywheel and to the equivalent inertia of the vehicle.

-;. K,. q (Nm/rad) is an aggregate stiffness related to the driveshafts. the clutl:h, the gearbox and the tires stiffness. -,, A ,.11 and A vt (Nm.s/rad) include all existing frictions.

-;. R,q (m) is the equivalent radius of the wheel through the reduction of the gearbox and of the drive axle. s (K eq (1 veq + I en11 ) + Aenx Aw )+ K eq (AV/ + Aenx )

These equations are representative of a system with a badly damped mode.

The purpose of the various design methods is to reduce the oscillations on vehicle acceleration. It consists in improving the damping ratio i;, of the function G.,.,{ s).

The strong constraint lies in the delay introduced by the four-stroke engine and its close control. It varies from 20 to 200 ms following the value of the engine speed.

Moreover the robustness of the controller must he significant with regard to neglected dynamics and uncertainties on parameters. The numerical nmtroller runs with variable step and is deduced according to Richard [START_REF] Richard | Synthese d'un regulateur pour un systeme a retard et a periode d' echantillonnage variables : Application au domaine automobile[END_REF] from the continuous controller synthesized with the maximum delay. The delay margin must he higher than one period of sampling in the worst case.

Two design methods have successfully fulfilled this particular problem. The l" one is based on an approach in term of pole placement [START_REF] Richard | Amelioration de l'agrement de conduite par amortissement actif des oscillations longitudinales de transmission[END_REF] and the 2"d one is the tL optimization [ 11 ]. Each proposed method presents a low number of independent tuning parameters.

w ---�-----•••-••-------r-�-•----,-- . _: i /, � 0 -•••••••••--•-• • -..... - -..... • --.
-�-... : . :-::. :-.. ._ ... . /. ; . . . '":.: .":'-.•--: • ..:... ;

I :l r : .__�_ •---_ ••• _ • • •• _ •� -' "-- " �-_ -_ :-: _ _ • . :-:_ . - .:: _ " .__ : ---'°_ -- -=-_ ----=-_ ----1 - _ ml.__ l_ "'--_ •=----_ -"'-- _ -=- 10• • 10 ° 10 1 10 2
Frequency (rad/s)

A . The describing function method

The describing function method is well known for the analysis of the behavior of the closed loop system with a separable non-linearity. The analysis is based on the harmonic study of the signals in the loop (see figure 3).

The assumption is made that the linear part of the system R(s) filters sufficiently the high frequencies so that only the I '1 harmonic of the signal u = lf1y) is considered.

R(s)

u= 'l'(y) y 'l'(. )

! N(x,)=1-� [ asin ( � )+ � � ( L1 )2] 7r 2x1 2x1 vi-l hi) if X1 �rz and N(x, )=o if X1 < fz ( 2)
The connection of the linear and nonlinear part will be unstable if the following relation is checked :

Or if l + N(x1 , m)R(jm)= 0. R(Jm)= � (x1,m) ( 3) L(x1,m)= � (x i , m
) is called the critical locus. For the threshold, this locus corresponds in the Nyquist plan to the negative real axis lower than (-1). The Nyquist plot of R(jm) and the critical locus plot allow to know the points for which a risk of instability exists. Indeed the equation ( 3) is verified when R(jm) cuts U.x1;m) in the Nyquist plan. The LOEB criterion is used to determine the stability of these points. The coordinates of the contact point provide information on the characteristics of the limit cycle. represents the smallest "size" of the matrix L1 for which the matrix 1-MLJ becomes singular.

The µ. analysis requires a formalization of the uncertain system as in figure 5.

Fu(H(s),0) = F1(P(s),K(s)) represents the nominal closed loop system without the non linearity where P(s) is the synthesis model and K(s) is the controller. Fu(H(s),L1(s)) = TzvCs) is the disturbed closed loop system. This theorem forms the basis of µ.-analysis. See [START_REF] Zhou | Robust and optimal control[END_REF] for the theory and [START_REF] Young | Computation ofµ with real and complex uncertainties[END_REF][START_REF] Balas | µanalysis and synthesis toolbox[END_REF] for instance, for the computation of µ4.

L1(s) H(s)

This theorem can be formulated for the generalized stability see [START_REF] Ferreres | Application of real/mixed µ computational techniques to a missile autopilot[END_REF]: If H( s) has only poles in the Q domain, the poles of the system of figure 5 will remain in the Q domain for any uncertainty LJ(s) of type ( 4) such that l!Lt(st <a if and only if:

'rfseoQ µjH Y ,,., .,(s))sa-1 [START_REF] Ferreres | Application of real/mixed µ computational techniques to a missile autopilot[END_REF] where ()Q is the border of the Q domain.

Here is an example of domain parameterized by a damping ratio c;. If uncertainties are real, µ.,J(H(s)) can be discontinuous [START_REF] Packard | Continuity properties of the real/complex structured singular values[END_REF] and the maximum value of µ.A is difficult to find. The v analysis can be applied for avoiding the frequency grid see [START_REF] Fan | Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics[END_REF][START_REF] Friang | Approche generique Hoo/v pour la synthese et !'analyse d'un pilote automatique de missile soumis a une aeroelasticite importante[END_REF] for more details.

IV. ANALYSISTOOLS

The objectives of the feedback are:

• to preserve the stability of the system,

• not to modify the dominant real mode,

• to ensure a minimal attenuation of transmission oscillations despite the uncertainties on parameters .

Moreover, the parameters of the model vary or are badly known. It is thus significant to be able to evaluate the good behavior of the objectives with regard to all uncertainties with simple indicators. A. Indicators of stability robustness 1) Traditional margins

The gain, phase and module margins are significant indicators to guarantee the stability of the closed loop. The delay margin is representative of the robustness of the loop with regard to parasitic pure delays and neglected time constants. This point is all the more important to check that a variable delay is present in the loop.

2) Dedicated margins

The parametric robustness is significant to check.

We consider an uncertainty ,1L on the system bounded by the pulsation wand the gain of the system L : ILIL(jm � < P�L(jm � Ujw) = K(jw)G(jw) is the transfer function in open loop. K(jm) is the controller and G(jm) is defined by the equation ( 1). According to the Nyquist criterion, stability will be ensured if uncertainty remains lower than the distance from L(jm) to the point (-1). We can then write that the system will be stable if v w E 9r PwjL(jw � < 1 1 + L(Jw � Or if jT(Jm � < JP w fJ can be calculated with the following expression: p sup �T(Jm� lls.T(s�L-1 wE 9\+ (

7)

It should be noted that fJ is the dynamic margin Tdyn• introduced by de Larminat [START_REF] Larrninat | Le placement de p6les[END_REF] which gives an approximation of the delay margin. A great value of Tdyn will be the proof of a good robustness.

B. Nominal perfonnance indicators

The reduction of the acceleration oscillations is equivalent to reduce the maximum value of the frequency response of the transfer function acceleration Gach) defined in equation [START_REF] Balas | µanalysis and synthesis toolbox[END_REF] (see figure 2). For the nominal system, the reduction ratio a, of this peak is representative of the level of performance. Its expression is given by: a, = log(2�a ) [START_REF] Friang | Approche generique Hoo/v pour la synthese et !'analyse d'un pilote automatique de missile soumis a une aeroelasticite importante[END_REF] where � is the damping ratio of the pole responsible for the resonant peak of the acceleration function Gacc(s) in open loop.

The dynamics of the system must not be modified. The controller must intervene the least possible in the frequential bandwidth. The control is supposed active when the input torque !input is modified of more than 10% by the corrective torque i1I' (see figure 7). This can be expressed by the following relation: The engine speed is used for the feedback (see figure 7).

I iJI' I = !TI � 0,
This data is measured on the engine flywheel, source of parasitic noises. The closed loop sensitivity to the measurement noises can be analyzed with the help of the indicator N introduced by de Larminat [START_REF] Larrninat | Le placement de p6les[END_REF]:

N _ ll S (s ) s.

C. Robust performance indicators I) Calculation of guaranteed damping ratio

The small µ theorem on the generalized stability, presented in part III, is applied to cakulate the damping ratio which is guaranteed for the worst case of parametric uncertainties. The area study Q is represented in figure 6.

Q is parameterized by the damping ratio �8 and the border J.Q is given by :

ag(� 8 ) = {-�gmn ± j�l-� 8 2 wnlmn E 9\+}
4'8 is calculated by dichotomy. The µ-analysis on a border J.Qf. 4'1) provides a value of µmax • If µmaxis lower than the desired value, ;1 is increased and µm ax is calculated with the new value of J.Q( 4'1). The calculus is executed until the desired precision on ;1 is obtained. This method allows to guarantee a minimum damping ratio in relation to the wished uncertainties intervals.

Inversely, the small µ theorem can be applied to determine the variation intervals of parameters which will ensure the stability.

2) Nonlinear analysis

The describing function method gives good results on this model. But the result is only valid for the set of parameters considered. The concept of parametric robustness misses.

To fill this lack, the robustness margin introduced by The non-linearity 1P is represented by his describing function N(X, m) defined in ( 2). There will be existence of a limit cycle if the following equation is verified :

det{I -Fu {H c(j w ), t1)N (X ,w ))= o.
The margin is then defined as follows :

r(X, w) = inf (cr(t1)/ det (l -Fu ( H c (jw), t1) N(X ,w ))= 0).

ilE,d

It is homogeneous to the inverse of a singular value.

The weaker the margin r(X, w), the closer to the limit cycle of characteristic (X, m) is the system. r(X, m) characterizes the distance from the closed loop system to the cycle limit (X, m). It allows to deduce the way in which the limit cycle will be modified when the parameters of the model change.

I) H�method

The method is based on frequential weightings on sensibility functions. Two independent tuning parameters are sufficient to obtain the objectives, see [11) for a detailed methodology. The 1 '1 controller called type I is synthesized with a direct method. The reduction of the oscillations is ensured by the 1" tuning parameter via the transfer function G acd s )S( s ). 2) Robust pole placement method

The method constraints the closed loop poles to be in a and damped by the desired damping ratio �(see [START_REF] Richard | Amelioration de l'agrement de conduite par amortissement actif des oscillations longitudinales de transmission[END_REF]).

B. Results

The guaranteed damping ratio is calculated on realistic intervals of parameters.

Here is the comparative analysis of all the indicators previously presented, applied to the model and the 3 a detailed frequency analysis but don't require to be an expert to be understood. Their examination will constitute for control designer in PSA a determinate stage before making more extensive simulation or experimental tests. Some of the proposed indicators are classical while some others take advantages of recent development in µ. or v-analysis. Some are "universal" while others are more or less dedicated to the controlled system considered here.

Nevertheless, each presented indicator can be extended to the multivariable control case (see Scorletti [START_REF] Scorletti | Robustness analysis with time-delays[END_REF] for example for multivariable delay margins). This is an important feature because future vehicle will include multi-torque providers in addition to the four-stroke engine (electric motor, piloted clutch) leading to multivariable control problem.

Finally, the proposed control analysis tools complete the methodological effort made in [START_REF] Richard | Amelioration de l'agrement de conduite par amortissement actif des oscillations longitudinales de transmission[END_REF]11) for model reduction and control design.

Fig. 1 .

 1 Fig. 1. Simplified model of powertrain The model of the powertrain is a system of high order including many non-linearities. This model is too complicated for applying simple indicators. The model must be reduced. The modal reduction is not adapted because it doesn't keep a physical identity on each parameters. The mechanical reduction is more appropriate. The reduction of the complete model allow to obtain a model of order 3 and to preserve the main non-linearity. This model corresponds to an inertia in rotation involving another inertia by the way of a flexible connection and a threshold. The input of the model Teng (see figure 1) corresponds to the torque provided by the engine. The angular velocity ill eng of the 1 st inertia Ieng is the image of the engine speed while Yveh represents the vehicle longitudinal acceleration.

  J.. (/ and (J (rad) are the angular bounds associated with the backlashes brought at the back gearbox input. Some assumptions were made to establish this simplified model. It is supposed that the clutch is stuck and that the differential is blocked. The inertias of the gearbox driveshafts and of the wheels are neglected, thus the high frequency dynamics are not modeled. Numerical values of each parameter are derived from the experimental data analysis and from information provided by the • manufacturer. Each coefficient is uncertain. The frictions A mi( and A vt are badly known. The value of the engine inertia l,.R is precisely known while the equivalent inertia of the vehicle fw•q can change depending on the vehicle load. The backlash is measured directly on the vehicle. It will be neglected during the control design whkh is based on the simplified model. Therefore, the engine speed w .. 11 and the vehicle acceleration Yveh are related to the torque input r, .• R by the two transfer functions:G(. ,• }= w,n, 30 s2/vn1 + s;\, + K,.

sR., 1 G

 1 ,.,. . •-)=-. -= K,.,,-- I '"' D(s) with D(s) = s 3 / VN/ I en /( + S 2 (ten!( Avt + A,,nx I veq ) + ( l )

  :;: :=:::::t==========:J.!.�---L ..... . G ecc (s) Vehicle acceleration G(s) Engine speed -100 '-----==========='...._ _L ____ ..J ...... : � '. ) ........... . .

Fig. 2 .

 2 Fig. 2. Bode diagram of G(s) function engine speed and G0,. ,{s) function acceleration

Fig. 3 .

 3 Fig. 3. Non linear feedback connection This study is valid if the non-linearity 'ff is symmetrical compared to the origin and if it does not depend on time. Then, the non linearity can be replaced hy his describing function defined by: M(x1,w)+ jQ(x1,m) N(x1,w) with !M (x1 ,w) = * r 'P(x1sin(wt )�in(wt )dt Q(x1 ,m) = * f 'ff(x1sin(wt )�os(wt }It T and x1 are the period and the amplitude of the input signal y. The non-linearity included in the model of the powertrain and allowing to take into account the looseness . is a threshold with an amplitude of L1 = 1£f -BI. His dcscrihing function is real and corresponds to:

Fig. 4 .

 4 Fig. 4 . Graphical method for detecting the existence of limit cycles B. µ analysis definition The concept of structured singular value was first introduced by Doyle and Safonov [3,15]. Let us recall its definition and some properties: !µ!!( M >=('.;:�\cr( LI)/ det( 1-MLI) = olf µ !! (M )=0, if \fLIE!d,det(l-MLl);tO where the set L1 is defined as follows:{L1(s)=diag�11, , .. .,o,I, ,E11c , .•• ,EJc ,L11(s),. . .,L1q(s)Ji( 4) L1�

Fig. 5 .

 5 Fig. 5. Standard form for µ. analysis Small µ. theorem (stability) If H(s) has only left half plane poles, the system of figure 5 is stable for any uncertainty L1( s) of type ( 4 ) such that J JL1(s t <a if and only if: Vwe9\+ µ.!!.. (Hy,,u)iw))sa-1

Fig. 6 .

 6 Fig. 6. Study area in the complex plan

Fig. 7 .

 7 Fig. 7. Representation of the closed loop

1 rinput

 1 We introduce B10 = m M 10 -mmw where WM10 and m,,,10 are defined by:{m M 10 = sup { m } �QIO { } QIO = {m1 1 7'(Jw � � 0,1} ( 9) mmw = mf m wEQ10The greater B10, the wider in frequency domain the action of the controller is.

  (s �1 2 ll s. (s �1 2 ( 10) S.(s) is the command sensibility function defined by K(s)S(s) and S(s) is the sensibility function of the loop. The function S(s)Su(s) characterizes the capacity of the closed loop system to transmit the noises when the control is active, while Su(s) is representative of the total transmission of the noises. The lower N, the less sensitive the closed loop is to the measurement noises.

Ferreres [ 5 ]Fig. 8 .

 58 Fig. 8. Study diagram for the calculation of the robustness margin

  Controllers designTwo methods will be compared : the robust pole placement method[START_REF] Richard | Synthese d'un regulateur pour un systeme a retard et a periode d' echantillonnage variables : Application au domaine automobile[END_REF] and IL optimization [11). The two methods are based on an iterative design. All the objectives are not taken into account a priori, this being explained partly by the reduced number of tuning parameters. The same model of synthesis is used for calculating the 3 regulators with a delay of 0,2 s approximated by a 4 order Pade function. The aim of the feedback is to reduce the oscillations and to assure the biggest parametric robustness. All the controllers obtained are stable.

  The 2"ct parameter guarantees the complementary module margin via the transfer function T(s) = l-S(s). These constraints don't avoid the cancellation poles-zeros between the system and the controller. The type II controller is obtained with the same tuning parameters but the criterion is widened. It takes into account a weighting on the transfer function K(s)S(s). The parametric robustness is increased and it avoids the cancellation poles-zeros.

  particular locus in the complex plan. The dominant real pole is kept. A first part of the closed loop poles are deduced from the open loop poles projection on the vertical fr! . The second part of the poles are calculated with the open loop zeros previously stabilized

  The indicators allow to turn down the regulator of type I. Indeed it has the lowest robustness with regard to parametric uncertainties. Its noise sensitivity is comparatively worse. It does not guarantee any performance and the characteristics of the limit cycle are unreasonable. This controller didn't intentionally fill the specifications. It is confirmed and proves the accuracy of the indicators. On the other hand the indicators do not allow to decide between type II and RPP controller. Each one has advantages in his field. The RPP controller presents better margins, but type II controller is more robust compared to the parametric variations and has a weaker limit cycle. The application of the robustness margin gives an additional indication to evaluate the quality of the two controllers ..

5 �Fig. 9 .

 59 Fig. 9. Robustness margin for pole placement controller and H.. type II controller The dark lines on the figure 9 correspond to low value of robustness margin. It confirms the results of the describing function method. Indeed, the margin is very low near the characteristics of the limit cycle found previously. If the parameters are modified, the size of the uncertainty increases and the characteristics of the limit cycle change: the pulsation °'NL will vary in little proportion whereas the amplitude xNL depends largely on the parametric variation. The pole placement controller tolerates weaker variation on parametric uncertainties in comparison with the H.. controller. VI. CONCLUSION Some control quality indicators have been proposed in the context of the vehicle driveline oscillations attenuation. They concentrate information obtained from

  Control analysis tools for active attenuation of vehicle longitudinal oscillations D. Lefebvre+*, P. Chevrel+ and S. Richard* + IRCCyN, UMR C.N.R.S 6597, 1 rue de la Noe, BP 92101, 44321 Nantes Cedex 3, France . if : +(33) 2 51 85 83 40, Fax: +(33) 2 51 85 83 49, E-mail : philippe.chevrel@emn.fr PSA Peugeot-Citroen, Direction de la Recherche et de !'Innovation Automobile (DRIA), Route de Gisy 78140

	Velizy Villacoublay France if : +(33) 1 41 36 71 94, Fax: +(33) 1 41 36 76 28, E-mail: lefebvOO@mpsa.com


Introduction

Studies on the interaction between societies and mangroves are quite recent, appearing slowly in the 1980s, and part has been done by natural scientists, noticing the knowledge gap. It partially explains why the social research on the mangroves has seldom used concepts or theories applied by social scientists to other ecosystems such as social representation theory (SRT). SRT considered that some objects become "social" when they create a system of opinions, beliefs and knowledge socially elaborated and distributed among social groups. They can explain and orient practices and define the identity of an individual or a group, creating a feeling of belonging.

Better understanding the SR of the mangrove by a society helps to understand what can be implemented easily from a management plan and which points would need more discussion and compromise with and between the inhabitants.

Conclusive discussion

The social representation of the mangrove in Mayotte Island LONGÉPÉE Esméralda 1, 2* , AHMED ABDALLAH Anliati 2, 3

1 Centre Universitaire de Mayotte (CUFR), France 2 ESPACE-DEV, Univ. Montpellier, IRD, Univ. Antilles, Univ. Guyane, Univ. Réunion, Montpellier, France 3 PRODIG, CNRS, Univ. Paris 1 Panthéon Sorbonne, AgroParisTech, Paris Sorbonne Université, IRD, Univ. de Paris, France *esmeralda.longepee@univ-mayotte.fr

Study sites

Mayotte Island (374 km 2 ) is part of the Comoros Archipelago and a French department since 2011, counting 256,000 inhab. ( 2017). In 2018, mangroves occupied a surface of 725 ha and 26% of the coastline. The social representation of an object can be analyzed considering three dimensions:  Cognitive: the mangrove is considered in a positive way regarding the elements structuring the ecosystem and its function, but a strongly visible anthropogenic impact (rubbish) on the ecosystem is also highly mentioned as characterizing the mangrove  Normative: the functional value given to the mangrove is strong with both the answers on ecosystem services and the quotation describing the mangrove as a protective person; the esthetical value is present too with inhabitants enjoying the view of the mangrove  Expressive: there are both positive emotions, such as the quietness and nice memories procured by the mangrove, and negative ones, such as an aversion because of rubbish, waste water and slum houses

DEMBENI MANGROVE BANDRELE MANGROVE

Free listing question on "mangrove"

On the free listing question asking for three words linked to the object "mangrove", 29% did not give any word.

For the 310 respondents, the most mentioned words are linked to knowledge describing natural (13% sea, 9% trees, 6% crabs, 5% fish) and anthropogenic (17% rubbish, 4% houses) structures and two are on the mangrove functions (15% protection against sea, 3% coolness). The expressive dimension is illustrated by beautiful (13%), quiet (4%) and dirty (4%). The words beautiful (13%) and landscape (4%) highlight an esthetical value. When respondents were asked what they liked and disliked about the mangrove of their village (see graphs), we can see that ecosystem functions are highly valued by almost half of respondents but there is a higher consensus on what they dislike: the anthropogenic impacts were mentioned by 67% of them, with the main one being rubbish (58%). They mentioned in second the landscape-ecosystem features, showing that some attributes of the mangrove such as trees, animals can be liked and others can be disliked such as mud, crabs and mosquitoes.

  you like in the mangrove of your village?" "What do you not like in the mangrove of your village?" N=310, 760 words mentioned at least 3 time N=439, multiple answers allowed  669 answers Results c.: Jeanson M. c.: Jeanson M.

"

  The mangrove for me is like a big person, a mother who looks after her children. It protects the villages closed to the sea from the strength of waves and treat the inhabitants. It is a pharmacy for new-born babies. The mangrove feeds inhabitants too."


 Overall, respondents have a limited knowledge of the health of mangroves in Mayotte but few are aware of it.

 Focus on causes of health degradation rather than consequences.

 Understanding that natural causes (seaward erosion) can explain mangrove degradation (south mangroves) based on knowledge of previous research.

 Concern (no local studies) that mangroves neighbouring highest urbanised area are in bad health because of anthropogenic impact. Interviewees considered that waste impacts aesthetics: "a mangrove in good health is a clean mangrove" (E3, Marin Natural Park).

Introduction

The management of mangrove is decided and applied, in western territories, essentially, by planners, managers and decision-makers under the influence of non-governmental associations. Regarding the mangroves of the Mayotte Island, it is the premises both for scientific research and for the development of management plans.

Mayotte is a tropical Island of 375 km² located in the south western Indian Ocean which counts 700 ha of mangrove covering 29% of the coastline. Since 2011 when Mayotte became a French department public institutions have to produce management plans imposed by French and European rules.

Quickly after its emergence in ecological scientific literature, the concept of ecosystem state is used in strategic documents. Indeed, in 2008, the Marine Strategic Framework Directive demands to achieve the good ecological status without defining the notion. Defining criteria of ecosystem health that incorporate an understanding of uncertainty or threshold limits is needed to trigger management intervention (O'Brien et al 2016). In this context, we examine how this concept is understood by stakeholders in Mayotte and applied to mangroves and make a comparison with indicators from the scientific literature.

Conclusion

-Although stakeholders are involved in ecosystem health evaluation, their understanding of the concept is limited. The format of the interview, however, relies only on their memory to define a complicated concept. -The ARESMA project implements knowledge exchange between scientists and stakeholders in order to improve mangrove health evaluation.

The health of mangroves in Mayotte Island (Indian Ocean)

Comparison of stakeholders and scientific knowledge

Reference

Out of 43 mangroves located on the background map presented to the interviewees, 11 were not assessed. All of them are less than 2.15 ha, however, some small mangroves have been assessed, the ones mostly located in a protected or labelled area.

Overall, respondents more easily chose bad (87 responses) or good (73 responses) to characterize the mangroves they knew rather than the fair state response (6 answers).

The mangroves for which more than half of the managers have expressed an opinion are:

 the island's largest mangroves  those known for their rapid evolution (high growth at Badamiers, see box below ; rapid decrease at Dapani).

It appears that the mangroves in the south of the island that undergo a rapid decrease and those adjacent to the Mamoudzou urban area were more easily designated as mangroves in poor condition.

On comments given by the 16 interviewees on their maps, only 3 questioned the notion of "health state" and 4 their ability to judge the health state of the mangroves of Mayotte.

The interviewees cited on average nearly 3 criteria to characterize a mangrove in good condition. For half of the respondents, the criteria mentioned are associated with the state of the plant community and the low anthropogenic impact.

  Indicate on a map presenting the mangroves of Mayotte their health state (good, fair, bad) for the ones that the interviewees knew  Comment the mental map produced  Explain which criteria can be used to decide that a mangrove is in good health

Synthesis of perception of mangrove health in Mayotte by interviewed stakeholders

Expansion of the mangrove of Badamiers "The mangrove of Badamier, I think it is in expansion, I put a plus [good health]" (E15, birds NGO) "It's really heaps and heaps of waste piled up in this place but paradoxically the mangrove likes it because it obviously proliferates. […] And it get directly the waste water. […] So, I do not know if I can say it is in bad health. […] a priori you might be tempted to say "oh my god it's the sickest because it's swimming in this waste" but nevertheless this one, it doesn't disappear..." (E2, Coastal Protection Agency)

The mangrove of Badamiers, reported on 19th century maps, they no longer existed in 1950, completely cut down. Regeneration was further prevented by the construction of a road and dike in 1848 that stopped saltwater from the lagoon attaining the marsh. The remarkable recovery of this mangrove stand (37.31 ha in 2016), despite the highly urbanized context, is related to renewed seawater intrusion following progressive degradation of beachrock outcrops that hitherto formed a natural barrier between the mangroves and the lagoon.  Focus on plant community ("tree height, density, juvenile regeneration") (n=17) but few at scale of ecosystem (assemblage composition and zonation) (n=4). Types of criteria and wording reflect limited knowledge of scientific literature on ecosystem health:  Criteria of human impact/desired state, not health ("clean", "no wood harvesting") (n=8)  Double-sided criteria that can respond positively to wastewater ("big tree", "green mangroves") or that might not only have a linear tendency ("natural cycle" of surface dynamics) are used as indication of good health (n=7)  Poorly defined criteria ("living mangrove", "green mangrove", "equilibrium state") (n=5)

Perception of mangrove health

1

  Condition of the plant community: living or dead mangrove trees (n=7), living mangrove (n=2), green mangroves (n=2)  Low anthropogenic impacts: clean (n=5), low anthropogenic impacts (n=3)  Structure of the plant community: tall trees (n=1), complete zonation pattern (n=3), presence of the back mangrove (n=1), dense (n=3)  self-regeneration: self-regenerates (n=3), many juveniles (n=3)  Fauna: presence (n=4), no exotic species (n=1)  Surface dynamics: increasing (n=2), stable (n=2)  Performs all functions (n=2)  Significant biodiversity (n=1)  State of equilibrium (n=1) LONGEPEE Esméralda 1,2 , JEANSON Matthieu 1,2 , DE SOUZA Cyrielle 1 , GOLLETY Claire 1,3 University center of Mayotte, France 2 UMR 228 Espace Dev, Montpellier, France 3 UMR 248 MARBEC, Montpellier, France * Contact : esmeralda.longepee@univ-mayotte.fr Example of a particularly detailed mental map on the health state of mangroves in MayotteMethod 16 semi-structured interviews with stakeholders influencing the conservation and management of the island's mangroves in order to better understand their knowledge We focus here on 3 questions/exercises of the interview guide:

  Scientists themselves have a hard time to define precisely what is ecosystem health. Yet, recent contributions on the debate agree it must include multiple ecosystem functions in addition to measures of ecosystem structure (O'Brien et al 2016, Palmer & Febria 2012).Most interviewees perceived that health is a multi-

  "I don't know how to say that a mangrove is in a good or bad health. I am not sur that we have the indicators to say what is the good health and the bad health. And personally, I don't know the mangroves enough, I went inside of very few mangroves[…]. I think that scientifically we do not know enough to say what is in good health and in bad health" (E4, Marin Natural Parc).

	Good health criteria

O'Brien, A., Townsend, K., Hale, R., Sharley, D., Pettigrove, V., 2016. How is ecosystem health defined and measured? A critical review of freshwater and estuarine studies. Ecol. Indic. 69, 722-729. Palmer, M.A., Febria, C.M., 2012. The Heartbeat of Ecosystems A Boron-Boron Triple Bond. Ecology 336, 1393-1394.


Fabien Theron
email: fabien.theron@onera.fr

Olivier Carraz

Geoffrey Renon

Nassim Zahzam

Malo Yannick Bidel

Cadoret ⋆⋆

Alexandre Bresson

Narrow linewidth single laser source system for onboard atom interferometry

Keywords: 37.25.+k, 42.60.-v, 42.65.Ky, 42.81.Wg, 42.62.Eh, 06.20.-f

A compact and robust laser system for atom interferometry based on a frequency-doubled telecom laser is presented. Thanks to an original stabilization architecture on a saturated absorption setup, we obtain a frequency-agile laser system allowing fast tuning of the laser frequency over 1 GHz in few ms using a single laser source. The different laser frequencies used for atom interferometry are generated by changing dynamically the frequency of the laser and by creating sidebands using a phase modulator. A laser system for Rubidium 87 atom interferometry using only one laser source based on a frequency doubled telecom fiber bench is then built. We take advantage of the maturity of fiber telecom technology to reduce the number of free-space optical components (which are intrinsically less stable) and to make the setup compact and much less sensitive to vibrations and thermal fluctuations. This source provides spectral linewidth below 2.5 kHz, which is required for precision atom interferometry, and particularly for a high performance atomic inertial sensor.

Introduction

Atom interferometers have demonstrated excellent performances for precision acceleration and rotation measurements [1,2]. Many applications of these sensors, as tests of fundamental physics in space [3] or gravimetry ⋆ Present address: European Space Agency -ESTEC Future Missions Division (EOP-SF) P.O. Box 299, 2200 AG Noordwijk, The Netherlands. ⋆⋆ Present address: Laboratoire Commun de Metrologie CNAM, 61 rue du Landy, 93210 La Plaine Saint Denis, France. [4], need the setup to be compact, transportable and robust in order to operate in relevant environments (satellite, planes, boats). Intensive research has been carried out over the last few years to develop transportable laser systems [5,6,7,[START_REF] Ménoret | Dual-wavelength laser source for onboard atom interferometry[END_REF] and, in particular, laser systems as compact and immune to perturbations as possible.

Atom interferometers usually operate with alkali atoms by driving transitions in the near-IR spectrum (852 nm for Cs, 780 nm for Rb, and 767 nm for K). A light pulse atom interferometer sequence consists usually of three stages. First, a gas of atoms is cooled, trapped and selected in a non sensitive magnetic state. Second, these cold atoms are illuminated by a sequence of three light pulses driving stimulated Raman transitions performing a Mach-Zehnder type atom interferometer [9]. Finally, the phase shift of the interferometer is deduced from fluorescence measurements. In these experiments, we need to use several stable laser frequencies relative to the atomic transitions of the alkali species considered: one cooling and trapping frequency, one repumping frequency, two Raman frequencies and one detection frequency. It is noted that the spectral linewidth of the laser needs to be smaller than the linewidth of the atomic transition for the cooling stage (6 MHz for Rb). Additionally, in order to realize stimulated Raman transitions, an even narrower linewidth laser is required because the frequency noise of the laser induces a noise on the atom interferometer measurement [10]. This aspect is very important for gravity gradiometers for which the sensitivity is not limited by vibrations.

State of the art

Different technologies of laser sources are available for addressing alkali atoms. For example, laser sources emitting directly at the same wavelength as the atomic transition can be used: Distributed FeedBack lasers (DFB), Distributed Bragg Reflector lasers (DBR) and Extended-Cavity Diode Lasers (ECDL) [11]. However, the disad-vantage of these technologies is that large efforts are required to obtain robust systems, immune to mechanical misalignments caused by vibrations, for onboard applications. Another appropriate solution for Rb and K, is to use frequency-doubled telecom lasers operating around 1.5 µm [12,13]. This technique is based on the maturity of the fiber components in the telecom C-band to reduce the amount of free-space optics and to make the setup more compact and less sensitive to misalignments. Moreover, many types of narrow linewidth laser sources are commercially available such as DFB laser, DFB with whispering-gallery-mode resonator [14], integrated ECDL diodes [15] and Erbium Fiber DFB Laser (EFL). In this article, we will present results obtained with an EFL source which has already been used for atom interferometry [5].

Different architectures are possible to obtain all the laser frequencies needed for an atom interferometer experiment. The most common one uses at least two lasers: a master laser and a slave laser [5,6,7]. The master laser has a fixed frequency and is locked on an atomic transition. The slave laser is locked relatively to the master laser thanks to a beat note. By changing the set point of the beat note lock, it is possible to change dynamically the frequency of the slave laser and to address all the functions needed for atom interferometry. However, for onboard applications where a compact and robust laser system is needed, the use of two laser sources is not optimal. Indeed, by using only one laser source, the size of the laser system is limited, the risk of failure of the system due to laser source breakdown is reduced and the electrical consumption is lower.

3 Description of the laser system

Laser system

In this article, we present a laser system for Rubidium 87 atom interferometry using only one laser source based on a frequency-doubled telecom fiber bench. A laser system, tunable in few ms, within a frequency range of typically 1 GHz, can generate cooling, detection and the first Raman frequencies. The repumping frequency and the second Raman frequency can be obtained by creating sidebands on the laser source. In our system, the laser source is a narrow linewidth EFL (IDIL fiber laser, output power: 20 mW, linewidth < 2 kHz), which can be frequency tuned (20 MHz/V) thanks to a piezoelectric actuator (PZT) (Fig. 1). In order to frequency stabilize the laser, part of the laser output goes through a phase modulator (PM1, Photline, RF level: 23 dBm) which generates side bands, then goes through a PPLN waveguide crystal (NTT Electronics, conversion efficiency: 225 %/W) which performs second harmonic frequency conversion, and finally goes in a Rubidium saturated absorption setup [16] where the 1 st order sideband of the laser spectrum is locked to the cross over F = 3 → F ′ = 3 c.o. 4 of the 85 Rb-D 2 line (Fig. 2). With a modulation at ν 1 = 1070 MHz, the laser carrier (0 th order sideband) is at resonance with the detection transition F = 2 → F ′ = 3 of the 87 Rb-D 2 line. By changing the frequency modulation ν 1 on PM1, the 1 st order sideband of the laser remains locked on the atomic transition whereas the frequency of the carrier is varied. In that case, the new carrier is detuned relative to the detection transition by 1070 MHz -ν 1 . In summary, the 1 st order sideband is locked on the deepest saturated absorption peak of the Rubidium 85, while the carrier is tuned to the Rubidium 87 transitions. With this technique, a frequency tuning range of at least 1 GHz at 780 nm can be obtained, which therefore addresses all the functions needed for atom interferometry.

Frequency stabilization

Frequency stabilization is achieved by modulating at 50 kHz the frequency ν 1 driving the PM1, and by collecting saturated absorption signal with a photodiode. The laser beam used for this saturated absorption has a power equal to 28 µW @780 nm (3.56 mW @1560 nm before frequency doubling) with a beam diameter of 1.8 mm. This signal is then amplified through a transimpedance amplifier, demodulated at 50 kHz and low-pass filtered, hence providing at this point the dispersive error signal of the lock system proportional to the frequency difference between the 1 st order sideband of the laser spectrum and the atomic frequency transition. Finally, it is integrated and amplified by a high-voltage amplifier, and sent to the PZT of the EFL with a feedback bandwidth of 3 kHz (Fig. 1). The amplitude of the 50 kHz modulation is adjusted in order to have the steepest slope (0.218 V/MHz). With this architecture, a peak-to-peak deviation of 7.8 MHz, and a capture capture range of 45 MHz on the error signal is obtain (Fig. 3). This finite range is due to the presence of absorption peaks located before and after the peak where we are locked. Fig. 3 Error signal as a function of the frequency scan of the laser (@ 780 nm).

In order to improve the stability and reduce the response time of the lock system during frequency steps, a feedforward (Fig. 1) proportional to the frequency of the VCO is added to the corrected signal driving the PZT. During the maximum frequency step of 1 GHz, the frequency deviation is too large compared to the lock range for the laser to remain locked. Therefore, a lock switch (ADG201A) is upstream from the integrator to open the feedback loop during 1 ms after the frequency step. A second integrator stage is also added to reduce the stabilization time. As a result, after a frequency step of 1 GHz, the laser frequency is stabilized with an error below 100 kHz after 3 ms only (Fig. 4).

With this laser architecture, to obtain the second Raman frequency and the repumper required to realize the atom interferometry experiment (Fig. 2), the laser is modulated at a frequency of 6.5 or 6.8 GHz with the PM2 (Photline, RF level from -1 to 23 dBm) (Fig. 1) [17]. Finally, the laser is amplified in an Erbium Doped Fiber Amplifier (IPG Photonics, input power: 3 mW, output power: 5 W) and sent in a frequency-doubling unit. The frequency doubling can be implemented either with a PPLN waveguide [18] or with free-space doubling in a bulk PPLN [5].

4 Frequency noise and influence on the atom interferometer

Estimation of the frequency noise of the laser

In order to determine the frequency noise of the laser, it is necessary to analyse the noise of the error signal (Fig. 5), which is proportional to the laser frequency within a bandwdith of 10 kHz (cut-off frequency of the LPF on Fig. 1). First, this noise is measured when the laser is unlocked and "out of resonance" from the atomic transition (in blue). In this configuration only the noise of our lock system (i.e. the noise on the error signal which does not come from the frequency noise of the laser) is measured. When the laser is "locked" (in green), the noise on the error signal is much lower than the noise of the lock. As a result, the frequency noise of the laser is given by the frequency noise of the lock system up to a frequency of 3 kHz. We investigate then the origin of the noise of the lock system which determines the frequency noise of the laser. The noise of the lock system can come mainly from electronic noise, intensity noise of the laser at 50 kHz and etalon effects in the saturated absorption setup which lead to a temporal fluctuation of the error signal offset. The origin of the noise can be determined by analyzing the noise of the error signal for different configurations (Fig. 6). In the configuration "laser off" (in red), the only contribution comes from the electronic noise. In the configuration laser on, out of resonance and PM1 off ("unmod laser" in black), both the electronic noise and the intensity noise of the laser are present. In the configuration laser "out of resonance" (in blue), all the noise sources are taken into account.

Comparing these configurations shows that the noise of the lock system comes mainly from the intensity noise of the laser between 1 Hz and 10 kHz, whereas the noise comes from etalon effects below 1 Hz. As a summary, the frequency noise of our laser below 10 kHz comes from the noise of the lock which is converted into frequency noise in the feedback loop. The noise of the lock is mainly due to intensity noise of the laser and etalon effects in the saturated absorption setup. In order to estimate the frequency noise of our laser at frequencies higher than 10 kHz, we perform a beat note measurement between our locked EFL and an integrated ECDL (RIO ORION laser source) which linewidth is below 10 kHz according to the manufacturer. Because the linewidth of the ECDL is not infinitely narrow, the analysis of the beatnote gives an upper limit of the frequency noise of the EFL. We notice that the wings of the beatnote fit well with a lorentzian function with a FWHM of 2.5 kHz (Fig. 7). As we know that a white frequency noise leads to a lorentzian spectrum with a FWHM equal to π.S 0 ν , we suppose that the frequency PSD of our laser at high frequencies is below ∆ν/π. Thus, for the determination of the atom interferometry noise, we will consider that the frequency noise of our laser is equal to (S 0 ν ) 1/2 = 28 Hz/Hz 1/2 for frequencies above 10 kHz.

Estimation of atom interferometry noise induced by the laser frequency noise

From the two previous measurements, we can model our laser spectrum by (S 0 ν ) 1/2 = 10 4 Hz/Hz 1/2 for f < 1 Hz, coming from etalon effects, (S 0 ν ) 1/2 = 400 Hz/Hz 1/2 for 1 Hz < f < 10 kHz, due to intensity noise, and (S 0 ν ) 1/2 = 28 Hz/Hz 1/2 for f > 10 kHz (red dash line in Fig. 5). Therefore we can estimate the noise on the atom interferometer measurement induced by the frequency noise of the laser. From the results of [10] and considering classical experimental parameters for a vertical atom accelerometer (the distance atom-mirror for gravimetry, or the distance between the two atom clouds for gradiometry: L = 1 m, the duration of a Raman pulse: τ R = 10 µs, and the time between two pulses T = 100 ms), a single shot rms noise equal to σ a = 2.6 × 10 -9 g is obtained, the main contribution coming from low frequency noise between 1 Hz and 10 kHz.

Improvements

For more demanding applications, frequency noise could be decreased by improving the saturated absorption setup (i.e. removing intensity noise of the laser and etalon effect coming from the fiber [19]). The electronic noise could also be decreased by using a higher laser power or a more efficient lock system. With these improvements, it should be possible to obtain a frequency noise of 28 Hz/Hz 1/2 over the whole frequency range of the laser, leading to single shot rms noise equal to σ a = 6.9×10 -10 g for typical atom accelerometers. In our laser architecture, the EFL could also be replaced by an integrated ECDL which is more compact and has comparable laser linewidth. Finally, a very compact system, immune to external disturbances could be built from an all-fibered bench with a fibered amplifier and a wave guided PPLN after the PM2 [18].

Conclusion

We have developed a tunable narrow linewidth single laser source system for atom interferometry. This system combines the reliability of fiber components and the agility allowed by phase modulators. These features can lead to plug-and-play laser sources for laboratories developing cold atom experiments. These sources could be developed for commercial devices, onboard system or space missions. Finally, the use of a single laser source significantly reduces failure risks and the amount of additional components for redundancy, particularly critical for space projects.

Fig. 1 Fig. 2

 12 Photodiode

Fig. 4

 4 Fig.4Behavior of the laser system locked during a frequency step : voltage to the PZT of the EFL (black); error signal (blue). Voltage step of 46.4 V on the PZT, i.e. frequency step of 965.12 MHz on the laser at 780 nm.

Fig. 5 Fig. 6

 56 Fig. 5 Square root of the Power Spectral Density (PSD) of the error signal noise : when the laser is locked (green) and when the laser is out of resonance (blue). The noise of the error signal when the laser is out of resonance (blue) determines the low frequency part (f < 10 kHz) of the frequency noise. The red dash line represents the model used to determine the atom interferometry sensitivity.

Fig. 7

 7 Fig.7Beat note between the EFL and an integrated ECDL with a linewidth lower than 10 kHz. The beat note determines the high frequency part (f > 10 kHz) of the frequency noise.

Fabien Theron, Olivier Carraz, Geoffrey Renon, Nassim Zahzam, Yannick Bidel, Malo Cadoret, Alexandre Bresson

ACKNOWLEDGEMENTS We thank F. Nez, from the Laboratoire Kastler Brossel (LKB), for his help on the project. We acknowledge funding support from the Direction Scientifique Générale of ONERA, the Direction Générale de l'Armement (DGA), and the Centre National d'Etudes Spatiales (CNES).