Cédric Lopez
email: lopez@lirmm.fr

Violaine Prince
email: prince@lirmm.fr

Mathieu Roche
email: mroche@lirmm.fr

Automatic Titling of Electronic Documents with Noun Phrase Extraction

Keywords: titling, noun phrases, information retrieval, application, morphosyntactic characteristics

Automatic titling (i.e. providing titles) is one of key domains of Web site accessibility. This paper provides an approach allowing the automatic titling of texts (e.g. emails, fora, etc.) relying on the morphosyntactic study of human written titles in a corpus of various texts. The method is developed in four stages: Corpus acquisition, candidate sentences determination for titling, noun phrase extraction in the candidate sentences, and finally, selecting a particular noun phrase to play the role of the text title (ChTITRES approach). The method has been evaluated by ten users, and the satisfaction enquiry shows that the titles selected through this process are relevant.

I. INTRODUCTION

A title definition met in any dictionary is 'word, expression, sentence, etc., serving to indicate a paper, one of its parts, to give its subject'. So it seems that a title role can be assumed by a well formed word group, an expression, a topic or a simple word, related to the text content, in one way or another. It ensues that some groups of well formed words can be convenient for a title, which means that a text might get several possible titles. A title varies in length (i.e. number of words), form and local focus. So, the human judgment on a title quality will always be subjective and several different titles might be judged as relevant to a given content.

This paper deals with an automatic approach providing a title to a document, which meets the different characteristics of human issued titles. So, when a title is absent, for instance in e-mails without objects, the described method enables the user to save time by informing him/her about the content in a single glance. In addition, it is designed to meet at least one of the criteria of the standard W3C. Indeed, titling web pages is one of key fields of the web page accessibility, such as defined by associations for the disabled. The goal is to enhance the page readability.

The originality of this method is that it relies on the morphosyntactic characteristics of existing titles to automatically generate a document heading. So the first step is to determine the nature of the morphosyntactic structure in titles and check whether it depends on the text style (e.g. e-mails, scientific papers, news) or if it is style independent.

The state-of-the art in automatic titling (section II) and our own corpus study (section III) have stressed out the following hypothesis: It seems that the first sentences of a document, most of the time regardless of its style (except maybe for novels, but this is not the mainstream of web pages), tend to contain the relevant information for a possible title. Our ChTITRES approach (section IV) extracts crucial knowledge in these selected sentences and provide a title. An evaluation by human judgment, obtained on real data is presented in section V.

II. RELATED WORK

Titling is a process aiming at relevantly representing the contents of documents. It might use metaphors, humor or emphasis, thus separating a titling task from a summarization process, proving the importance of rhetorical status in both tasks [START_REF] Teufel | Sentence extraction and rhetorical classification for flexible abstracts[END_REF]. Titles have been studied as textual objects focusing on fonts, sizes, colors, etc. Also, since a title suggests an outline of the associated document topic, it is endowed with a semantic contents that has three functions: Interest and captivate the reader, inform the reader, introduce the topic of the text.

It was noticed that elements appearing in the title are often present in the body of the text [START_REF] Zajic | Automatic headline generation for newspaper stories[END_REF]. [START_REF] Baxendale | Man-made index for technical literature -an experiment[END_REF] has showed that the first and last sentences of paragraphs are considered important. The work of [START_REF] Jacques | Titres et structuration des documents[END_REF] supports this idea and shows that the covering rate of those words present in titles, is very high in the first sentences of a text. [START_REF] Vinet | L'aspet et la copule vide dans la grammaire des titres[END_REF] notices that very often, a definition is given in the first sentences following the title, especially in informative or academic texts, meaning that relevant words tend to appear in the beginning since definitions introduce the text subject while exhibiting its complex terms.

A title is not exactly the smallest possible abstract. While a summary, the most condensed form of a text, has to give an outline of the text contents that respects the text structure, a title indicates the treated subject in the text without revealing all the content [START_REF] Wang | Multi-document summarization using sentence-based topic models[END_REF]. Summarization might rely on titles, such as in [START_REF] Goldsteiny | Summarizing text documents: Sentence selection and evaluation metrics[END_REF] where titles are systematically used to create the summary. This method stresses out the title role, but also the necessity to know the title to obtain a good summary.

A title is not an index: A title does not necessarily contain key words (and indexes are key words), and might present a partial or total reformulation of the text (what an index is not).

Finally, a title is a full entity, has its own functions, and titling has to be sharply distinguished from summarizing and indexing.

A rapid survey of existing documents helps to fathom some of title characteristics such as length, and nature of part-of-speech items often used. The first step is to determine the text type, i.e., its category (scientific article, newspaper article, e-mail, forum question or comment, . . . ), and to examine a possible relationship between a text type, and its title characteristics. Therefore, next section is devoted to this study.

III. TEXT TYPES IDENTIFICATION: A STEP PRIOR TO TITLING

A. Type Identification Protocol

The statistical analysis of titles is an essential preliminary stage that helps to understand which kind of title one has to assign to a given type of texts. To ascertain the impact of text type on title form (and length) we have selected five categories of documents: Wikipedia articles, scientific papers, news (the French newspaper 'Le Monde', for the year 1994), e-mails, research mailing lists, and fora. Since French was the main working language, we selected 100 french texts in each category.

Two items were chosen for analysis: What POS (partof-speech) tags were the most frequent in titles, and how many words contained in the title were also frequent in the text. The POS tagging was performed by TreeTagger [START_REF] Schmid | Probabilistic part-of-speech tagging using decision trees[END_REF]. It allowed to know the titles composition according to the types of texts. The number of words present in both text body and titles inform us about the place of the relevant information in the text and indicates if titling is possible from text chunks.

Next section tackles the morphosyntactic characteristics of titles according to the types of considered texts.

B. Analysis and discussion

The results show that the noun is the most used POS: Nouns are present in almost 90% in the titles of all categories. Within a title, the noun represents approximately 31% of the terms. Named entities (NE) appear in 45% of the titles (all categories merged). If the titles of Wikipedia articles which use NE only in 7% of the cases are not taken into account, the average of presence of NE in titles is 60%. Its presence in a title enables to specify the sense evoked by the other terms. 44% of the retained titles contain adjectives. The main function of an adjective is to appoint in the noun to express a quality (qualificative adjective). Its strong presence in the titles indicates the same intention as the NE, i.e., specifying the nature of the subject.

Verbs are not as widely spread as nouns, NE and adjectives (or noun phrases (NP) in general). Moreover, it seems that verbs in a title are more representative of the journalistic style and the scientific articles (26%), where titles are long, close to a complete sentence [START_REF] Mitra | An analysis of statistical and syntactic phrases[END_REF], and thus contain verbs, whereas in Wikipedia articles, e-mails, mailing lists, or fora, verbs occur in only 6% of the titles. So this result is the first clue that title POS composition and text type might be related to each other.

Another interesting feature is the punctuation. It is present in almost 50% of the scientific articles titles. A more detailed analysis showed that 50% of the scientific titles contain the word and. The strong presence of internal punctuation and coordination marked by conjunction indicates a will of bipartition such as it was described in [START_REF] Ho-Dac | Sur la fonction discursive des titres[END_REF].

C. What Type of Title, for Which Text?

According to the first rapid survey presented above, it seems that titles depend on text types, and the most important clues are the following: The nature of the effort in writing the text body, the presence of a verb in the title. Thus, we have splited the documents into two main groups. The first one (G1) contains those texts, in the titles of which, verbs are rare or absent: Mailing lists, fora, and e-mails. The second group (G2) contains the other texts, whose titles present a more complex syntax (related to longer titles), where verb(s) are more likely to appear. This involves a better representation of the semantic contents according to [START_REF] Mitra | An analysis of statistical and syntactic phrases[END_REF].

In this paper, we will focus on G1 documents, since titling procedures would not be the same in both groups. In this group, the expected titles to produce are noun phrases (if we want to stick to the existing titles characteristics studied in the collected corpus). The issue is then how to determine at least one relevant noun phrase that would be an acceptable title.

IV. THE AUTOMATIC TITLING APPROACH A. A Global process of Automatic Titling

The statistical analysis of titles in the various categories of our corpus led to the design of a global process for automatic titling, composed of the following steps:

• Step 0: Corpus Acquisition: Determining the characteristics of the texts to be titled; Described in the previous section. Last, a few candidate noun phrase remain, and they are ranked according to a score, for which we propose several computing procedures. In the following sections, Steps 1 to 3 are described. The software of our system is described in [START_REF] Lopez | Text titling application (demonstration session, to appear)[END_REF]

B. Candidate Sentences Determination and Extraction

The first elementary step consists in determining the textual data from which we will build a title. These data have to contain the information necessary for the titling of the document. As said before, the title words can be often found in the first sentences of the text. In our corpus, when selecting the first two sentences, we potentially access 73% of the semantic content of the title. During our study, we will stick to the first two sentences as a mining field for titling.

Corpus analysis showed that the titles of group G1 documents contain few verbs and are short (between approximately two and six words). Our aim is to extract the most relevant noun phrases in order to provide a title. We shall begin by proposing a list of noun phrases based on their size.

C. Selecting of the maximal noun phrases (N P max )

The step 2 of our approach begins with the extraction of noun phrase (NP). For that purpose, texts are tagged with TreeTagger, and we have determined syntactical patterns allowing noun phrase (NP) extraction, e.g. Adjective1 -N oun1, N oun1 -Det1 -N oun2, N oun1 -N oun2 etc. New syntactical filters can be easily added.

This step process consists in selecting among this list of NP, the most relevant one. A first preselection allows to choose a NP based on its length, with lengths equivalent to L max and L max -1 where L max is the longest local candidate 1 . This technique prevents from pruning interesting candidates too quickly. These candidates are called N P max . If there only one N P max preset, then it is presented as a title. Otherwise, to extract among this preselection the most relevant NP to exploit it as title, two methods are studied. These two methods will rely on a very popular measure in NLP, the TF-IDF [START_REF] Salton | Term-weighting approaches in automatic text retrieval[END_REF].

D. Selecting a Title Among the Candidate NP, the ChTitres Approach

Step 3 consists in selecting the most relevant NP for its use as title. In the following sections, we shall use the measure TF-IDF to calculate the score of every NP. This score can be the maximal TF-IDF obtained for a word of the SN (T M AX ) either the sum of the TF-IDF of every word of the NP (T SU M ).

1) T M AX : For each word of the candidate NP, the TF-IDF is calculated. The score for every candidate NP is the maximum TF-IDF of the words of the NP. With this method, discriminant terms are highlighted. It is obvious that the T M AX method values named entities (NE), these being generally more discriminant than any other type of word in the corpus. During our study, we shall use this method on the first sentence only (T M AX1 ) either on the first two sentences (T M AX2 ).

2) T SU M : For each word of the candidate NP, the TF-IDF is calculated. The score of every NP candidate is the sum of each term TF-IDF. This method favors long noun phrases. However, this method still allows to distinguish between noun phrases of the same size. The benefit of this method is to extract the noun phrase containing the most information, without worrying about the relevance of its words. In this paper, we use T SU M 1 being the first sentence T SU M score, and T SU M 2 , which is the first two sentences scoring.

E. Lexical selection

We locate Named Entities (NE) mainly by the presence of capital letters, i.e., words or word groups designating names (such as names of persons, names of organizations or companies, names of places, and so forth), can be excellent keywords allowing to quickly encircle the content of the text. If a NE is located among three first ones N P max , then it favors selecting it as a title. Otherwise, the N P max retained will be the one of higher score with T M AX or T SU M .

V. EXPERIMENTS

A. Data Description

The experiments have been run on G1 group documents extracted from: the LN mailing list messages2 , fora, and e-mails. For each of these three categories, ten texts were selected. Text are variable in size (i.e. number of words), topics, technicality, and effort of writing.

B. Experimental Protocol

Thirty titled texts are proposed to the experts, by three groups of ten texts from G1. For every text, eight titles were suggested3 among all the titles determined according to the methods T M AX1 , T SU M 1 , T M AX2 , and T SU M 2 as well as the real title T R. Three other titles (A1, A2, A3) are exposed in a random way from the list of noun phrases extracted among those that were rejected by the process. Comparing the evaluation of rejected NP with selected ones will allow, in particular, the estimation of the selection process accuracy.

For every 'candidate' title, the user has to apreciate its relevance to the document contents with the following scale: Very relevant (C1), Relevant (C2), I don't know (C3), not very relevant (C4), not relevant at all (C5). For each of these C n judgements, a digital value is assigned: -2 for C5,-1 for C4, 0 for C3, +1 for C2 and+2 for C1. The final note obtained for a title is the average value of the experts given grades.

C. Results and Discussion

Generally, the four score computing methods determine relevant titles according to the human experts average opinion (see Table I). The disparity in results can be explained by the fact that experts compare all candidate titles and determine the most relevant one, and then after, assign a judgement to the others. So, even if two titles are very relevant, only one will be privileged by being assigned the label very relevant, while the other one will get relevant. The evaluation experiment also shows that it seems better to use T M AX2 as a filtering method in order to title e-mails and mailing lists. The method T SU M 1 seems to be more appropriate for forum message titling. In the Forum category, results indicate that it is better to extract the first sentence, to avoid noise. However, in a general way, the score computing methods taking into account the first two sentences often offer better results (for two categories out of three).

The four methods enable to extract the most relevant N P max . Titles A1, A2, and A3 are always judged as little relevant (even not relevant at all) while score computing methods determine relevant titles (even very relevant). The titles built by the automatic titling process are thus of good quality, even if they obtain results slightly weaker than the real titles, for two categories on three. Two remarks are appropriate: 1) Real titles get an average of 1.17, all categories merged, which means that they are generally relevant, but not necessarily very relevant. Moreover, deviation is quite high in evaluation when browsing the text titles in a same category. 2) E-mail real titles get rather a low grade from the human judges. This tends to indicate a possible benefit of an automatic method that might build a more relevant title than a 'real' one, and is a time saving procedure for an e-mail writer...

VI. CONCLUSION

The quality of automatically computed titles strongly depends on the care brought to the text writing. Nevertheless, the ChTITRES approach 4 proposes relevant titles for the G1 group documents (i.e. e-mails, fora, mailing lists). The results show all the same that improvements can be brought. Even if a part of the performance of this approach depends on TreeTagger, it seems possible to improve results. As seen here, selection methods scores depend on the text type. A combination of methods is contemplated, as a technique more robust to type variation. Naturally, G2 group texts, i.e., newspapers, scientific articles, and encyclopedias texts will be also studied and their titling experimented. However, this group requires a detailed syntactic analysis that we shall lead in our next work. According to our statistics, group G2 document titles must be built by taking into account the more significant presence of verbs, and the peculiarities of text goals. Finally, we also plan to study automatic subtitling, as a natural sequel to automatic titling.

the average size of the extracted NP candidates is 3 words. We preset the NP length to L max-1 , and this allows to remove single words NP from the possible list.

http : //toliste.cines.f r/bow/ln

Identical titles obtained with different approaches are not given.

Available on http : //www.lirmm.f r/ ∼ lopez/


Pierre-Antoine Gédouin
email: gedouin@enib.fr

Emmanuel Delaleau
email: delaleau@enib.fr

Jean-Matthieu Bourgeot
email: bourgeot@enib.fr

Cédric Join
email: cedric.join@cran.uhp-nancy.fr

Shabnam Arbab Chirani

Sylvain Calloch
email: sylvain.calloch@ensieta.fr

Experimental comparison of classical pid and model-free control: position control of a shape memory alloy active spring 6

Keywords: Nonlinear control, Model-free control, Shape memory alloy, Derivative estimation, Nonphysical modelling

Shape memory alloys (sma) are more and more integrated in engineering applications. These materials with their shape memory effect permit to sim-6

Introduction

Shape memory alloys (sma) actuators offer the possibility to recover a known shape after a thermomechanical cycle. This property, known as the "shape memory effect", is due to the transition between the two crystallo-graphic phases in their composition (i.e., the transition between martensite and austenite). This variation of shape, controlled by temperature variation, can be used in the development of actuators (see e.g., [START_REF] Jänker | New actuators for aircraft and space applications[END_REF][START_REF] Peirs | A retrospective evolution of a sma microactuation[END_REF][START_REF] Kohl | Sma microgripper system[END_REF]. sma can easily be heated by Joule effect, but their control is not completely solved and it is principally due to their hysteretic behaviour (see [START_REF] Patoor | Micromechanical modelling of the shape memory behavior[END_REF]. Another difficulty is that the characteristics of the material are time-varying, especially during cyclic loadings. Phases kinetic transformations and 3-dimensional models are proposed in Šittner et al. (2000), Arbab [START_REF] Arbab Chirani | Superelastic behavior modeling in shape memory alloys[END_REF] and [START_REF] Bouvet | A phenomenological model for pseudoelasticity of shape memory alloys under multiaxial proportional and nonproportional loadings[END_REF].

These complex models can render very subtle properties of sma, but often need to compute a finite element code, what is not suitable for real time control. On the opposite side, Robotics research has been done on sma actuator by using simpler model and classical control method. A lot of control strategies have been applied to sma actuators, classical pid loop are used in [START_REF] Calin | Microrobots realized by microstereophotolithography and actuated byshape memory alloys[END_REF], [START_REF] Shameli | Stability analysis and nonlinear control of a miniature shape memory alloy actuator for precise applications[END_REF] and Da Silva (2007). A feedforward path is added in [START_REF] Majima | Modeling of shape memory alloy actuator and tracking control system with the model[END_REF] and [START_REF] Ahn | Internal model control for shape memory alloy actuators using fuzzy based Preisach model[END_REF]; the feedforward command is obtain by using a Preisach model for the hysteretic effect. In [START_REF] Dutta | Modeling and control of a shape memory alloy actuator[END_REF] feedforward scheme is also used but the hysteretic behaviour is described by a Duhem differential hysteresis model. In [START_REF] Song | Precision tracking control of shape memory alloy actuators using neural networks and sliding mode based robust controller[END_REF] and [START_REF] Romano | Modeling, control and experimental validation of a novel actuator based on shape memory alloys[END_REF] a feedforward loop is combined with sliding mode control to obtain robustness, the feedforward path is respectively given by neural network and by a physical model. In [START_REF] Jayender | H∞ loop shaping controller for shape memory alloy actuators[END_REF], the gains of a PI controller have been tuned by H ∞ loop shaping considering a physical model linearised around some operating points.

Passivity property of the system is used in [START_REF] Madill | Modeling and l2-stability of a shape memory alloy position control system[END_REF] to prove stability of a proposed proportional law. Nonlinear control techniques based on the Lie algebra are also used in [START_REF] Benzaoui | Nonlinear motion and force control of shape memory alloy actuators[END_REF]. Even if the model is good enough, using a dynamic model for computing the control law implies the identification of the model parameters. As already mentioned, the model parameters of sma vary during cycling, then a classical model-based control is ineffective or particularly complex. We report our experience, where industrial partners still explain that in order to realise the process control, the part of process modelling represents 90 % of project global time and requires a true know-how in control and about the process to-be controlled.

Indeed, the choice of the physical model structure, the identification of the model parameters, the experimental validation of model are never simple.

pid controllers are often tuned with a simple (linear) non physical model (an extensive literature about this subject is available see e.g., Åström andHägglund, 2005, 1995;[START_REF] Visioli | Design and tuning of a ratio controller[END_REF][START_REF] Ang | PID control system analysis, design and technology[END_REF][START_REF] Panagopoulos | Design of PID controllers based on constrained optimisation[END_REF][START_REF] Khodabakhshian | A new robust PID load frequency controller[END_REF][START_REF] Gyöngy | On the automatic tuning and adaptation of PID controllers[END_REF][START_REF] Tavakoli | Tuning of decentralised PI (PID) controllers for TITO processes[END_REF], that is why pid control is often preferred to physical modelbased control in the industry. However, pid controllers could render poor results when a process has a large operating domain. So, how is it possible to efficiently control a complex process:

• without complex physical model ?

• with easy-to-use and theoretically understandable controllers ?

In this paper, a solution2 to this difficult problem is proposed. The proposition is based on some new results in the framework of "model-free control" (see [START_REF] Fliess | Model-free control and intelligent PID controllers: towards a possible trivialization of nonlinear control?[END_REF][START_REF] Fliess | Vers une commande multivariable sans modèle[END_REF][START_REF] Join | Commande sans modèle pour l'alimentation de moteurs : résultats préliminaires et comparaisons[END_REF]. The approach uses derivative estimations (see [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF][START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF][START_REF] Fliess | Non-linear estimation is easy[END_REF] which provides good results even if the measured signals are corrupted by noise. Thus a non-physical model valid a very short part of time is estimated and permits classical control design.

The present work constitutes an extension of two previous articles: In [START_REF] Gédouin | Model-free control of shape memory alloys antagonistic actuators[END_REF], the advantages of model-free control of sma has been high-lighted by simulations while preliminary experimental results have been presented in [START_REF] Gédouin | A new control strategy for shape memory alloys actuators[END_REF].

The paper is organised as follows: The next Section gives an introduction to the new "model-free control" and explains the design of a control law within this framework; Sec. 3 develops the model-free control of a shape memory alloys actuator and gives experimental results. Sec. 4 develops a simple improvement of the model-free controller and the PI controller; this modification results in position dependent gain of the control; this gain is obtain by steady state identification of an input-output nonphysical process model; experimental and robustness results of the modified controller are presented in great details. Sec. 5 concludes the paper and raises some perspectives.

Model-free control

Model-free control is a very recent approach to nonlinear control that has been introduced in [START_REF] Fliess | Vers une commande multivariable sans modèle[END_REF], (see [START_REF] Fliess | Model-free control and intelligent PID controllers: towards a possible trivialization of nonlinear control?[END_REF], for a thorough presentation). A first industrial and application is reported in [START_REF] Join | Étude préliminaire d'une commande sans modèle pour papillon de moteur[END_REF].

Derivatives of noisy signals

We recall basics of derivative estimation. Interested reader might refer to [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF] for a complete presentation. The method presented here, consists in designing FIR filters resolving a classical polynomial approximation of the signal. The polynomial approximation is obtained by algebraic manipulation of signals in the operational domain. We consider a signal y that is available through a measurement y m corrupted by some additive noise ̟, i.e. y m = y + ̟. The objective is to estimate time derivatives of signal y(t), up to a finite order, from its measurement y m .

The Taylor expansion of y around 0 reads:

y(t) = ∞ n=0 y (n) (0) n! t n (1)
Approximate y(t) in the interval [0, T ], T > 0, by the polynomial y N (τ ) = N n=0 y (n) (0) τ n n! of degree N. The operational 3 analogue (see [START_REF] Mikusiński | Operational calculus, 2nd Edition[END_REF]) Y N (s) of y N (τ ) is given by:

Y N (s) = y(0) s + ẏ(0) s 2 + • • • + y (N ) (0) s N +1 (2)
It is possible to isolate each coefficient y (i) (0) appearing in the previous 3 Reader not familiar with operational calculus can just think in terms of Laplace transform to understand the development of the derivatives estimators.

expression by applying a convenient operator to Y N (s) (see [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF], for details4 ). Indeed:

∀i = 0, . . . , N, y (i) (0) s 2N +1 = (-1) i i!(N -i)! • 1 s N +1 • d i ds i • 1 s • d N -i ds N -i s N +1 Y N (s) (3)
The two following formulas of operational calcul [START_REF] Mikusiński | Operational calculus, 2nd Edition[END_REF]) are very useful:

• the operator 1 s α corresponds to the function t → t α-1 (α-1)!

• the operator d ds corresponds to the multiplication in the time domain by -t

Moreover, the Cauchy formulae to transform a multiple integral in a simple one:

T 0 τ α-1 0 • • • τ 1 0 f (µ)dµ dτ 1 . . . dτ α-1 = T 0 (T -µ) α-1 (α -1)! f (µ)dµ (4)
Consequently, one obtains in the time domain the expression of y (i) (0) as:

y (i) (0) = T 0 P (µ; T )y N (µ)dµ (5)
where P (µ; T ) is polynomial in µ and T . Notice that (5) gives the calculation of y (i) (0) from an integral on the time interval [0, T ] for a given small T > 0.

As d i y(t-µ) dµ i

| µ=0 = (-1) i y (i) (t) it is possible to express y (i) (t) as an integral which involves values of y N on the time interval [t -T, t]:

y (i) (t) = (-1) i T 0 P (µ; T )y N (t -µ)dµ (6)
A simple estimator, y Taylor expansion), of the derivative y (i) (t) is then obtained from the noisy signal y m by:

y (i) N (t) e = 1 0 R i N (σ)y m (t -σT ) dσ (7) = (N + i + 1)! (N + 1)! T i 1 0 P i N (σ)y m (t -σT ) dσ (8)
P i N (σ) = N -i k=0 Q N,i k (σ) (N -i -k)! k! (i + k + 1) (9) Q N,i k (σ) = i j=0 (-σ) k+j (1 -σ) N -j-k j! (i -j)! (k + j)! (N -j -k)! (10)
which is deduced from (5) by replacing y N by y m in (6) with a change of variable σ = µT and by direct application of the rules of operational calculus previously given. Note that the integral operation plays the role of a low-pass filter and reduces the noise that corrupts the signal y m . The choice of T and N results in a trade-off: the larger is T , the smaller is the effect of the noise (the larger is T the better is integrals low pass filtering) and the larger is the error due to truncation. The larger is N, the smaller is the error due to truncation and the larger is the error due to noise.

In practice, the integral in ( 7) is evaluated using a classic composite Newton-Cotes approximation exact for a polynomial of degree N using an odd number of samples n s + 1. So, y

(i) N (t)
e is evaluated at each sample time t = k.T s , k = 0, 1, . . . as the output of a FIR filter:

y (i) (kT s ) e = ns j=0 w(j) R i N ( j n s ) y m ((k -j)T s ) (11)
where w(j) are the weights due to integral approximation and n s T s = T . We refer the reader to Zehetner et al. (2007a,b); [START_REF] Reger | Algebraische Ableitungsschätzung im Kontext der Rekonstruierbarkeit[END_REF] for helpful complements on the implementation of these estimators.

Remark 1. When N = i, one can see that R i i (σ) in (7) is symmetric ( i.e, R i i ( x+1 2 ) = (-1) i R i i ( 1-x 2 )
) the weights due to Newton-Cotes approximation are also symmetric so the filter in (11) is a linear phased FIR filter with constant delay equals to n s T s 2 . In the sequel, we assume that the delay is negligible for

N = i + 1 and n s = 10.

Model-free control design

Assume we have a plant for which we do not know any model. For the sake of simplicity of the presentation we assume that this plant is single-input and single-output. The control input is denoted by u and the output is denoted as y. As seen in the previous section, we are able to estimate on-line some derivatives of y and u. Model-free control consists in trying to estimate via the input and the output measurements what can be compensated by control in order to achieve a good output trajectory tracking. This implies the construction of a purely numerical model -also called "local model5 "of the plant that can be written as:

y (ν) = F + αu (12)
where α ∈ R is a non-physical constant design parameter;

F ∈ R represents
all what is unknown on the system and can be compensated from the knowledge of the input-output behaviour of the system. As we have assumed that we do not know any model of the plant, the order ν ∈ N of the numerical model ( 12) is necessarily a design parameter that can be arbitrarily chosen.

But if we assume that the relative dominant order of the plant is known then ν will be equals to this order.

Equation ( 12) should not be confused with a "black-box" identified model.

In the present approach, the quantity F in ( 12) is updated at each sampling time from the measurement of the output and the knowledge of the input:

At sampling time k (i.e. t = kT s , where T s denotes the sampling period), the estimation of F reads:

[F (k)] e = [y (ν) (k)] e -αu(k -1) (13)
where [y (ν) (k)] e is the estimation of the ν-st derivative of the output that can be laid at time k and u(k -1) is the control input that has be applied to the plant during the previous sampling period.

Based on the numerical knowledge of F the control for sampling period k is calculated on (12) as a simple cancellation of the non-linear terms F plus a closed loop tracking of a reference trajectory t → y * (t):

u(k) = - [F (k)] e α NL Cancellation + y * (ν) (k) + ∆(ǫ(k)) α Closed loop tracking (14)
where ǫ(k) = y(k)y * (k) is the tracking error and ∆(ǫ(k)) is a closedloop feedback controller based on the tracking error. Note that the term

-[F (k)]e α + y * (ν) (k)
α is also the "nominal control" in the "flatness-based6 " control of (12) (see e.g., [START_REF] Fliess | Flatness and defect of non-linear systems: introductory theory and examples[END_REF]Hagenmeyer and Delaleau, 2003a[START_REF] Jänker | New actuators for aircraft and space applications[END_REF][START_REF] Hagenmeyer | Robustness Analysis with Respect to Exogenous Perturbations for Flatness-Based Exact Feedforward Linearization[END_REF]. When the closed loop controller is of "pid" type, model-free control can be named as "intelligent pid" (i-pid) (see [START_REF] Fliess | Model-free control and intelligent PID controllers: towards a possible trivialization of nonlinear control?[END_REF]. This control scheme is summarised in

Prototype of SMA Actuator

Description of the actuator

A simple prototype of sma spring actuator has been built and can be seen in Fig. 2. It is constituted of a shape memory alloy spring7 in Ni-Ti from the Mondotronics company and an aluminium mass. At room temperature the length of the spring is about 20 mm but the weight of the mass (450 g) deforms the spring to an initial deflection of 160 mm, i.e. the range displacement of this actuator is about 140 mm.

To generate a vertical displacement the spring is alternatively heated by Joule effect and cooled by free air convection:

• During heating, the martensitic phase changes into the austenite phase; the length of the spring reduces and the mass moves upward.

• During cooling, the austenite phase changes into the martensite phase; the length of the spring increase and the mass moves downward.

A fan (Not shown on Fig. 2) positioned at 110 mm from the spring allows to simulate a disturbance by modifying the rate of air convection.

The sma spring is electrically isolated from the frame and a simple electronic circuit (not shown on Fig. 2) allows to regulated the current flowing through the spring. The heating current is measured with a shunt precision resistance and an instrumentation amplifier. The voltage applied to the spring is also measured with an instrumentation amplifier. The vertical position of the mass is measured with a laser sensor 8 with a resolution of 0.3 mm over a range of 200 mm at a sampling frequency of 100 Hz. The actuator is interfaced to a computer via a data-acquisition board 9 . The computer is running under Linux/rtai 10 which is a low latency kernel that allows to implement hard real-time control and measurements from C language codes.

Position control development

For this application, a first-order local numerical model

ẏ = F + αu (15)
is considered. The choice of a first order comes from the literature about shape memory alloy modelling in which the order is usually one (see e.g., [START_REF] Leclerc | A general macroscopic description of the thermomechanical behaviour of shape memory alloy[END_REF]. The control input u corresponds to the electric 8 OWLE 4025 FA S1

9 Humusoft mf-604.

10 http://www.rtai.org/ Fig. 2. Prototype of a sma spring based actuator.

power crossing the sma spring and the output y is the measured vertical position of the mass. According to section 2.2, the control is given by:

u = 1 α (-[F ] e + ẏ⋆ -K p (y -y ⋆ )) (16)
where K p is a positive gain. Note that as ( 15) is here first order, a simple proportional controller is enough to ensure convergence of the error to zero.

Contrarily to classical feedback control there is no need here to add integrators in the stabilisation controller to ensure converge of the position error to zero as the model free control implicitly involves integrators. This controller is hence simply called an i-P controller.

For comparison purpose, we also have implemented a classical PI control:

u(t) = -K P (y(t) -y ⋆ (t)) -K I t 0 (y(τ ) -y ⋆ (τ ))dτ (17)
The classic control does involve an integral term in order to avoid static error.

The controller parameters have been manually tuned and are given in Tab. 1. The dynamic of the process is quite slow and, moreover, the control effect is "not symmetric" in the sense that there is no control to cool the sma spring (cooling relies only on free air convection). Consequently, this is not possible here to apply most of the pi or pid tuning algorithms, like

Ziegler & Nichols rule. So, the tuning of the pi gains have been done in two steps: Firstly, tuning of the proportional gain on the undisturbed process (cooling fan off), secondly, tuning of the integral gain in order to achieve a good perturbation rejection. The obtained gains are, to our best expertise, the larger we can obtain in order to achieve the trade-off between dynamic precision and stability.

Similarly, the gains of the i-P controller have been tuned in three steps:

first choose a very large α and a large K p to have a small steady-state error, then decrease α to obtain an oscillatory quick response of the system. Finally decrease K p to stabilise the system. Note that for step responses, the first derivative of the trajectory to track, ẏ⋆ equation ( 16), has been chosen identically zero.

The comparison is laid off in two scenarios presented in the two following sections.

First experimental results

3.3.1. Scenario 1 (Fig. 3)

We present classical closed loop step response with thermal disturbance.

In this scenario the mass has to reach a reference position of 70 mm. At time t = 60 s a thermal perturbations is applied using a fan (the effect of the fan on cooling speed is shown in Fig. 4) and at time t = 90 s the fan is switched off. Displacement of the mass (output) and electric power crossing sma spring (input) are plotted in Fig. 3. When the actuator is controlled with the i-P controller, we observe that the mass reaches the reference position in approximately 8 seconds without overshoot whereas a 17 mm overshoot and 34 s response time are obtained when the actuator is controlled by a classical pi controller. This overshoot should be reduced (reducing the integral gain) nevertheless perturbations rejection would be bad. We can observe in Fig. 3

Scenario 2 (Fig. 5)

As the control effect is "not symmetric", overshoots due to set-point jumps imply poor performances of the controllers (see for example, the response time of the plant with the PI controller). A solution to improve performances is to generate a transient profile between two set-points that the output of the plant can reasonably track. So, we present classical polynomial tracking. We can observe that tracking is slightly better when the actuator is controlled by an i-P (see RMS error and maximal error given in We have seen that the i-P controller was able to reject perturbations faster than the pi controller yet, the tracking performances are slightly disappointing. In the next section, we will modify nonlinear model-free and PI controllers using a simple off-line nonlinear nonphysical process model identification. Then we will compare robustness performances of the two controllers.

Table 1. Controllers parameters for experiments of Sec. 3.3.

K p (proportional gain) K i (integral gain) α pi 5 × 10 -2 4 × 10 -3 - i-P 0.9 - 600

Nonlinear model-free and PI controllers

Nonphysical model process have been used in control and particularly in industrial applications since many years because models of physical processes are often difficult to obtain or are too complex to be used in real-time control.

The well known Broïda and Strejc models (linear models with delay) are very popular and are still used to tune pid controllers (see e.g., [START_REF] Åström | The future of PID control[END_REF][START_REF] O'dwyer | Handbook of PI and PID controller tuning rules, 2nd Edition[END_REF][START_REF] Visioli | Practical PID Control. Advances in Industrial Control[END_REF]. However linear models often become insufficient for a process with a large operating domain. In this section, we will see how the use of a very simple nonphysical input-output model, can improve the control performance of model free control on the SMA actuator. To provide a fair comparison, we also modify the PI controller in the same manner, i.e. the control law is applied through a position dependent gain. This modified PI controller is named as "non linear PI controller" in the sequel.

Input-output nonphysical process model

Some physical models of sma actuators used in control have relative degrees equals to three and used Preisach, neural network or Prandtle-Ishlinskii model to describe hysteresis (see e.g., [START_REF] Ahn | Internal model control for shape memory alloy actuators using fuzzy based Preisach model[END_REF][START_REF] Asua | Neural network based micropositioning control of smart shape memory alloy actuators[END_REF][START_REF] Romano | Modeling, control and experimental validation of a novel actuator based on shape memory alloys[END_REF] but these models are difficult to identify and to use in control (see e.g., [START_REF] Ru | A hysteresis compensation method of piezoelectric actuator: Model, identification and control[END_REF][START_REF] Wang | Robust adaptive control of a class of nonlinear systems including actuator hysteresis with Prandtl-Ishlinskii presentations[END_REF][START_REF] Tan | Modeling and control of hysteresis in magnetostrictive actuators[END_REF][START_REF] Ram | Approximate inversion of the Preisach hysteresis operator with application to control of smart actuators[END_REF].

Generally, the industrialists use linear with constant delay models (LCDMs) because these models are easily and quickly identifiable, they permit to tune pi/pid controllers and to use Smith predictors. But as we have already mentioned the linear approximation could be bad for processes with large operating domains. We will try to convince the reader that mfc controllers combined with a nonlinear process model could be a good alternative to pi/pid controllers combined with LCDMs. Indeed, in some cases nonlinear models could be more precise than LCDMs and mfc controllers more robust than pi/pid controllers. That is why in this subsection we propose a type of nonlinear nonphysical process model, easy to combine with model free control, keeping in mind that the identification process must be quick and easy.

For a given order ν, the simplest linear model used to describe a SISO system is:

y (ν) = - ν-1 i=0 A i y (i) + Bu (18)
where the A i 's, B and ν are constant parameters. From simulations results, we found that if B is in fact significantly time varying the MFC controller yields poor performances. Indeed, in this case, the local model approximation (equation ( 12)) is too far from the real behaviour. One of the simplest way to improve control performances is to enrich the local model with information about the variations of B. To do so, we first propose a general model which is an extension of the model (18):

y (ν) = - ν-1 i=0 A i sign(y)|y (i) | n i + g(y)u (19)
where g is a polynomial11 in y, the A i 's and the n i 's (n i ≥ 1) are constant parameters which will not be necessary identified as the main information we need to improve the local model is the knowledge of g. Some example of response of ( 19) are given in Fig. 6-(b). The easiest way to identify g is to use the static relation between the steady-state displacement, y ss , and the applied constant input u ss .

In steady-state equations ( 19) yields:

-A 0 sign(y ss )|y ss | n 0 + g(y ss )u ss = 0 (20)

Then g(y) = s j=0 a j y j and the parameters A 0 , n 0 are obtained by fitting the experimental static relation between steady-state output and the corresponding input step. According to equations ( 20) fitting has been done by resolving12 the following minimisation problem: min

(A 0 ,n 0 ,a j ) m i=1 s j=0 a j y j ss i u ss i -A 0 y n 0 ss i 2 (21)
where the set (u ss i , y ss i ) i = 1, • • • m consists of m measured data pairs where u i is a constant input and y ssi is the corresponding steady-state output. After some trials, we have found that a polynomial of degree 3 was a good trade off between model complexity and minimal fitting residue.

g(y) = a 0 + a 1 y + a 2 y 2 + a 3 y 3 (22)
In Fig. 6-(a) we remark a good correlation between measured data and simulation. Nevertheless, the obtained solution could be a local minimum and the model has been tested in only one situation. However the idea is not to obtain an excellent model of the process but just to have some easilyobtainable insights on its input-output behaviour to design a mfc-based robust controller and a nonlinear PI controller. Now we will see, how to modify the two controllers using the identified function g. Then, in Section 4.4, we will expose some experiments in closed loop.

Remark 2. Some other tests could permit to fully identify the model (19). It would then be possible to study the stability of this model controlled with the MFC controller. But as we have already mention, this model is too simple to describe the entire nonlinear behaviour of the actuator, so it seems that a stability analysis would be in this case without guarantee on the real stability of the control on the real plant.

Design of the nonlinear model-free controller

According to the purely numerical model (equation ( 12)) identified at where α ∈ R is a non-physical constant design parameter. The quantity F ∈ R represents all what is unknown on the system and is updated at each sampling time, replace α in equation ( 13) by αg(y). The control is now given by:

u = 1 αg(y) (-[F ] e + ẏ⋆ -K p (y -y ⋆ )) (23)
The new control takes advantage of information obtained with the previous simple nonphysical model and of the robustness of mfc (there is still an estimation of F at each sampling time).

Design of the nonlinear PI controller

A simple nonlinear PI controller is simply derived from the classical PI controller using the previously identified weighting function g(y). It reads:

u(t) = 1 g(y) -K p (y(t) -y ⋆ (t)) -K i t 0 (y(τ ) -y ⋆ (τ ))dτ (24)
4.4. Robustness analysis of the two nonlinear controllers

Robustness with respect to thermal disturbance and mass variation

In this subsection, the objective is to test the robustness of the two nonlinear controllers (equations ( 23) and ( 24)). Assume that for some set-points, we reasonably want a response time of the plant equals to 10 seconds 13 . So we design a polynomial reference trajectory with a transfer time equal to 10 seconds between the two set-points. The controller has to ensure a good tracking of this trajectory in order to achieve a response time equal to 10 seconds. In the next experiments the trajectories between set-points are polynomials of degree 5 with zero speed and acceleration at starting and ending points. We first choose a sixty-seconds scenario in which the 450 g mass has to reach a steady state level of 65 mm with a response time equals to 10 seconds and to reject a thermal disturbance in steady-state (at time t = 35 s a thermal perturbations is applied using the fan). Then the gains adjustment of the two controllers is done manually with the same methods as those explained in subsection 3.3. The controller parameters are given in Table 2. • response time of the plant calculated before the thermal perturbation (time after which the error is lower than 5 % of the total displacement)

• RMS error on t ≥ 10 s

• maximal error on t ≥ 10 s

• mean input power on the whole experiment These numbers have been calculated for the previously described experiments and are given table 3.

We can see in Table 3 that the ten-seconds response time is always achieved in the case of the nonlinear i-P whereas in the nonlinear PI case, the response time is greater than 10 s in three cases. The RMS error and the maximal error are smaller for the nonlinear i-P controller. However, the mean power is often slightly smaller for the nonlinear PI controller than for the nonlinear i-P controller. • nonlinear PI case : RMS error = 1.76 mm and mean power = 0.14 W

• nonlinear i-P case : RMS error = 0.51 mm and mean power = 0.13 W From these experiments, we can conclude that a trajectory profile is really necessary to avoid problems due to set-points jumps and that information from a very simple model of the plant nonlinearities can improve significantly the performances of a simple linear feed-back.

Conclusion

The contribution of the paper is twofold: it presents a convincing application of the new model-free control in the area of sma actuators control, a field in which model-based control is especially difficult to develop.

Secondly, in order to improve performances of a i-p and a pi controller, the paper exposes a general input-output nonphysical model, off-line identified, which is used to design a nonlinear mfc-based controller and a nonlinear pi controller.

Experimental results on a sma spring based actuators show good tracking performances and good robustness towards thermal disturbances and mass variations for the two controllers. Nevertheless, it seems that the mfc-based

  to the order of the truncated

  Fig. 1. Model-free control.

Fig. 4 .

 4 Fig. 3. (a): Output vs time for a 70 mm step with thermal disturbance when

Fig. 5 )

 5 Fig.5) however tracking performances are unsatisfactory for the two con-

  Fig. 6. (a): Experimental steady-state output (displacement) as a function of

Fig. 7 -

 7 Fig. 7-(a) shows the output of the plant as a function of time after the

Fig. 7 .

 7 Fig. 7. (a): Output vs time for a trajectory with 65 mm steady-state level

Table 3

 3

	. Performance-indicating numbers obtained for experiments of sub-
	section 4.4.
	mass displacement indicating numbers nonlinear PI nonlinear i-P
	[g]	[mm]
	280	20	response time [s]	13.87	8.05
			RMS error [mm]	1.02	0.32
			maximal error [mm]	3.18	1.47
			mean power [W]	0.23	0.25
		40	response time [s]	8.75	9.51
			RMS error [mm]	1.03	0.37
			maximal error [mm]	3.57	1.78
			mean power [W]	0.36	0.40
		60	response time [s]	9.51	9.71
			RMS error [mm]	1.31	0.50
			maximal error [mm]	4.33	2.36
			mean power [W]	0.39	0.42
		80	response time [s]	9.46	9.64
				-Continued on next page

See, e.g., Han (2009) for another approach.

Note that those operators are not unique, we have chosen here to use the ones with the least order of integration for the sake of simplicity of the presentation.

Local must be here understood in the time domain, i.e., this model is valid on a short-time horizon.

See Sira-Ramírez and Agrawall (2004) or[START_REF] Lévine | Analysis and Control of Nonlinear Control Systems: A Flatness-based Approach[END_REF] for a complete overview of flatness-based control.

Coil outside diameter: 6 mm and wire diameter: 750 µm

Note that g can also be another kind of function of y.

The authors have chosen a classical quasi-Newton algorithm implemented in Scilab software

This work was supported in part by the Agence nationale de la recherche (France) -Project "

controller outperforms nonlinear pi. Finally, we are quite confident to be able to efficiently control sma micro actuators as micro servo-motors in a near future.