Agnès Tutin
email: agnes.tutin@u-grenoble3.fr

For an extended definition of lexical collocations

In this paper, we address two main problems raised by the classical definitions of restricted lexical collocations: 1) the binary status of collocations; 2) the problem of function words which are often implicitly excluded from the definition of collocations We show that collocations are fundamentally argument-predicate structures and that most ternary collocations can be considered as combinations of binary collocations. We think that content grammatical words should be considered as parts of collocations, while pure functional words are not central elements of the collocations.

Introduction

In European tradition, two main conceptions of collocations co-exist: in the British contextualist framework [START_REF] Firth | [END_REF]Halliday & Hasan 1978;Sinclair 1993;[START_REF] Williams | Les collocations et l'école contextualiste[END_REF], collocations can be broadly defined as recurrent lexical elements which contribute to the text cohesion. In the "continental" tradition [START_REF] Williams | Les collocations et l'école contextualiste[END_REF], collocations are also called "restricted lexical collocations" and considered as lexicalised phrases where two recurrent lexical elements have a syntactic relationship. In this paper, we will address several problems related to the continental definition of collocations.

Restricted lexical collocations have now been studied and encoded in dictionaries for over than twenty years, and stable definitions have been provided for this notion by numerous scholars working on collocations (e.g. [START_REF] Hausmann | Le dictionnaire de collocations[END_REF][START_REF] Mel | Collocations and Lexical Functions[END_REF][START_REF] Heid | On Ways Words Work Together -Topics in Lexical Combinatorics[END_REF][START_REF] Tutin | Collocations régulières et irrégulières : esquisse de typologie du phénomène collocatif[END_REF]. They are roughly defined as recurrent combinations of two linguistic elements which have a syntactic relationship. One of the elements of the collocation, called "base" keeps its usual meaning (autosemantic words [START_REF] Hausmann | Wortverbindungen -mehr oder weniger fest[END_REF])) while the other, the "collocate" is dependent on the other (synsemantic words) and usually has a less transparent meaning.

Nevertheless, even though such a definition is operational for a large number of lexical associations, it raises several problems. The first one is the binary status of the collocation and the unequal status of the two parts of the collocation, which has been questioned by several linguists (inter alia [START_REF] Siepmann | Collocations et dictionnaires d'apprentissage onomasiologiques bilingues : questions aux théoriciens et pistes pour l'avenir[END_REF][START_REF] Bartsch | Structural and functional properties of collocations in English, A corpus study of lexical and pragmatic constraints on lexical co-occurrence[END_REF]) who suggest expanding the definition to associations of three or more elements. A second problem concerns the grammatical status of the collocations. Should functional words -and to what extent -be included in the definition of collocation? For example, in expressions such as for fear of, the whole combination can be analysed as a preposition, and not as a phrase contrary to prototypical collocations such as pay attention (verb phrase), major problem (noun phrase), seriously injured (adjective phrase). However, fear in for fear of can be considered as relatively transparent, and according to us should be considered as a collocation.

In this paper, we address these two issues in detail and call for an extended typology of restricted collocations. We examine the lexicographical consequences of such an extended definition.

Are collocations fundamentally binary associations?

Collocations are considered as binary associations in classical definitions of lexical restricted collocations in two ways:

1. Collocations are associations of two lexical units, or broadly speaking, two linguistic elements :

2. The two elements of the association have a different status: the base is the prominent element while the collocate depends on the base.

These two characteristics are present in these classical definitions of Hausmann, Mel'cuk and Heid.

On appellera collocation la combinaison de deux mots1 … (Hausmann 1989 :10010) [We will call collocation a two-word combination].

[…] collocations are combinations of exactly two lexemes (of category noun, verb, adjective or adverb), realizing two concepts where the choice of one of them depends on (or : is restricted by) the other. (Heid 1994 : 228) A COLLOCATION AB of language L is a semantic phraseme of L such that its signified 'X' is constructed out of the signified of one of its two constituent lexemes -say, of A -and a signified 'C' ['X' = 'A⊕C'] such that the lexeme B expresses only 'C' contingent on A. (Mel'čuk 1998 : 30) These two characteristics seem to be operational for a large number of multiword units since in a large number of dictionaries (e.g. BBI, OCDSE, LTP, DC) provide collocational information within the entries of the base, and most collocations (all of them in LTP and DC2 ) are binary lexical collocations. For example, pay attention is recorded in the BBI, OCDSE or LTP dictionaries within the entry dictionary of attention. We observe the same facts for the French equivalent prêter attention in DC.

It must be emphasized that the two elements are not necessarily simple words. The base and the collocate can be fixed idioms, for example: a) travailler d'arrache-pied ('to work hard') where travailler is the base and d'arrache-pied is a collocate; b) point de vue classique ('a classical viewpoint') where point de vue is the base and classical the collocate. More interestingly, some expressions like similes, used as collocates, (to sleep (base) like a child (comparison phrase as a collocation) cannot really be considered as lexicalized. These cases show that the notion of lexical element in the definition should be extended to idioms and even to expressions like similes for collocates.

However, several linguists have questioned this binary status [START_REF] Hausmann | Lexicographie française et phraséologie[END_REF][START_REF] Siepmann | Collocations et dictionnaires d'apprentissage onomasiologiques bilingues : questions aux théoriciens et pistes pour l'avenir[END_REF][START_REF] Bartsch | Structural and functional properties of collocations in English, A corpus study of lexical and pragmatic constraints on lexical co-occurrence[END_REF]), because 1) some apparent collocations are larger than two constituents, and because 2) no dominant constituent seems to emerge in some examples.

2.1 Some collocations are larger than two constituents 2.1.1 Some collocations like to pay close attention apparently include three or more elements. Nevertheless, most of them in this case can be analysed as a merging of two or three binary collocations ("collocational chains" for [START_REF] Hausmann | Wortverbindungen -mehr oder weniger fest[END_REF][START_REF] Hausmann | Lexicographie française et phraséologie[END_REF], "collocational cluster" for Spohr (2005)). For example, the sequence to pay close attention can be decomposed into two collocations: to pay attention (a support verb construction) and close attention (an intensive, "Magn-like" collocation in ECD words, according to Mel'čuk's terminology). Though close attention very often collocates with pay, this collocation can be encountered in other contexts: a rapid Google search yields for example close attention is required, to recommend close attention, this issue needs close attention …

Even tricky cases can often be analysed as merged collocations. For example, D. [START_REF] Siepmann | Collocations et dictionnaires d'apprentissage onomasiologiques bilingues : questions aux théoriciens et pistes pour l'avenir[END_REF] gives the example of avoir un geste déplacé as an example of a ternary collocation (Lit. "to have an inappropriate gesture") and shows that the adjective cannot be removed: *avoir un geste (Lit. "to have a gesture"). But, for us, this problem is largely due syntactic constraints, viz. to the presence of the indefinite determiner. In this context, other determiners il a eu ce geste, le geste qu'il a eu … seem perfectly appropriate with this collocation without any modifier. On the other hand, the verb can easily be deleted and the collocation un geste déplacé (lit "inappropriate gesture") can be encountered in other contexts: se permettre un geste déplacé, l'auteur d'un geste déplacé, à la suite d'un geste déplacé …

In other words, there is a syntactic constraint on the collocation avoir un geste and a modifier is required, but there is no lexical constraint on this modifier (geste d'horreur, geste rapide …). The sequence avoir un geste déplacé can thus be analysed as the merging of avoir un geste (+ Modifier) and geste déplacé. This case shows that close attention should be paid to syntactic constraints on collocations, and that in this case, specific constraints should associated to the determiner. As suggested by [START_REF] Heid | A model for a multifunctional electronic dictionary of collocations[END_REF], we think that morphosyntactic properties of collocations should be accounted for in detail in lexicographic subentries ("second level treatment units").

Syntactic constraints of collocations could also explain why two collocations cannot merge into a collocational cluster in some cases. For example, while avoir peur (Lit. "to have fear") and une peur bleue (Lit. "a strong (blue) fear") can be merged into avoir une peur bleue), this is not the case for prendre peur (Lit. "to gain fear") and peur bleue : *prendre une peur bleue. The difference lies in the syntactic structures of the collocations avoir peur and prendre peur. While the first one also has the structure avoir une peur + Modifier, this is not the case with prendre peur, where the noun cannot be modified, *prendre une peur + Modif, probably because it has an inchoative value which precludes modification.

2.1.2

In some examples, analysing a complex collocation as a cluster is not possible. For example, in sequences like in other words, the sequence in Adj words only appears with other, and if other words is possible, it does not have exactly the same meaning as for the collocation. This collocation can be considered as a kind of real ternary collocation with words as a base, while in and other would be the collocates. If one wants to keep the binary status of the collocation (since words is intuitively the "base" of this complex collocation), one may suggest that in other is a kind of complex collocate.

2.2.2

The two previous cases must be distinguished from "recursive collocations", when collocations are inserted into collocations. In this case, collocations can themselves be used as a base or as a collocate. In the sequence to fall in love, in love can be considered as a collocation for love and exists independently: a woman in love (with). fall, the collocate, would be the specific inchoative verb associated to this adjectival collocation.

In the sequence freshly baked bread (example of Bartsch (2004: 67)), freshly baked can be considered as a collocational collocate of bread, freshly being the collocate of the collocation freshly baked that can be encountered in other contexts: freshly baked cakes, freshly baked buns, freshly baked cookies … The existence of recursive collocations also provides a strong argument to develop collocational subentries in dictionaries as advocated by [START_REF] Heid | A model for a multifunctional electronic dictionary of collocations[END_REF].

The table 1 summarizes these different kinds of ternary combinations.

Phenomenon Definition Example (collocates are underlined) Merged collocations into clusters (if syntactic patterns unify)

Two collocations which have the same base and can syntactically combine.

pay attention + close attention pay close attention

Recursive collocations

Case 1 : the base is a collocation.

fall (in( love))

Recursive collocations

Case 2 : the collocate is a collocation (freshly baked) bread

True ternary (or more) collocations

Two or more collocates can be associated to the base. The collocation cannot be decomposed

In other words

Table 1: Different kinds of ternary combinations

In dictionaries, merged collocations and recursive collocations should be decomposed whenever they can be, but very frequent clusters could be mentioned if necessary within the base entries. True ternary collocations could be mentioned within the base entry.

Are all collocations dissymmetric?

2.2.1 Most dictionaries of collocations (e.g. BBI, OCDSE, LTP) provide collocational information within the base entries: s.v. attention for pay attention or close attention, s.v. bachelor for confirmed bachelor, s.v. injured for seriously injured or severely injured and the dissymmetry of collocations is de facto institutionalized in this lexicographic practice, even though it may be assumed that the collocation dictionary user probably uses more syntactic clues than the notions of base and collocate to access lexical information. She knows that she will find verb-noun collocations within the noun entries and verb-adverb collocations within verb entries, and probably uses more this syntactic hint than the complicated notion of base and collocate.

According to us, the question of the dissymmetry between the two elements of the collocation should not only be analysed in terms of psycholinguistic salience of one element. We think that this notion is better analysed in terms of semantic properties than with lists of syntactic patterns. Collocations can in general be analysed as predicateargument structures, where the base is the argument and the collocate the predicate, for example the modifying adjective of a noun (heavy (pred) smoker (arg)) or the subject or the object of a verb (la tristesse (arg) envahit (pred) ; mourir (pred) de tristesse (arg)).

The semantic content of the argument is generally more precise (it is often a noun) than the semantic content of the predicate (generally an adjective or a verb) whose meaning is specified when it applies to a given argument.

2.2.2 According to several linguists [START_REF] Siepmann | Collocations et dictionnaires d'apprentissage onomasiologiques bilingues : questions aux théoriciens et pistes pour l'avenir[END_REF][START_REF] Hausmann | Lexicographie française et phraséologie[END_REF]Gonzales Rey 2002;[START_REF] Bartsch | Structural and functional properties of collocations in English, A corpus study of lexical and pragmatic constraints on lexical co-occurrence[END_REF], in some lexical associations though, it is difficult to decide what element should be considered the base, since no dominant constituent seems to emerge. Interestingly, one can notice that the tricky cases often examined in the literature have the same syntactic properties, as pointed out by Hausmann (1996): (a) N prep N pattern, e.g. in a pack of dogs3 (is the base pack or dogs?) or an outburst of anger. If the quantitative interpretration is chosen ("Mult" in the language of Lexical Functions for pack or "Sing" for outburst), dogs or anger could be considered as the bases, while if the non quantitative interpretation is chosen, they would be considered as collocates of content words. (b) V Prep N pattern like in blush with shame, where Prep N can be interpreted as an adverbial (the manifestation) or a causative complement. In both structures, two predicate-argument structures can be pointed out, hence the ambiguity concerning the identification of the base. As advocated by [START_REF] Hausmann | Lexicographie française et phraséologie[END_REF], we think that these two-base collocations should be stored within both dictionary entries.

2.2.3

Apart from these cases, a difficult issue is the case of conjoined collocations, such as rich and famous, sain et sauf, safe and sound, slowly but surely … These associations share several characteristics with collocations: the sequence includes two recurring content words and it is quite transparent from a semantic viewpoint. The two elements are irreversible: *sauf et sain, *surely but slowly… From a syntactic viewpoint, the coordination suggests that no dominant element emerges, and from a semantic viewpoint both elements seem to have an equal importance. If such combinations could be considered as (atypical) collocations in a way -due to their binary character and their transparency -they do not have a prominent element like a base. Several collocation dictionaries store these two-base collocations within both entries. For example, safe and sound in the OCDSE is recorded s.v safe and s.v. sound, which seems the best solution.

Should collocations include function words, and to what extent?

Determiners

Function words are the poor relation of phraseology. In studies about collocations, they are often regarded as secondary. Definitions of collocations generally assume that the syntactic relation holds between two content words or constituents, what would implicitly exclude function words from being part of a collocation. Nevertheless, as pointed out by [START_REF] Bartsch | Structural and functional properties of collocations in English, A corpus study of lexical and pragmatic constraints on lexical co-occurrence[END_REF] or [START_REF] Siepmann | Collocations et dictionnaires d'apprentissage onomasiologiques bilingues : questions aux théoriciens et pistes pour l'avenir[END_REF], function words are essential in the syntactic description of collocations. In examples such as to commit a crime4 or faire une promenade ("to take a walk"), the determiner is essential in the collocations (the determiner cannot be omitted: *to commit crime, *faire promenade (Lit. 'to make walk')).

However, one can notice that the information about the determiner is highly likely to vary according to contexts (to commit crimes, he committed this horrific crime …). In other words, the word a is not a stable element in the collocation to commit a crime. Collocational elements are here inserted in a syntactic construction where a determiner is compulsory and this determiner can vary (to commit DET crime). As pointed out above, the syntactic construction of the collocation (constraints about determiners, syntactic alternations, subcategorization of the collocation5 ) should be detailed, but according to us, variable determiners and subcategorized elements should not be considered as essential parts of the collocations, except in specific cases.

Prepositions

All prepositions do not have the same status. This is very clear in some classical syntactic models like Lexical Functional Grammar [START_REF] Bresnan | The mental representation of grammatical relations[END_REF]: prepositions which are used to introduce arguments (i.e. I send that to him) cannot be considered as predicates (they generally have a weak meaning (or no meaning at all) and their choice is determined by the verbal predicate, here to send), while prepositions used to introduce modifiers are generally considered as predicates. In they did that for fear, for has a causative meaning. The same prepositions can generally be used to introduce either arguments or predicates, even if they do not have the same meaning in these two contexts. For example, de in je me souviens de ça (Lit. "I remember of that") de is a pure function word and can be considered as an empty word while in de dépit ("out of pique") de introduces a causative meaning.

Functional empty words should not be considered as main elements of the collocations (even if the subcategorized prepositions of the collocation should be adequately described). Meaningful grammatical words should, however, be included in collocations such as for fear, out of love, as a result where they are predicates and can be considered as collocates of the base argument6 . This is already the case for ECD-like collocation dictionaries like the DAFLES and the LAF, but the inclusion of meaningful grammatical words is not yet generalized in most collocation dictionaries. In OCDSE, subcategorized (empty) prepositions like of in fear of failure and meaningful prepositions like out of (fear) are stored within the same PREP paragraph, although they fulfil a very different function from a semantic viewpoint.

Complex prepositions and conjunctions including content words

Some sequences such as for fear of, with the aim of, de peur que ("for fear of"), in which the noun keeps its usual meaning, are problematic. A large set of these collocations are stored in some dictionaries of collocations like the DAFLES, the BBI or the OCDSE, as can be seen in table 2. (1) Because they had a terrible fear of failure, they did not take the exam.

(1') Parce qu'ils avaient très peur d'un échec, ils n'ont pas passé l'examen.

(2) For fear of failure, they did not take the exam.

(2') De peur d'échouer, ils n'ont pas passé l'examen.

Moreover, most of these expressions can be encountered without subcategorized elements:

(3) They did that for fear.

(3') De peur, elle a fait ça.

(4) They did that with this aim.

These properties seem to show that sequences like for fear of, with the aim of or de peur que should not really be considered as complex conjunctions or prepositions, but like prepositional collocations in which the noun can subcategorize conjunctive sentences (peur que P (Lit 'fear that S')) or prepositional phrases (fear of NP). Due to their syntactic properties, the sequences should be considered as prepositional collocations (equivalent to a PP) and their syntactic properties (in particular, subcategorization) should be adequately accounted for.

Conclusion

Collocations can be considered as predicate-argument structures, and as such, are prototypically binary associations, where the predicate is the collocate and the argument is the base. Most ternary (and over) collocations are merged collocations (collocational clusters) or recursive collocations. For language users, however, it can be useful to have access to these productive clusters or recursive collocations in dictionaries. Some atypical collocations do not show a dissymmetric base-collocate structure because they do not have a clear predicate argument structure.

Content grammatical words should be considered as parts of collocations (collocates), while other grammatical words like determiners and subcategorized prepositions or conjunctions should be adequately described but are not central elements of the collocations.

Dictionaries of collocations like the DAFLES, the BBI and the OCDSE are less timid than linguists in their description of collocations. They already include atypical collocations like conjoined collocations and collocations including grammatical words. Defining clear linguistic properties could probably enhance the microstructure organization of such dictionaries and enable adequate computational treatment of collocations.

Table 2 :

 2 Complex prepositions and conjunctions including content words in dictionaries of collocations

	These collocations involve two function words which are discontinuous. The first ones,
	for (in for fear of) or with (in with the aim of), are meaningful prepositions, while the
	second ones (a preposition or a conjunction) are in a way subcategorized by the noun,
	and can be encountered in other contexts, as can be observed in (1) and (2) (Cf. Gross,
	1986):

The underlining is ours.

In the LTP, collocations are provided within the base entries. Syntactic models are associations of two content words: V-N, N-V, A-N, N-N, V-ADV. We find the same kinds of associations in Beauchesne's DC.

(Bartsch 2004 : 36)

Example ofBartsch (2004:36).

For example, (Someone) gives a talk (about something) (to an audience).

Several linguists do not include this kind of syntactic pattern (Prep + N) within the set of collocational patterns[START_REF] Heid | A model for a multifunctional electronic dictionary of collocations[END_REF][START_REF] Spohr | A Description Logic Approach to Modelling Collocations[END_REF][START_REF] Hausmann | Le dictionnaire de collocations[END_REF]).

Acknowledgements

Special thanks to Robert Barr and Iva Novakova for their valuable comments.

Dictionaries

BBI :

The BBI dictionary of English word combinations (1997) (1984,1988,1992,2000)


Marcelo Menezes Morato
email: marcelomnzm@gmail.com

Julio E Normey-Rico

Olivier Sename

A Tracking Model Predictive Control for Input-Output LPV Systems using Parameter Extrapolation

Keywords: Model Predictive Control, Linear Parameter Varying Systems, Input-Output Representation, LMI and Convex Optimisation, Stability and Performance

whether they are published or not. The documents may come

INTRODUCTION

Model Predictive Control (MPC) is a very well established control method, based on a receding-horizon optimisation problem [START_REF] Camacho | Model predictive control[END_REF]. MPC schemes using state-space (SS) process models have had considerable research focus over the last years, with a wide variety of results [START_REF] Limon | Nonlinear MPC for tracking piece-wise constant reference signals[END_REF]. Moreover, exploiting the capacity of Linear Parameter Varying (LPV) SS models to describe nonlinear and time-varying dynamics, LPV MPC schemes have been developed for a wide range of applications, e.g. [START_REF] Cisneros | Fast nonlinear MPC for reference tracking subject to nonlinear constraints via quasi-LPV representations[END_REF][START_REF] Morato | A fast dissipative robust nonlinear model predictive control procedure via quasi-linear parameter varying embedding and parameter extrapolation[END_REF]; refer also to the survey [START_REF] Morato | Model predictive control design for linear parameter varying systems: A survey[END_REF] and references therein.

Yet with great theoretical value, the standard SS MPC design requires the availability of state measurements in real time. In turn, state variables are often difficult to measure or estimate with precision. Moreover, state estimation schemes can significantly deteriorate closed-loop performances of MPC in the presence on disturbances and constraints. Due to these issues, SS realisations fall short of industrial expectations, which are seldom anchored in input-output (IO) process descriptions. We also highlight that powerful LPV IO identification tools nowadays exist for a great variety of applications and system classes [START_REF] Bachnas | A review on data-driven linear parametervarying modeling approaches: A high-purity distillation column case study[END_REF][START_REF] Den Boef | LPVcore: MATLAB toolbox for LPV modelling, identification and control[END_REF].

For the previous reasons, we focus henceforth on IO MPC schemes for LPV systems. Obtaining reliable SS descriptions of an LPV IO model is numerically very This work has been supported by CNPq (304032/2019 -0), CAPES (001) and ITEA3 European project (15016) EMPHYSIS.

tough1 , since the dynamic-dependency problem hinders such translation (Tóth et al., 2011a). Therefore, it is desirable to design MPC schemes directly using IO LPV models, without the need for any IO-SS conversion. Up to the Authors' best knowledge, there are only a few papers which propose MPC schemes for LPV systems described in the IO form: [START_REF] Abbas | An MPC approach for LPV systems in input-output form[END_REF][START_REF] Abbas | A robust MPC for input-output LPV models[END_REF][START_REF] Abbas | A new approach to robust MPC design for LPV systems in input-output form[END_REF]. Although these works provide closed-loop stability guarantees, they assume that the future scheduling behaviour is known, which is false for the vast majority of applications, or resort to the worst-case solution, robustifying the MPC by considering the scheduling variables as bounded uncertainties, which often leads to excessive conservativeness.

Motivated by the fact that recent works have proposed (numerically-cheap) recursive linear schemes that are able to extrapolate the future scheduling trajectories with accurateness, e.g. [START_REF] Morato | Novel qLPV MPC design with least-squares scheduling prediction[END_REF][START_REF] Morato | A fast dissipative robust nonlinear model predictive control procedure via quasi-linear parameter varying embedding and parameter extrapolation[END_REF], we formulate herein a novel robust MPC scheme for IO LPV systems using such parameter extrapolation laws. With their inclusion, we are able to enhance the previous IO LPV MPC schemes from [START_REF] Abbas | An MPC approach for LPV systems in input-output form[END_REF][START_REF] Abbas | A robust MPC for input-output LPV models[END_REF][START_REF] Abbas | A new approach to robust MPC design for LPV systems in input-output form[END_REF], while benefiting from their stability frameworks. Specifically, the contributions of this paper are as follows:

• Using an explicit integral action scheme, we propose an IO LPV MPC formulation (Sec. 2), which enables offset-free output tracking of piece-wise constant reference trajectories. The scheme includes the future scheduling parameter estimation scheme from [START_REF] Morato | A fast dissipative robust nonlinear model predictive control procedure via quasi-linear parameter varying embedding and parameter extrapolation[END_REF][START_REF] Morato | Sufficient conditions for convergent recursive extrapolation of qLPV scheduling parameters along a prediction horizon[END_REF], which uses a Taylor argument to conceive a recursive extrapolation law.

• Benefiting from the stability framework from [START_REF] Mayne | Constrained model predictive control: Stability and optimality[END_REF] as expanded to the LPV IO setting in [START_REF] Abbas | A robust MPC for input-output LPV models[END_REF], we generate terminal ingredients that ensure a recursively feasible optimisation, and an asymptotically stable closed-loop (Sec. 3). These ingredients are enabled through a sufficient Linear Matrix Inequality (LMI) constraint. • A numeric benchmark example is used to demonstrate the effectiveness of the proposed method, as well as its advantages and capabilities of the algorithm for real-time nonlinear applications (Sec. 4).

Notation. The index set N [a,b] represents {i ∈ N | a ≤ i ≤ b}, with 0 ≤ a ≤ b.
The identity matrix of size j is denoted as I j ; I j,{i} denotes the i-th row of I j ; -→ I k j denotes I j with k columns shifted to the right. 1 n denotes the stacked vector of n unitary entries. col{•} denotes the vectorisation of the entries and diag{v} denotes the diagonal matrix generated with the line vector v. The predicted value of a given variable v(k) at time instant k + i, computed based on the information available at instant k, is denoted as v(k + i|k). K refers to the class of positive and strictly increasing scalar functions that pass through the origin. A C 1 function f : R m → R n is such that it is differentiable with continuous derivatives.

• denotes the 2-norm. A denotes the transpose of A, while ( ) denotes corresponding symmetrical transposes in LMIs.

SETUP

Preliminaries: IO LPV Model

We consider the following discrete-time multi-input multioutput (MIMO) LPV system given in an IO form:

I ny + na i=1 a i (ρ(k))z -i y(k) = n b j=1 b j (ρ(k))z -j u(k) .(1)
Moreover, we include an implicit integral action, taking the vector of control inputs as follows: u(k) := δ(k) + u(k -1) ∈ R nu , where δ(k) stands for the corresponding control increments. In (1), y(k) ∈ R ny is the process outputs vector, z -1 is the one-sample backward shift operator, n a , n b ≥ 0, while a i ∈ R ny×ny and b i ∈ R ny×nu are coefficient functions.

This LPV system is scheduled by the vector of timevarying parameters ρ(k) ∈ R nρ . Specifically, we assume that these parameters are output-dependent, bounded and measured online, this is: ρ(k) = f ρ (y(k -1)) ∈ P ⊆ R nρ , being P a known set. Moreover, we consider that the nonlinear scheduling map f ρ (y) is algebraic, class C 1 for all y; P := {ρ ∈ R nρ : ρ j ≤ ρ j ≤ ρ j , j ∈ N [1,nρ] } defines a compact, convex scheduling set. By definition, the future scheduling parameters ρ(k + j), ∀j ∈ N [1,∞] are unknown at time instant k. Remark 1. Nonlinear systems can be embedded to a LPV realisations such as Eq. (1), as long as a differential inclusion property is satisfied. This topic is further explained in [START_REF] Morato | Model predictive control design for linear parameter varying systems: A survey[END_REF], with experimental examples given in [START_REF] Hoffmann | A survey of linear parameter-varying control applications validated by experiments or high-fidelity simulations[END_REF].

Impulse Response

The IO realisation from Eq. ( 1) has an equivalent infinite impulse response form, which is given by:

y(k) = +∞ i=0 h i (ρ(k), . . . , ρ(k -i))u(k -i) , (2)
where h i (•) ∈ R ny×nu are known as the Markov coefficients of the LPV systems. These coefficients can be computed recursively as follows, where h i (k) := h i (ρ(k), . . . , ρ(k-i)):

h i (k) = b i (ρ(k)) - min(i,na) j=1 a j (ρ(k))h i-j (k -j), i ≤ n b , - min(i,na) j=1 a j (ρ(k))h i-j (k -j),
else.

SS Representation for Analyses

Now, we give a non-minimal, non-unique SS representation of Eq. ( 1), taking δ(k) as input. This SS model, used for the stability analyses presented in Sec. 4, is as follows:

x(k + 1) = A(ρ(k))x(k) + B(ρ(k))δ(k) , (3)
y(k) = C(ρ(k))x(k) + D(ρ(k))δ(k) , where x(k) = [y(k -1) , . . . , y(k -n a ) , u(k -1) , . . . , u(k - n b ) ] ∈ R nx
defines the state vector with n x = n a n y + n b n u . Note that in this SS representation, all states are known variables (current and past values of the inputs and outputs). The corresponding matrices

A(•) B(•) C(•) D(•) are:            -a 1 (•) . . . -a na (•) (b 0 (•) + b 1 (•)) . . . b n b (•) b 0 I ny . . . 0
(•) + b 1 (•)) . . . b n b (•) b 0 (•)           
.

Full-horizon IO Predictor

The IO LPV model from Eq. ( 1) is used by an MPC algorithm and, thus, a full-horizon prediction equation is required to compute control law at each sample. Consider that

P (k|k) = col{ρ(k + j|k) } , ∀j ∈ N [0,Np-1]
is known (we will discuss how this vector can be generated later on), being N p the prediction horizon. Thus, it follows that:

Y (k|k) = H(P (k|k))∆(k|k) + Θ(P (k|k))x(k) , (4) where Y (k) = col{y(k + j|k) } , ∀j ∈ N [0,Np-1] ∈ R nyNp and ∆(k) = col{δ(k + j|k) } , ∀j ∈ N [0,Np-1] ∈ R nuNp .
In MPC theory H(P (k|k))∆(k) is usually named the "forced response" of the system, giving the dynamics implied by the control effort, where Θ(P (k|k))x(k) is called the "free response", which gives the behaviour in the absence of future control inputs. The prediction matrices are:

H(P (k|k)) =    h 0 (k) . . . 0 . . . . . . . . . Np-1 0 h i (k + N p -1) . . . h 0 (k + N p -1)    , Θ(P (k|k)) = θ(k + 1) . . . θ(k + N p ) .
The elements of Θ(P (k|k)) are found using θ(k

+ j) = - min(j,na) i=1 a i (ρ(k + j|k))θ(k + j -1) + - → I j θ(k + j) , ∀j ∈ N [1,Np] , with θ(k + j) = [-a 1 (ρ(k + j|k)) . . . -a na (ρ(k + j|k))b 1 (ρ(k+j|k)) . . . b n b (ρ(k+j|k))] and - → I j = diag{ -→ I j na ,
-→ I j n b }. Finally, each θ(k+j) is given by θ(k+j) with its (n a +1)-th element added to a correction bias j i=1 h i (k + j).

Process Constraints

We consider that the LPV system in Eq. ( 1) is subject to hard compact polyhedral constraints on outputs and inputs, which define an admissible operation. Specifically, we use y(k) ∈ Y and u(k) ∈ U, ∀k ≥ 0, where:

Y := {y ∈ R ny : |y j | ≤ y j , ∀j ∈ N [1,ny] } , (5)
U := {u ∈ R nu : |u j | ≤ u j , ∀j ∈ N [1,nu] } .
Due to the explicit integral description of the control input, the following constraint is also used:

δ(k) ∈ D :=, {δ ∈ R nu : |δ j | ≤ δ j , ∀j ∈ N [1,nu] } ∀k ≥ 0.
Through the sequel, we assume that the scheduling proxy is locally Lipschitz whenever these constraints are satisfied: Assumption 1. The nonlinear scheduling parameter map

f ρ : Y → P agrees to a local Lipschitz condition around any arbitrary point y ∈ Y, i.e. f ρ (y) -f ρ (ŷ) ∞ ≤ γ ρ (y -ŷ) ∞ , ∀ y, ŷ ∈ Y,
where the smallest constant γ ρ is known as the Lipschitz constant for f ρ (•).

Scheduling Parameter Extrapolation

The concept of MPC is based on spanning a prediction of the process variables along a future horizon window. For such, we use the IO LPV prediction Eq. ( 4). As previously stated, the future scheduling sequence P (k|k) is required.

For this, we benefit from the framework from [START_REF] Morato | Sufficient conditions for convergent recursive extrapolation of qLPV scheduling parameters along a prediction horizon[END_REF], which is based on a first-order Taylor expansion of the scheduling proxy f ρ (y(k -1)). The main advantage of this extrapolation procedure is that it guarantees a convergent guess with a small, bounded residual error, while only resorting to linear operators.

Denote δy(k) = (y(k) -y(k -1)) as the incremental output deviation. By definition, δy is bounded to a compact and convex box-type set δY := {δy ∈ R ny :

|δy j | ≤ δy j , ∀j ∈ N [1,ny] }.
The method is as follows: consider that the static scheduling map f ρ (y) can be approximated, at the first derivative order, by f ρ (y) = f ρ (y)| y + ∂fρ(y) ∂y y (y-y)+ξ ρ , being y the expansion point and ξ ρ a residual which inherits the discrepancy between the real static map and its approximate.

Since f ρ (y) is assumed class C 1 , it is direct that the partial derivatives f ∂ ρ (k) := ∂fρ(y) ∂y y(k)
are ultimately bounded for all y(k) ∈ Y. From this development, we obtain:

ρ(k + 1) = ρ(k) + f ∂ ρ (k)δy(k) + ξ ρ (k) , . . . ρ(k + N p -1) = ρ(k + N p -2) + f ∂ ρ (k + N p -2)δy(k + N p -2) + ξ ρ (k + N p -2) .
Note that ρ(k) and δy(k) are known variables at each instant k, whereas f ∂ ρ (k) can be numerically evaluated. In practice, f ∂ ρ (k+j) is unknown for j ∈ N [1,N -2] , but we can assume that it shows an approximately uniform variation rate, i.e.

f ∂ ρ (k+j) = f ∂ ρ , ∀ j ∈ N [1,N -2]
, where f ∂ ρ denotes the partial derivative evaluated at instant k (refer to the discussion in [START_REF] Morato | Sufficient conditions for convergent recursive extrapolation of qLPV scheduling parameters along a prediction horizon[END_REF]). By doing so, it is implied that ρ(k

+ j) ≈ ρ(k + j -1) + f ∂ ρ δy(k + j -1|k
). Therefore, the estimate for the future scheduling variables can be written as the sum of the estimate from the previous sample corrected with an adjustment term f ∂ ρ δy(k +j -1). Accordingly, we can write the vector-wise extrapolation in a recursive fashion:

P (k|k) = P (k -1|k -1) + f ∂ ρ δY (k|k) , (6)
where the sequence of output increments is given by δY 1) is stable in closedloop. Then, the recursive extrapolation algorithm in Eq. ( 6) converges. Proof 1. This proof is reduced for brevity. Complementary discussions are available in [START_REF] Morato | Sufficient conditions for convergent recursive extrapolation of qLPV scheduling parameters along a prediction horizon[END_REF]. Consider that the residual term ξ ρ (k + j) should turn null. Thus, use lim k→∞ y(k) = y e holds (stability) and take

(k) = col{δy(k + j|k)} , ∀j ∈ N [0,Np-2] . Lemma 1. Assume that f ρ (•) is class C 1 and that f ∂ ρ is ultimately bounded. Assume that (
ξ ρ (k) = f ρ (y(k -1)) -f ρ (y(k -2)) -f ∂ ρ δy(k).
Due to the stabilisation, it directly follows that lim k→∞ f ρ (y(k -1)) = lim k→∞ f ρ (y e ) and lim k→∞ δy(k) = 0, which implies in lim k→∞ ξ ρ (k) = -lim k→∞ f ∂ ρ δy(k) → 0. This concludes the proof. Lemma 2. The estimation error (residual) is ultimately bounded, i.e. ξ ρ (k) ∞ ≤ γ ρ + f ∂ ρ δy, ∀k ≥ 0. Proof 2. The residual term in the extrapolation law is given by ξ

ρ (k + j + 1) = f ρ (y(k + j + 1)) -f ρ (y(k + j)) -f ∂ ρ δy(k + j).
Using a triangular inequality, we obtain

ξ ρ (k + j + 1) ∞ ≤ f ρ (y(k + j + 1)) -f ρ (y(k + j)) ∞ + f ∂ ρ δy(k + j) ∞ . Finally, due to Assumption 1, we state that ξ ρ (k + j + 1) ∞ ≤ γ ρ δy(k + j) ∞ + f ∂ ρ δy(k + j) ∞ . Since f ∂ ρ is ultimately bounded, it follows that: ξ ρ ∞ ≤ γ ρ + f ∂ ρ δy.
This concludes the proof. Remark 2. The extrapolation procedure presented in this section is completely detailed in [START_REF] Morato | Sufficient conditions for convergent recursive extrapolation of qLPV scheduling parameters along a prediction horizon[END_REF]. The Hessian-based estimates from [START_REF] Hanema | Stabilizing non-linear model predictive control using linear parameter-varying embeddings and tubes[END_REF] and the iteratively refined estimated from [START_REF] Cisneros | Fast nonlinear MPC for reference tracking subject to nonlinear constraints via quasi-LPV representations[END_REF] are comparable methods to the one considered herein. Nevertheless, the main advantage of the proposed scheme is that converging estimates with reduced error bounds, formulated by the means of linear laws. Thereof, the corresponding model-process uncertainties that are derived when applying MPC became very reduced, enabling less conservative control synthesis.

The MPC Design

As previously stated, we consider an MPC design for LPV processes with IO descriptions. Accordingly, we use the recursive extrapolation procedure in Eq. ( 6) to generate the scheduling sequence P (k|k) that is used to compute the future output predictions Y (k) through Eq. ( 4). In this paper, we assume that the residual errors ξ ρ are null. The MPC terminal ingredients are presented later, in Sec.

3.

Consider the following cost function:

J = Np-1 i=0 (e(k + i), v(k + i)) + V (x(k + N p ) -x r ) ,
where the stage cost (e, v) := e Q + v R is given with respect to the output tracking error e(k) = r(k) -y(k), under the assumption that r(k) is piece-wise constant s.t.

r(k+j) = r(k), ∀j ∈ N [0,Np-1] . The terminal cost V (x-x r )
requires x, which is the state description given through Eq.

(3), and x r , the envisioned state reference target, defined in terms of r(k); this cost is used to penalise the distance of non-minimal states at the end of the prediction horizon to a given target, whereas (•, •) weights the performance along the horizon. Consider Q and R as positive definite weighting matrices, used to imply the envisioned trade-off between control effort and output reference tracking.

Taking J into account, the proposed MPC resides in solving the following optimisation, at each instant k:

min ∆(k) Np-1 i=0 (e(k + i), v(k + i)) + V (x(k + N p ) -x r ) ,
s.t. : Y (k|k) = H(P (k|k))∆(k) + Θ(P (k|k))x(k), ( 7)

y(k + j|k) ∈ Y , u(k + j|k) ∈ U , δ(k + j|k) ∈ D , (x(k + N p ) -x r ) ∈ X f ,
where X f is a terminal invariant set for the controlled IO LPV system. Let J (x(0), r(k), P (k|k)) be the optimal solution of the optimisation in Eq. ( 7), from which ∆(k) is the optimal sequence of control inputs. Then, the MPC law at time instant k considers in apply the first entry of ∆(k) , i.e. δ (k|k), to the process using u(k) = u(k -1) + δ (k). Definition 1. Positive Invariant Set Assume that there exists a terminal set X f . X f is a positively invariant set for stability-related SS LPV model from Eq. (3) iff, for any x ∈ X f and ρ ∈ P, it follows that x + ∈ X f , where the successor state x + is given by A(ρ)x + B(ρ)δ.

CLOSED-LOOP STABILITY AND RECURSIVE FEASIBILITY

Next, we discuss how to generate the MPC terminal ingredients V (•) and X f in order to render an asymptotically stable closed-loop, as well as a recursively feasible optimisation.

First, let us define the admissible steady-state targets for the IO LPV system in Eq. ( 1). Consider r as a constant output reference target. We say that r ∈ R is an admissible target if and only if there exists a control input u r ∈ U such that

I ny + na i=1 a i (f ρ (r)) r = ( n b i=1 b i (f ρ (r)
)) u r . Accordingly, we introduce the set of all admissible terminal state-input targets (x r , u r ), that is:

X r := {x r ∈ R nx | x r = [1 na r, 1 n b u r ] , ∀(r, u r ) ∈ R × U}.
Due to the box-type constraints over y and u, X r can be equivalently described as

{x ∈ R nx | |x j | ≤ x j , ∀j ∈ N [1,nx] }.
Assumption 2. The reference trajectory R(k) = col{r(k + j)}, ∀j ∈ N [0,Np-1] is known, piece-wise constant, and admissible at each sampling instant k, i.e. r(k) ∈ R. Assumption 3. The scheduling parameters take a constant value ρ r in steady-state, i.e. ρ = ρ r , ∀ (x -x r ) ∈ X f . Assumption 4. The stage cost function is positive definite and uniformly continuous such that (e, δ) ≥ α ( e ) and | (e 1 , δ 1 )-(e 2 , δ 2 )| ≤ λ e ( e 1 -e 2 )+λ δ ( δ 1 -δ 2 ), where α , λ e and λ δ are K-functions. It is implied that (0, 0) = 0. Assumption 5. (1) There exists an admissible terminal feedback law u(k -1) + κ t (x(k), r(k), P (k|k)) ∈ U.

(2) The terminal set X f is closed, contains the origin, and represents admissible positive invariant set.

(3) The terminal cost V (x-x r ) is continuous and positive for all x-x r ∈ X r . Moreover V (•) represents a control Lyapunov function for the unconstrained LPV system in Eq. ( 1), meaning that there exist constants b > 0 and σ > 1 such that

V (x -x r ) ≤ b|x -x r | σ . This implies that V (A(f ρ (r))(x -x r ) + B(ρ)κ t (•)) -V (x - x r ) ≤ (y -r, κ t (•)), for all r ∈ R and V (x 1 -x r ) - V (x 2 -x r ) ≤ α r (|x 1 -x 2 |) (i.e. V (•) is a K function).
The following Theorems ensure that the error dynamics e(k) = r(k)-y(k) converge to the origin, which conversely means that offset-free reference tracking is ensured. Theorem 1. Stability and Recursive Feasibility Consider that Assumptions 2-5 hold. Suppose that there exists a terminal control law δ = k t (x, r, P ). Consider that the IO LPV system in Eq. ( 1) is controlled by the MPC, as rendered through Eq. ( 7). Then, asymptotic output stability is ensured if the following conditions hold ∀ρ ∈ P:

(C1) The origin x(k) -x r (k) = 0 lies in the interior of X f ; (C2) X f is positively invariant under the terminal feed- back controller κ t (•); (C3) The discrete Lyapunov equation is verified within this invariant set, this is, ∀ x -x r ∈ X f and ∀ ρ ∈ P: V (x + -x r , ) -V (x -x r ) ≤ -(e, κ t (•)), where x + = A(ρ)x + B(ρ)κ t (•); (C4) The image of the terminal control is admissible, i.e. κ t (•) + u ∈ U , ∀ρ ∈ P; (C5) The terminal set X f is a subset of X r .
Assuming that the initial solution of the MPC problem ∆ (k) is feasible, then, the MPC is recursively feasible, steering e(k) = r(k) -y(k) to the origin. Proof 3. This proof is standard and follows [START_REF] Mayne | Constrained model predictive control: Stability and optimality[END_REF][START_REF] Morato | Model predictive control design for linear parameter varying systems: A survey[END_REF].

In order to satisfy the conditions required by Theorem 1, we choose the following quadratic terminal cost function V (x -x r ) = (x -x r ) T (x -x r ), where T = T is a positive definite weight. Accordingly, the terminal set X f is taken as a sub-level set of the terminal cost, i.e.:

X f := {x ∈ R nx | x T x ≤ 1}.
By definition, X f is an ellipsoidal set constraint, which should be positively invariant for the terminal feedback κ t (•). Thus, the following Theorem gives a numerically solvable sufficient solution that can be used to generate the terminal ingredients. Theorem 2. Terminal Ingredients Consider that Assumptions 2-5 hold. Then, conditions (C1)-(C5) from Theorem 1 are satisfied if there exist a symmetric positive definite matrix T ∈ R nx×nx and a rectangular matrix W ∈ R nu×nx such that Y = T -1 > 0, W = KY and that LMIs ( 8)-( 10) hold under the minimisation of log det{Y } for all ρ ∈ P. The terminal feedback is then given by κ

t (•) = K(x -x r ).    Y (A(ρ)Y + B(ρ)W ) Y Y 0 Q-1 W 0 0 R -1    ≥ 0 , (8)
x

2 j I {j} Y I {j} Y Y ≥ 0, j ∈ N [1,nx] , (9)
δ 2 i I {i} W Y ≥ 0, i ∈ N [1,nu] . ( 10
) Proof 4. Consider r(k) is piece-wise constant, thus sat- isfying Assumption 2. Note that (e, v) = e Q + v R is equivalent to x -x r Q + v R , using Q = diag{QI nyna , 0 nun b }.
Consider P (k|k) is known due to the extrapolation procedure from Eq. ( 6). Since ρ(k) = f ρ (y(k -1)), it follows that ρ(k) = ρ r in steady-state, iff lim k→+∞ y(k -1) = r. Then, by applying a Schur complement to LMI (8), we obtain condition (C3), which suffices for (C2). By definition, an ellipsoid ensures (C1). ( C4) and (C5) are respectively satisfied by applying Schur complements to LMI ( 9) and (10). In turn, the terminal feedback κ t (•) = K(x -x r ) ensures that the SS representation in Eq. ( 3) is asymptotically stable, which conversely implies in the output tracking of r(k). This concludes the proof.

Remark 3. The terminal ingredients provided through Theorem 2 ensure recursive feasibility and asymptotic stability of the tracking error trajectories (refer to Propositions 1-2). Note that LMI ( 8) is infinite-dimensional, having to hold ∀ ρ ∈ P. In practice, the solution can be found by enforcing the inequalities over a sufficiently dense grid of points (ρ) along P, then verifying it for a denser grid. Remark 4. An alternative formulation to the prior can be used in order to drop the parameter-dependency of LMI (8): the full-block S-procedure over a corresponding Linear Fractional Transformation (LFT) of the LPV system [START_REF] Scherer | LPV control and full block multipliers[END_REF], leading to similar LMIs to those presented in [START_REF] Abbas | A new approach to robust MPC design for LPV systems in input-output form[END_REF]. Proposition 1. (Recursive Feasibility). Consider Y as a solution to Theorem 2. Then, given any x ∈ X f , x r ∈ X r , r ∈ R and δ = κ t (x, r, •), we have x + = Ax+BKx ∈ X f . Con-sider an optimal sequence ∆ = (δ 0 , δ 1 , . . . , δ Np-1 ) and an reference target r. Then δc = (δ 1 , . . . , δ Np-1 , κ t (x, r, •)) define feasible (candidate) solution of the MPC problem in Eq. ( 7) for any r ∈ R, which means that the optimisation is recursively feasible. Proof 5. Let Assumptions 2-5 hold. Consider there exists a solution Y to Theorem 2. Then, from conditions (C1), (C2), (C4), and (C5) from Theorem 1, we can infer that the generated control signal provides recursively feasible solutions to the MPC optimisation in (7). Take the feedback2 κ t (•) = K(x -x r ) and ρ r = f ρ (r). Then, the if the initial condition x(0) generates a feasible input sequence ∆ , all future iterations of the optimisation will also be feasible: the generated control law control is admissible (condition (C4)) and all state variables x ∈ X f generate successor state variables x + which are also inside X f (condition (C2)), which contains the origin (condition (C1), terminal condition for (x-x r ), and X f being sub-set of X r (condition (C5), which ensures that x, x + are admissible). This concludes the proof. Proposition 2. (Asymptotic Stability). Consider Y as a solution to Theorem 2. Then, the LPV system (1) in closed loop with the MPC input κ t (•) exhibits an asymptotic stable tracking error dynamics. That is, for any feasible initial condition x 0 and constant set-point r ∈ R, it is implied that x(k) -x r ≤ β( x(0) , k), where β is a Kfunction which passes through the origin. Proof 6. Let there be a terminal stage cost V (•) such that Assumption 4 holds. Let Assumption 5 also hold and Proposition 1 be verified. Note that since (•, •) is a quadratic stage cost, α , λ e and λ u indeed exists. Consider there exists a solution Y to Theorem 2. Then, the SS closed-loop is stable due to (C3) of Theorem 1, which conversely ensures that V (x(k)-x r )-V (x(k -1)-x r )) ≤ -y(k) -r Q . Thus, we obtain y(k) -r Q ≤ β( (x(0)x r ) , k). Since Q > 0 (positive definite), we have y(k)r(k) Q ≥ x(k) -x r and thus asymptotic stability is established. This concludes the proof.

NUMERIC BENCHMARK

In this Section, we demonstrate the effectiveness of the proposed method for controlling LPV system in the IO form, without any state measure or estimation procedure involved. For such, we consider an adapted version of the unstable second order system from [START_REF] Abbas | An MPC approach for LPV systems in input-output form[END_REF], with additional nonlinearities, given as:

y(k) = -a1(ρ(k)) (0.2 -0.7 y(k -1) 2 10 ) y(k -1) (11) + -a2(ρ(k)) (-0.7 -0.4 y(k -1) 2 10 ) y(k -2) + b1(ρ(k)) (3.4 -1.2 y(k -1) 2 10 ) u(k -1) + b2(ρ(k)) (1.6 -2.8 y(k -1) 2 10 ) u(k -2) ,
where the scheduling parameter is endogenous and given by ρ(k) = y(k-1) 2 10

. This system should be controlled such that the output trajectory y(k) tracks a given piecewise reference signal, whilst the following constraints are respected: u(k) ∈ U := [-1, 1], δ(k) ∈ D := [-0.04, 0.04], y(k) ∈ Y := [0, 5], ∀k ≥ 0. We note that these constraints imply that ρ(k) ∈ P := [0 , 2.5], ∀k ≥ 0. This system is assumed to operate under a sampling rate of 40 Hz. The proposed MPC is tuned with a prediction horizon of N p = 10 discrete-time steps, thus previewing a window of 250 ms.

First of all, we show that there indeed exists a solution to Theorem 2, given by: Therefore, we can affirm that κ t (x -x r ) = K(x -x r ), with K = W T , V (x -x r ) = (x -x r ) T (x -x r ), and X f := {x-x r ∈ R nx | (x-x r ) T (x-x r ) ≤ 1} are suitable terminal ingredients for the MPC optimisation in Eq. ( 7), ensuring recursive feasibility and asymptotic stability (and, in turn, offset-free reference tracking). We stress that these ingredients are generated offline, with respect to the tuning parameters Q = 0.8I ny and R = 0.2I ny and the known sets U, D, Y, and P.

T -1 = Y =    15.
Taking into account that stability is ensured by design (and, thus, Lemma 1 holds), we now demonstrate how the scheduling parameter extrapolation procedure operates.

During the implementation, the recursive estimates are generated through the linear operator given in Eq. ( 6). As shows Fig. 1, the generated estimates P (k|k) are very accurate, thus passing to the MPC precise information regarding the future behaviour of ρ. In this Fig. , we can also see that the estimation error ξ ρ is indeed small (refer to Lemma 2) and converges rapidly, which means that it can be neglected for simplicity (treating this uncertainty robustly directly in the MPC synthesis step is a topic for future works).

Next, we show the obtained performances with the proposed method. Fig. 2 shows the system trajectories being steered to the terminal set X f , as ensured by the MPC. The left-side sub-figure gives a three-dimensional cut of X f , while the right-side shows a two-dimensional projec-tion. The system trajectories shown are the tracking error e(k) = r(k) -y(k) and the control input u(k). As one can see, the tracking error dynamics are repeatedly steered to the origin (for each new reference goal). Fig. 3 gives the control input trajectories (and the control increment), altogether with the piece-wise constant reference target signal r(k) and the output. Evidently, the integral-embedded MPC ensures offset-free tracking, which is significant.

The MPC requires, in average, 6.5 ms to solve the optimisation procedure, while the Taylor-based scheduling extrapolation takes only 0.07 ms, in average. This clearly indicates the relieved numeric burden of the proposed technique (6.57 ms, in average), which is ready for realtime embedded applications. We recall, once again, that no state measures or observers are necessary, making the proposed method coherent with industrial practices.

CONCLUSIONS

In this paper, a novel MPC algorithm for IO LPV systems is proposed. The future LPV scheduling parameters are extrapolated using a recursive Taylor expansion law, which generates the MPC prediction matrices at each sampling period. Terminal ingredients are offered through an LMIsolvable remedy, which ensures Lyapunov properties of the closed-loop. The method is able to ensure asymptotic offset-free reference tracking, also thanks to an explicit integral action and to these optimisation ingredients. In order to demonstrate the effectiveness of the method, it is to a numeric benchmark system, exhibiting good performances.

The proposed IO MPC method exhibits real-time capabilities, since its online implementation only requires a linear operation (scheduling parameter extrapolation) and the solution of a single quadratic optimisation problem. Also, the method is directly applicable to nonlinear systems, as long as if LPV embedding is satisfied. Moreover, the method does not require any additional reference tracking tool (such as the artificial reference variables).

For future works, we plan on generalising the proposed framework for data-driven predictive control synthesis, replacing the IO LPV description of the system by a databased IO realisation.

Fig. 1 .

 1 Fig. 1. Scheduling trajectory and Taylor-based extrapolation estimated (at different samples).

Fig. 2 .

 2 Fig. 2. Stability and terminal invariance. Terminal set X f in two-(right) and three-dimensional (left) projections along with system trajectories u(k), e(k) = r(k) -y(k).

Fig. 3 .

 3 Fig. 3. System performances.

This issue is exploited in details in recent key papers on the topic, e.g.(Tóth et al.,

2011a,b).

Note that this not a traditional state-feedback, since x relates to past input and output measurements only (refer to Eq. (3)).


P Bordet

E Suard

Magnetic structure and charge ordering in Fe 3 BO 5 ludwigite

Keywords: numbers: 75.25.+z, 71.27.+a Keywords: charge ordering, spin ladder, magnetic structure, neutron diffraction

The crystal and magnetic structures of the three-leg ladder compound Fe3BO5 have been investigated by single crystal x-ray diffraction and neutron powder diffraction. F e3BO5 contains two types of three-leg spin ladders. It shows a charge ordering transition at 283 K, an antiferromagnetic transition at 112 K, ferromagnetism below 70 K and a weak ferromagnetic behavior below 40K. The x-ray data reveal a smooth charge ordering and an incomplete charge localization down to 110K. Below the first magnetic transition, the first type of ladders orders as ferromagnetically coupled antiferromagnetic chains, while below 70K the second type of ladders orders as antiferromagnetically coupled ferromagnetic chains.

I. INTRODUCTION

The so-called homometallic ludwigite Fe 3 BO 5 is a mixed valence low dimensional semiconductor which shows a number of exciting electronic and magnetic properties [START_REF] Continentino | Frontiers in Magnetic Materials[END_REF][START_REF] Guimaraes | [END_REF] . The general chemical formula for ludwigites is M 2+ 2 M 3+ O 2 BO 3 where M 2+ or M 3+ are 3d transition metal ions. The M 2+ and M 3+ ions are located at the centers of edge sharing oxygen octahedra forming zig-zag walls. The crystal structure of the Fe 3 BO 5 high temperature phase is depicted in Figure 1, where the atom labeling is shown. The symmetry is orthorhombic, space group P bam, with a=9.462 Å, b=12.308 Å, c=3.075 Å. The F e 2+ and F e 3+ cations occupy four distinct metal sites. The quasi-two dimensional structure can be viewed as resulting from the presence of two types of three-leg spin ladders sub-units of F e cations. Ladders I are built upon the F e2 and F e4 F e 3+ cations having localized high-spins S=5/2 with one additional itinerant electron per rung. Ladders II involve the F e1 and F e3 F e 2+ cations with S=2 localized spins according to Mössbauer spectroscopy [START_REF] Guimaraes | [END_REF]3 . Ladders I show a structural and charge ordering transition at Tc = 283K such that long and short bonds alternate along the ladder caxis 4 . This transition is accompanied by an anomaly in the magnetization and a change of slope of the resistivity. The origin of this transition has been discussed in terms of excitonic instability 6 . F e 3 BO 5 shows an antiferromagnetic transition at 112K, weak ferromagnetism below 70K and another magnetic transition at 40K where ferromagnetism disappears [START_REF] Continentino | Frontiers in Magnetic Materials[END_REF][START_REF] Guimaraes | [END_REF] . It is of primary importance to determine the evolution of charge and magnetic ordering in the whole temperature range between 300K and 10K in order to gain insight into the complex physical behavior of this material. Previous x-ray diffraction studies 4,5 only provide structural information at 300K, 144K and 15K. Magnetic structure investigation at 5K was reported by Attfield et al. 9 , but nothing is known about magnetic orderings at intermediate temperatures. Here we present the results of a x-ray single crystal diffraction experiment and of a neutron powder diffraction experiment both as function of temperature, aiming at providing detailed information about the charge and spin behavior of this complex and original compound. Preliminary results were given in refs. 10,11 .

II. EXPERIMENTAL

Single crystal and powder sample used for this work were synthesized and characterized according to ref. [START_REF] Guimaraes | [END_REF] .

A. Single crystal x-ray diffraction

In order to investigate in close detail the charge ordering occurring at 283K, we performed a single crystal xray diffraction experiment as a function of temperature to follow the evolution of the structural parameters across the transition and in a large temperature range. The experiment was carried out with a Nonius KappaCCD diffractometer equipped with graphite monochromatized AgKα radiation (λ=0.5608 Å). The temperature was controlled with an Oxford Cryosystem nitrogen gas blower. A single crystal sample was ground to a sphere of 0.1 mm radius. 180˚omega scans with 2˚frame size were used to collect the data up to sinθ/λ=0.9, with a 35 mm sample to detector distance, every 20K between 320K and 110K on cooling. CCD data were processed, averaged and corrected for absorption with the Denzo-SMN [START_REF] Otwinowski | Methods in Enzymology[END_REF] and Maxus 14 packages, refinements and structure analyzes were performed using the Jana2000 software [START_REF] Petricek | The crystallographic computing system[END_REF] . Although the transition was reported to appear at 283K [START_REF] Mir | [END_REF] , we could detect superstructure spots characteristic of a doubling of the c axis parameter already on the 290K data. These data and those at lower temperature were therefore collected for the refinement on the basis of this double cell, while the 300K, 310K and 320K data were re-fined with the high temperature phase unit cell. For the low temperature phase, each experiment leads to about 2450 unique reflections of which approximately one half were superstructure reflections (l odd), and ≈ 1750 were considered as observed, i.e. had I > 3σ(I) (where I and σ(I) denote a Bragg reflection intensity and its standard deviation, respectively). For the high temperature phase, about 1260 unique reflections were obtained, of which about 1050 were observed. The structure of ludwigite has space group P bam with cell parameters a=9.465 Å, b=12.310 Å, c=3.077 Åat 320K. As stated above, all superstructure spots appearing below 290K could be indexed by doubling the c axis parameter. The systematic extinction condition lead to space group P bnm, as reported by Mir et al. [START_REF] Mir | [END_REF] . The refined atomic coordinates at 320K and 110K are shown in Table I. It can be seen that the main consequence of the phase transition on the structure is that some atomic positions split into two positions which are no longer equivalent by symmetry in the low temperature phase. It is also noteworthy that among the Fe cations, only the Fe4 cations undergo such a splitting, which suggests that the transition can be driven by charge ordering in the Fe4-Fe2-Fe4 ladders along which one electron is delocalized at high temperature. This is clearly visible in Figure 2 : above the transition, the Fe2 cations present a strongly anisotropic atomic displacement parameter (a.d.p.) along the rung, while below it, this a.d.p. drastically decreases and the Fe2 position is displaced toward Fe4a or Fe4b in an alternate way along the c axis. In order to characterize this effect, the ionic valences were calculated using the bond valence sum (b.v.s.) method 17 . In Figure 3 are represented as function of temperature the evolution of the Fe-Fe distances along the rungs of the Fe4-Fe2-Fe4 ladders, the Fe cation valences and the Fe2 a.d.p.'s. These parameters were found to display the most prominent modifications across the transition. At higher temperature, the Fe2-Fe4 distances is ≈ 2.79 Å; well below the transition, Fe2-Fe4a ≈ 2.61 Å, an extremely short distance (the Fe-Fe distance in metallic iron is ≈ 2.48 Å) while Fe2-Fe4b increases to ≈ 2.95 Å. At the same time, the valences of Fe4a and Fe4b become different. Above the transition, the Fe4 cation valence is found to be +2.67, which is the expected value if one extra electron is shared in a disordered way among the three F e 3+ cations (2 Fe4 and 1 Fe2) which consti-tute the ladder rung. However, the Fe2 cation valence is found to be close to +2.5. This low value, together with the large and anisotropic a.d.p. for Fe2 strongly indicate that this cation is in a disordered position above the transition, very probably dynamic, i.e. the cation vibrates between to positions symmetric about the rung center, coming alternately close to the two Fe4 cations at the rung extremities. Below the transition, ordering occurs and the Fe2 displacements toward the rung extremities (Fe4a cations) alternate in a zigzag way along the c axis. At the same time the Fe2 a.d.p.'s become isotropic and similar to those of the other atoms. The Fe4a valence decreases to ≈+2.53, close to the Fe2 cation valence, while the Fe4b valence increases to ≈+2.86. Clearly, this structural phase transition is related to the ordering of charges between Fe4a and Fe4b. A schematic picture can be proposed where the Fe2 and Fe4a cations form a pair by sharing the common electron which was delocalized over the whole rung in the high temperature phase, while the Fe4b valence becomes +3. However, as indicated by the anomalous behavior of the Fe2 cations, the Fe2-Fe4 pairs probably already exist in the high temperature phase but are dynamically disordered between the two sides of the rung. Therefore, the transition can also be viewed as an ordering of the pairs or of the electrons which are responsible for their existence. On cooling towards the transition, the decrease of the available thermal energy could be the driving force to induce the localization of the Fe2 cations (or Fe2-Fe4 pairs). The zigzag order along the c axis of the Fe2-Fe4 pairs would be a way to minimize the structural distortions effects induced by this localization.

As can be seen in Figure 3, the phase transition takes place over a quite wide temperature range of more than 50K before all parameter values are stabilized. This indicates that a large amount of disorder is still present down to ≈ 200K. Furthermore, the values at which the evolution of the Fe cation valences saturate do not correspond to complete charge ordering even down to 110K. This will have important consequences for the magnetic behavior of this compound, since the magnetic ordering process will take place on a context of dynamical charge disorder, at least within the Fe4-Fe2-Fe4 ladders.

B. Neutron powder diffraction experiment

In order to investigate the evolution of the magnetic ordering for Fe 3 BO 5 , a neutron powder diffraction (NPD) experiment was performed on the D20 instrument of the I.L.L. The 5g sample was put in a cylindrical vanadium can inside an orange cryostat. A wavelength of 1.3 Åfrom the Cu(200)monochromator reflection was used, which corresponds to the optimized flux configuration of the instrument. The sample was first heated to 320K, that is above the structural phase transition, and then cooled at an approximately constant rate of 1K/9 minutes down to 10K while diffractograms were continuously measured FIG. 4: Low angle part of the neutron diffractograms for Fe3BO5 at 125K, 75 and 14K. The two different sets of magnetic peaks corresponding to the two ordering transitions are clearly visible, as well as magnetic diffuse scattering at the higher temperatures. FIG. 5: Evolution with temperature of the diffracted intensity for Fe3BO5 at three positions corresponding to background (crosses), higher temperature magnetic phase (circles) lower temperature magnetic phase (squares). every 10 minutes. Each diffractogram covers a temperature range of ≈ 1.1K. The data were analyzed using the program Fullprof 18 . The structural phase transition is almost undetectable with these data. The only observable though very weak superstructure reflections are the (1 3 1/2) and (2 2 1/2) at about 23.4˚.

The thermal evolution of the low angle part of the neutron diffractograms is shown in Figure 4. The evolution of the diffracted intensities at three selected angles are shown in Figure 5. The chosen angles correspond to the positions of two magnetic peaks and of a background point in between them. At high temperatures, a broad hump centered at about 14˚is clearly visible in the background in Figure 4. Its intensity increases steadily with decreasing temperature, as shown in Figure 5. At 115 K, a first set of magnetic peaks appears, accompanied by a small decrease of background intensity. At 70K new magnetic peaks set in, and the background decreases to its high angle value. In the whole temperature range, the magnetic peaks can be indexed on the basis of the crystallographic unit cell. These observations indicate that the compound undergoes two successive magnetic ordering transitions on cooling. Since a strong diffuse scattering of magnetic origin persists between the two transitions, the higher temperature magnetic order must concern only part of the spin system, the rest remaining disordered down to the second transition. The two magnetic structures were solved and refined using data at 82K and 10K, and their evolutions were followed by sequential refinement using Fullprof. The treatments included a full structural refinement of the crystal structure, with a single isotropic a.d.p. for each atom type. The peak shapes were described with a pseudo-Voigt function with reflection widths following the Cagliotti law. The peak asymmetry evidenced in figure 4 is essentially due to axial divergence of the experimental setup. It was taken into account in the Rietveld refinement using the Berar empirical model [START_REF] Brar | [END_REF] . The results of the magnetic structure refinements at these two temperatures are given in Table II. The Rietveld plot of the 10K refinement is shown in Figure 6. The high temperature magnetic structure (HTMS) between 70K and 115K was refined to Rmag=6.6%. It involves only the Fe2, Fe4a and Fe4b cations from ladder I. As shown in Figure 7, it consists in ferromagnetically coupled antiferromagnetic chains running along the c axis within each ladder. All moments are aligned along the b down to the lowest temperatures. This arrangement is close to the one reported by Attfield et al., but in their case all moments were considered as equal. If we try to refine our data with the same constraint, the refinement becomes clearly worse, with Rmag≈12% compared to 5.3%. We can also obtain slightly better refinements by allowing some of the moments to rotate in the (a,b) plane. Only Fe4b and Fe3 rotations out of their direction along the b, resp. a axis yield an improvement of Rmag by ≈1%. However, these positions were quite unstable as function of temperature. We consider that they are not well demonstrated by the refinements and that the most probable structure is the one described in Figure 7. In order to check the consistency of the magnetic structures determined experimentally with the space group symmetry, we performed a group theory analysis based on the method of Bertaut 19 , using the Basireps software [START_REF] Rodriguez-Carvajal | program available with the fullprof distribution[END_REF] .

Magnetic

For the P bnm space group, there are 8 one-dimensional real irreducible representations for the little group Gk associated to the observed magnetic propagation vector (0,0,0). For sites Fe2, Fe4a and Fe4b, the magnetic arrangement is given by the basis vectors of the representation 4 : (u,v,0), (-u,-v,0), (-u,v,0), (u,-v,0), where only the component along the b axis is found non-zero experimentally. For sites Fe1 and Fe3, the magnetic arrangement is given by the basis vectors of the representation 5 : (u,v,w), (u,-v,w), (u,-v,-w), (u,v,-w),[(u,v,w), (u,-v,w), (u,-v,-w), (u,v,-w)]. This corresponds to a ferromagnetic component along the a axis. It could also imply an antiferromagnetic component in the (b,c) plane which is not observed experimentally. In the case of a magnetic structure with several magnetic sites (here 5 sites corresponding to the 5 non-equivalent Fe cations), the onset of the magnetic transitions depends on the relative strengths of the intra-site and inter-site interactions. In the present case, the Fe2, Fe4a, and Fe4b cations order first with basis vectors belonging to the same representation. This means that the inter-site interactions are dominant between these sites forming ladder I. The situation is similar for the Fe1 and Fe3 cations on ladder II which order at lower temperature with the basis vector of the same representation (but different from the one for ladder I). These different transition temperatures and representations for to the two ladders indicate that the magnetic interactions between them are weaker than those prevailing within each of them.

III. DISCUSSION

The single crystal x-ray diffraction analysis down to 110K indicates in ladder I an electron localization and Fe2-Fe4a pair formation with a very short Fe-Fe bond length. In view of the calculated bond valences, it seems however that the electron remains delocalized between the Fe2 and Fe4a cations. The 15K data reported by Mir et al. 5 are consistent with this picture. However the presence of a single data point at 15K prevents from detecting a possible further electronic localization correlated to the magnetic ordering transitions. Our neutron powder diffraction data, aimed at solving the magnetic structures, do not have sufficient resolution to tackle this question and are hindered by the presence of magnetic peaks. The present NPD experiment shows that a considerable dynamic magnetic scattering is already present a high temperature down to the 70K transition. This indicates the existence of strong magnetic correlations at temperatures 4 times higher than the ordering temperature. The magnetic ordering transition at 112K concerns only the Fe cations in ladders I, while ladders II remain disordered and still display very strong magnetic correlations. This is a direct proof of the very weak magnetic coupling between the two types of ladders in this system. In ladders I, the coupling along a rung is ferromagnetic, with negligible difference of moments between the 3 Fe cations. The rungs are antiferromagnetically coupled along the ladder (and c axis) direction. This is in contrast to the theoretical predictions of Whangbo et al. 7 based on the Strongly Interacting Spin Units approach, who proposed antiferromagnetic coupling in the ladders rungs. Recently, Vallejo et al. 8 have proposed an interpretation taking into account the prominent role of itinerant electrons in this system, which was able to predict the right magnetic structure. This is also supported by the relatively moderate values of the resistivity (≈10 Ω.cm at room temperature 4 ) which is only slightly modified by the charge ordering transition. The value of 2.3µB for the Fe cation moments is much smaller than what is expected for Fe cations (5 and 4 µB for F e 3+ and F e 2+ , respectively). This may be another indication that magnetic and electronic ordering is not complete even in ladders I in the 70K-112K temperature range. Below 70K, a new magnetic order sets in, where the Fe cations of lad-ders II form ferromagnetic chains along the c axis which are coupled antiferromagnetically. The magnetic order on ladders I is not modified, except for an increase of the Fe2 moments. Examining Figure 7, one can notice that the Fe2 antiferromagnetic chains of ladders I are weakly connected to ladders II through two Fe3 ferromagnetic chains. Therefore, this Fe2 moment increase is probably not a consequence of the magnetic ordering of ladders II. It might be related to a further ordering of charges between the Fe2 and Fe4a cations, but as stated above, we do not have any direct proof of this effect yet. The difference of magnetic coupling along the rung of ladders I and ladders II probably originates in the strong structural difference between the two types of ladders. The Fe2, Fe4a and Fe4b cations of ladders I are connected along a rung via edge-sharing at distances between 2.6 and 2.95 Å. The Fe1 and Fe3 cations of ladders II are connected via corner sharing at distances above 3.35 Å. At 10K, the Fe1 and Fe3 moment values (3.3 and 3.9µB) are reasonable for a F e 2+ cation, but those for Fe2, Fe4a and Fe4b cations of ladders I are still small compared to expected values. Since there are twice as many Fe3 than Fe1 cations per cell, the magnetic arrangement leads to a net moment along the a axis of 18.9µB per cell (i.e. 0.79µB per Fe cation). Indeed, the presence of ferromagnetism was demonstrated in this compound below 70K 2 by magnetization measurements. However, the same authors observed its disappearance below 40K. We do not detect any anomalous change in the NPD diffractograms around this temperature. If the magnetic structure is preserved, the disappearance of the net magnetic moment along the a axis would be possible only if the magnitude of the Fe1 moments becomes twice that of the Fe3 moments, which would certainly be detectable with our data. The origin of this magnetization change at 40K as to be searched elsewhere.

IV. SUMMARY

In this study, we have used single crystal x-ray diffraction and neutron powder diffraction as function of temperature to investigate the charge and magnetic ordering in Fe 3 BO 5 ludwigite. A charge ordering transition occurs close to room temperature. X-ray diffraction indicates that it takes place over a wide temperature range between 200K and 300K. It consists in an order-disorder transition within one of the two three-leg ladders contained in the structure and is related to the localization of an electron to form a pair of iron cations within the ladder's rung. Above the transition this electron is dynamically delocalized over the three iron cations forming the rung. To minimize the effects of the structural distortion created by this electron localization, the iron cation pairs order in a zig-zag way along the ladder's direction. At lower temperatures, ludwigite has been reported to undergo two successive antiferromagnetic and ferromagnetic ordering transitions at 112K and 70K, respectively. Refinements of neutron powder diffraction data indicate that the first transition is due to magnetic ordering of the ladders which charge-order around room temperature, forming ferromagnetically coupled antiferromagnetic chains running along the ladder axis. At 70K, magnetic order of the second type of ladders sets in. It consists in antiferromagnetically coupled ferromagnetic chains running along the ladder axis. This ferrimagnetic arrangement remains unchanged down to 10K and leads to a net magnetic moment of ≈19µB per cell at low temperature. We observe no modification of the NPD diffractograms around 40K, where the ferromagnetic component has been reported to disappear.

FIG. 1 :FIG. 2 :

 12 FIG.1: Structure of the high temperature phase of Fe3BO5 above 290K projected along the c axis. Light gray octahedra contain Fe cations with a +2 valence, dark gray ones Fe cations with an average valence of +2.66.

FIG. 3 :

 3 FIG. 3: Charge ordering and Fe cation pair formation in Fe3BO5 at the structural phase transition. Top : Fe-Fe distances along rungs of the Fe4-Fe2-Fe4 ladders. Middle : Fe cation valences calculated with the b.v.s. method. Bottom : Atomic displacement parameters for the Fe2 cation. Error bars are smaller than the marker sizes

FIG. 6 :FIG. 7 :FIG. 8 :

 678 FIG. 6: Rietveld plot of the NPD data refinement at 10K for Fe3BO5. The tick marks correspond to the nuclear (top) and magnetic (bottom) reflection positions.

TABLE I

 I

	: Positional parameters for ludwigite at 320K and
	110K from single crystal x-ray diffraction

Acknowledgments

The authors would like to thank V. Simonet, J. Dumas, M. Avignon and M. Continentino for fruitful discussions, J. Fernandes for providing us with the samples and T. Hansen for his help during the D20 experiment.