Anca Maria Nagy

Gilles Mourot

Benoı ˆt Marx

Jose ´ragot,

Georges Schutz

Systematic Multimodeling Methodology Applied to an Activated Sludge Reactor Model

In this article, an analytical method is used to obtain multiple model (MM) forms strictly equivalent to a given nonlinear model. The proposed method induces no information loss, contrary to existing order reduction methods. Several MMs formally equivalent to the initial model may be obtained, due to the intermediate quasi-LPV forms. Thus some criteria are given, using the linear matrix inequality (LMI) formulation, in order to chose the most suitable MM for analysis or control purpose. This method is applied to an activated sludge reactor model.

Introduction

The complexity problem of nonlinear dynamic systems appears in a great number of scientific and engineering fields. Numerous decomposition and simplification techniques were developed in the past years, in order to realize complexity reduction, according to objectives like identification, control, and stability analysis.

In most studies, the complexity reduction of nonlinear systems, represented in the state equation form, consists in the reduction of the system order that generally comes with an information loss. This order reduction is realized in different ways: parameter reduction of mathematical model from a sensitivity analysis and a comparison with the initial system, 1,2 neglecting certain phenomena (reactions and/or species in the case of complex chemical mechanisms) whose explanatory power is weak in the initial system; 3 elimination of model parts (parameters and/or variables) which have no significant importance on system dynamics; [START_REF] Moore | Principal component analysis in linear systems: Controllability, observability and model reduction[END_REF][START_REF] Sayesel | Model simplification and validation with indirect structure validity tests[END_REF] simplification by truncation and singular perturbation approximation. [START_REF] Andersson | Model comparison and simplification[END_REF][START_REF] Steffens | A systematic approach for reducing complex biological wastewater treatement models[END_REF] Another way to deal with complex systems is to use analytic transformations to obtain particular representations that are easier to study than a generic nonlinear form, in the sense that performance analysis, controller design, or supervision based on these new system representations is facilitated. In this paper, nonlinear systems are proved to be strictly equivalent to multiple models.

The multiple model [START_REF] Murray-Smith | Multiple Model Approaches to Modeling and Control[END_REF] actually constitutes a commonly used tool for dynamic nonlinear system modeling. The multiple model formalism is based on a nonlinear interpolation between certain linear submodels. The global model is a convex combination of the r submodels: [START_REF] Mourot | Modelling of ozone concentrations using a Takagi-Sugeno model[END_REF] where x ∈ R n represents the state vector, u ∈ R m the input vector, and y ∈ R l the output vector. A i , B i , C i , D i , are constant matrices of appropriate dimensions, and r denotes the number of submodels. The function µ i (x,u) quantifies the relative weight of the ith submodel {A i ,B i ,C i ,D i } in the global model. The weighting functions µ i (x,u) are generated by the so-called premise Variables, expressing different nonlinearities of the system, and they have the following property:

In the literature several equivalent terminologies are used to define this class of models: the multiple model (MM), [START_REF] Murray-Smith | Multiple Model Approaches to Modeling and Control[END_REF] the fuzzy Takagi-Sugeno model, [START_REF] Takagi | Fuzzy identification of systems and its application to modelling and control[END_REF] the polytopic linear model (PLM). [START_REF] Angelis | System Analysis, Modeling and Control with Polytopic Linear Models[END_REF] Different techniques exists to obtain a MM. The first technique is based on the linearization of the nonlinear model around one (or several) operating point(s). Dynamic linearization near an arbitrary trajectory is proposed as well. [START_REF] Johansen | On the interpretation and identification of dynamic Takagi-Sugeno fuzzy models[END_REF] The second method consists in a system identification using experimental data. [START_REF] Mourot | Modelling of ozone concentrations using a Takagi-Sugeno model[END_REF][START_REF] Abonyi | Modified Gath-Geva fuzzy clustering for identification of Takagi-Sugeno fuzzy models[END_REF] The loss of information constitutes the first drawback of these techniques. Second, the choice of the premise variables is not systematically realized. Third, the choice of different operating points or trajectories still remains very delicate.

In the present paper, the multiple model structure is used to reduce the complexity of an ASM1 model (activate sludge model 1) [START_REF] Olsson | Wastewater Treatment Systems. Modelling, Diagnosis and Control[END_REF] describing a biological degradation process of an activated sludge reactor. Most of the previous works devoted to the activated sludge systems were based on linearization techniques, [START_REF] Steffens | A systematic approach for reducing complex biological wastewater treatement models[END_REF][START_REF] Smets | A linear ASM1 based multimodel for activated sludge systems[END_REF] despite the drawbacks mentioned above.

Afterward, we will present a systematic procedure to transform a nonlinear system by rewriting it into a multiple model form, avoiding the major inconveniences: the transformation is realized without loss of information, the obtained system has exactly the same state trajectory as the initial system. Moreover, the choice of different operating points is no longer necessary. Starting with a general form of nonlinear system, a quasi-linear parameter varying (quasi-LPV) state representation is realized. In general, several equivalent quasi-LPV forms can be constructed from a given nonlinear dynamic model. Each quasi-LPV form is associated to a particular premise variable set: choosing a quasi-LPV form is equivalent to choosing the premise variable set. This quasi-LPV representation constitutes a polytopic form because the matrices with variable parameters * To whom correspondence should be addressed. E-mail: anca-maria.nagy@ensem.inpl-nancy.fr; gilles.mourot@ensem.inplnancy.fr; benoit.marx@ensem.inpl-nancy.fr; jose.ragot@ensem. inpl-nancy.fr; georges.schutz@tudor.lu.

† Centre de Recherche en Automatique de Nancy. ‡ CRP "Henri Tudor".

{ x ˙(t) ) ∑ i)1 r µ i (x, u)[A i x(t) + B i u(t)] y(t) ) ∑ i)1 r µ i (x, u)[C i x(t) + D i u(t)]
(1)

∑ i)1 r µ i (x, u) ) 1 µ i (x, u) g 0, ∀(x, u) ∈ R n × R n (2)
are convex combinations of constant matrices calculated from the polytope vertices. These vertices are obtained by using the convex polytopic transformation (CPT). [START_REF] Wang | An approach to fuzzy control of nonlinear systems: Stability and design issues[END_REF] Each vertex defines a linear submodel and the nonlinear parts are rejected into the weighting functions.

Most of the work done on the quasi-LPV systems in the framework of stability analysis or controller/observer design is only based on the matrices defining the vertices. As a consequence, even if the different quasi-LPV forms of a nonlinear system are formally equivalent, the obtained multiple model strongly depends on the chosen realization. The choice of the premise variable set is important, because it has an influence on the submodel number and on the global model structure. This degree of freedom will be used to ease the controllability, observability, stability analysis studies. So, it is possible to choose from different MM structures the one that ensures, for example, the convergence of the certain multiobserver forms which are based on the MM structure, with respect to given conditions. The convergence conditions of the state estimation error can be expressed in linear matrix inequality (LMI) formulation using the Lyapunov method [START_REF] Ichalal | State estimation of nonlinear systems using multiple model approach[END_REF] and they are at the root of the MM choice rules making-out.

The proposed method generalizes the sector nonlinearity approach frequently used when dealing with Takagi-Sugeno models. [START_REF] Tanaka | Fuzzy Control System Design and analysis[END_REF][START_REF] Ohtake | Fuzzy modeling via sector nonlinearity concept[END_REF] The contribution of this paper is to give a systematic procedure in order to choose the best quasi-LPV form of a given nonlinear system with regard to the study objective (such as stability or performance analysis, controller or observer design).

In the first section, the general methodology to transform a nonlinear system into a multiple model form will be presented. The notion of convex polytopic transformation will be introduced and, using an example, some characteristics of this method will be underscored. The general case of this method will be presented in subsection Analytical Method of Structure Simplification. In subsection "Choice Criteria for Quasi-LPV Form", we will detail the choice between different quasi-LPV forms of a nonlinear system.

Before closing the paper with some conclusions, the proposed method is applied to an activated sludge model.

Analytical Method of Structure Simplification of Nonlinear Systems

This section is dedicated to the general methodology of transforming a given nonlinear model into a multiple model, eq 1. The proposed method is analytical, and the obtained multiple model is equivalent to the initial nonlinear system. Considering a nonlinear system with bounded nonlinearities, a quasi-LPV form can be written from it. This form is a state and control affine representation. In general, for a nonlinear dynamic system, this representation is not unique, that is the reason why different criteria of choice for a quasi-LPV form have to be found according to the study objectives. To each quasi-LPV form, a premise variable set corresponds. These premise variables will be partitioned into two parts, using the convex polytopic transformation. The two partitions will contribute to the construction of multiple model submodels and to the corresponding weighting functions. The multiple model will be a convex combination of linear submodels, the nonlinearity being transferred into weighting functions related to each submodel.

Let us first introduce the notion of convex polytopic transformation that will serve afterward in the simplification methodology.

Convex Polytopic Transformation. Lemma 1. Let h(x(t),u(t)) be a bounded and continuous function from [x 0 ,x 1 ] × [u 0 ,u 1 ] to R, with x 0 ,x 1 ∈ R n , u 0 ,u 1 ∈ R m . Then there exist two functions [START_REF] Tanaka | Fuzzy Control System Design and analysis[END_REF] with F 1 (x(t),u(t)) + F 2 (x(t),u(t)) ) 1 such that for all h 1 g max x,u {h(x,u)} and h 2 g min x,u {h(x,u)}. The two functions F 1 and F 2 are defined by Let us note that this decomposition is not unique.

We can notice that the bigger h 1 is and the smaller h 2 is, the more the taken values of F 1 and F 2 are restricted in [0, 1] interval.

In particular, we can choose After the introduction of polytopic transformation, an example will be used in the following section to illustrate the proposed methodology to obtain a multiple model. Introductory Example. Let us consider the following nonlinear system: First, the system (3) can be represented in a quasi-LPV form: By realizing this transformation several quasi-LPV forms can be obtained, because this form is a state-and control-affine representation. For the first state eq 3a, this separation is clear because of the product between the function cos(x 1 ) and the second state variable x 2 . For the second term, x 1 3 u, we can either affect the nonlinearity x 1 3 in the control matrix B eq 4, or distribute this nonlinearity among the state vector (x 1 component) and the state matrix A eq 5:

For the second state eq 3b, at least two possible decompositions are observed. The most obvious decomposition is obtained by factorizing the two terms by x 1 and x 2 , respectively, eq 4.

h(x(t), u(t)) ) F 1 (x(t), u(t)) • h 1 + F 2 (x(t), u(t)) • h 2 F 1 (x(t), u(t)) ) h(x(t), u(t)) -h 2 h 1 -h 2 F 2 (x(t), u(t)) ) h 1 -h(x(t), u(t)) h 1 -h 2 h 1 ) max x,u {h(x, u)} h 2 ) min x,u {h(x, u)} x ˙1 ) cos(x 1 )x 2 + x 1 3 u (3a) x ˙2 ) 1 √ x 2 x 1 + x 1 2 x 2 (3b) x ˙) A(x, u)x + B(x, u)u A(x) ) [ 0 cos(x 1 ) 1 √ x 2 x 1 2 ] B(x) ) [ x 1 3 0 ] (4) A(x, u) ) [ x 1 2 u cos(x 1 ) 1 √ x 2 x 1 2 ] B(x) ) [ 0 0 ] (5)
Another possibility is to factorize the right-hand terms of eq 3b only by x 1 , reducing in this way the number of premise variables to three, eq 6.

In the following we only focus on the derivation of the MM form. The choice criteria between the possible forms of eq 4, eq 5 and eq 6 will be discussed later.

Retaining for example the representation eq 6, the premise variable set V z ) {z 1 ,z 2 ,z 3 } is defined as follows:

The choice of these variables z 1 , z 2 , z 3 is directly linked to the chosen quasi-LPV form.

Second, the convex polytopic transformation is applied, for each premise variable z j (x)(j ) 1, ..., 3) for x 1 ∈ [-2π,2π] and x 2 ∈ [0.1, 12]. Then, using lemma 1, each premise variable will be partitioned into two parts:

where We choose here:

The functions F j,1 and F j,2 represent, respectively, the first and the second partition of each premise variable. Let us note that A(x) contains as premise variables z 1 and z 3 , while B(x) contains as premise variable z 2 ; in this way, we will evaluate the matrices A and B at the vertices of the polytope defined by the partitions of the premise variables which occur in these matrices, as follows:

First, using the convex polytopic transformation (lemma 1) applied for z 1 ,e q8 a : Then, using the convex polytopic transformation (lemma 1) applied for z 3 8c:

In order to also include the partitions of the second premise variable z 2 , which is not involved in the matrix A(z 1 ,z 3 ), but in B(z 2 ), a multiplication with F 2,1 (x) + F 2,2 (x) ) 1 is realized as follows:

The same transformations are performed on the matrix B, where the premise variable z 2 is involved:

Finally, looking at eq 10 and eq 11 we have where µ i (x) are combination of F j,k (x)( j ) 1,2,3 and k ) 1,2) and where

A(x) ) [ 0 cos(x 1 ) 1 √ x 2 + x 1 x 2 0 ] B(x) ) [ x 1 3 0 ] (6) z 1 (x) ) cos(x 1 ) z 2 (x) ) x 1 3 z 3 (x) ) 1 √ x 2 + x 1 x 2 (7) z 1 (x) ) F 1,1 (z 1 )z 1,1 + F 1,2 (z 1 )z 1,2 (8a)
z 2 (x) ) F 2,1 (z 2 )z 2,1 + F 2,2 (z 2 )z 2,2 (8b)
z 3 (x) ) F 3,1 (z 3 )z 3,1 + F 3,2 (z 3 )z 3,2 (8c)
F 1,1 (z 1 (x)) ) cos(x 1 ) -z 1,2 z 1,1 -z 1,2 F 1,2 (z 1 (x)) ) z 1,1 -cos(x 1 ) z 1,1 -z 1,2 F 2,1 (z 2 (x)) ) x 1 3 -z 2,2 z 2,1 -z 2,2 F 2,2 (z 2 (x)) ) z 2,1 -x 1 3 z 2,1 -z 2,2 F 3,1 (z 3 (x)) ) 1 √ x 2 + x 1 x 2 -z 3,2 z 3,1 -z 3,2 F 3,2 (z 3 (x)) ) z 3,1 - 1 √ x 2 -x 1 x 2 z 3,1 -z 3,2 z j,1 ) max x {z j (x)} z j,2 ) min x {z j (x)} ∀j ) 1, ..., 3 (9)
A(z 1 , z 3 ) ) [ 0 z 1 (x) z 3 (x) 0 ] ) F 1,1 (x) [ 0 z 1,1 z 3 (x) 0 ] + F 1,2 (x) [ 0 z 1,2 z 3 (x) 0 ] A(z 1 , z 3 ) ) F 1,1 (x)F 3,1 (x) [ 0 z 1,1 z 3,1 0 ] + F 1,2 (x)F 3,1 (x) [ 0 z 1,2 z 3,1 0 ] + F 1,1 (x)F 3,2 (x) [ 0 z 1,1 z 3,2 0 ] + F 1,2 (x)F 3,2 (x) [ 0 z 1,1 z 3,2 0 ] A(z 1 , z 3 ) )F 1,1 (x)F 2,1 (x)F 3,1 (x) [ 0 z 1,1 z 3,1 0 ] + F 1,2 (x)F 2,1 (x)F 3,1 (x) [ 0 z 1,2 z 3,1 0 ] . + F 1,1 (x)F 2,2 (x)F 3,1 (x) [ 0 z 1,1 z 3,1 0 ] + F 1,2 (x)F 2,2 (x)F 3,1 (x) [ 0 z 1,2 z 3,1 0 ] . + F 1,1 (x)F 2,1 (x)F 3,2 (x) [ 0 z 1,1 z 3,2 0 ] + F 1,2 (x)F 2,1 (x)F 3,2 (x) [ 0 z 1,2 z 3,2 0 ] . + F 1,1 (x)F 2,2 (x)F 3,2 (x) [ 0 z 1,1 z 3,2 0 ] + F 1,2 (x)F 2,2 (x)F 3,2 (x) [ 0 z 1,2 z 3,2 0 ] (10)
B(z 2 ) ) [ z 2 (x) 0 ] ) [ z 2,1 F 2,1 (x) + z 2,2 F 2,2 (x) 0 ] ) F 2,1 (x) [ z 2,1 0 ] + F 2,2 (x) [ z 2,2 0 ]
.

) [F 1,1 (x) + F 1,2 (x)][F 3,1 (x) + F 3,2 (x)] { F 2,1 (x) [ z 2,1 0 ] + F 2,2 (x) [ z 2,2 0 ]} B(z 2 ) ) F 1,1 (x)F 2,1 (x)F 3,1 (x) [ z 2,1 0 ] + F 1,2 (x)F 2,1 (x)F 3,1 (x) [ z 2,1 0 ] . + F 1,1 (x)F 2,2 (x)F 3,1 (x) [ z 2,2 0 ] + F 1,2 (x)F 2,2 (x)F 3,1 (x) [ z 2,2 0 ] . + F 1,1 (x)F 2,1 (x)F 3,2 (x) [ z 2,1 0 ] + F 1,2 (x)F 2,1 (x)F 3,2 (x) [ z 2,1 0 ] . + F 1,1 (x)F 2,2 (x)F 3,2 (x) [ z 2,2 0 ] + F 1,2 (x)F 2,2 (x)F 3,2 (x) [ z 2,2 0 ] (11)
A(z 1 , z 3 ) ) ∑ i)1 8 µ i (x)A i , B(z 2 ) ) ∑ i)1 8 µ i (x)B i (12) A 1 ) A 3 ) [ 0 z 1,1 z 3,1 0 ] A 2 ) A 4 ) [ 0 z 1,1 z 3,2 0 ] . A 5 ) A 7 ) [ 0 z 1,2 z 3,1 0 ] A 6 ) A 8 ) [ 0 z 1,2 z 3,2 0 ] B 1 ) B 2 ) B 5 ) B 6 ) [ z 2,1 0 ] B 3 ) B 4 ) B 7 ) B 8 ) [ z 2,2 0 ] (13)
After this example it is interesting to present a systematic way of constructing the matrices A i and B i . For example, to determine A 3 and B 3 , the triplet σ 3 ) (1,2,1) is used. This triplet codes the variable partitions occurring in the 3rd submodel and σ 3 k denotes the kth value in the triplet σ 3 . According to the expression of A 3 and B 3 , these matrices may be noted: A 3 ) A(z 1,σ 3 1 ,z 3,σ 3 3 ) and B 3 ) B(z 2,σ 3 2 ), where z 1,σ 3 1 , z 2,σ 3 2 and z 3,σ 3 3 are the scalars defined in eq 9. In a more general way, A i and B i (i ) 1, ..., 8) are noted:

Those notations are in agreement with eq 13. Associated to A 3 and B 3 , the definition of eq 10 is obtained by using the triplet σ 3 . Indeed: Table 1 details the construction of all submodels (A i ,B i ) and the weighting functions µ i (x) of a MM with three premise variables.

Each function defining a premise variable being partitioned into two functions, there are 2 3 submodels and 2 [START_REF] Petzold | Model reduction for chemical kinetics: An optimization approach[END_REF] weighting functions. To each submodel i corresponds a triplet σ i which codes the variable partitions occurring in it. After multiplying the functions representing these partitions, the weighting function µ i (x) corresponding to the ith submodel is obtained.

To express the constant matrices A i and B i , characterizing each submodel i (i ) 1, ..., 8), we use the quasi-LPV form 6 of the system (eq 3), where A(x,u) and B(x,u) were defined in 6.

In the next section, the proposed methodology applied to the example of this section section will be generalized to any nonlinear systems.

The General Case. A large category of dynamic nonlinear systems can be represented by

where f(x( • ),u( • )) ∈ R n and g(x( • ),u( • )) ∈ R l .
Under the hypothesis that f(x(t),u(t)) and g(x(t),u(t)) are continuous and bounded in U ⊆ R n with f(0, • ) ) 0 and g(0, • ) ) 0, the system eq 15 can be represented in a quasi-LPV form:

where

A(F(x,u)) ∈ R n×n , B(F(x,u)) ∈ R n×m , C(F(x,u)) ∈ R l×n , D(F(x,u)) ∈ R l×m and where F(x,u) ∈ C 1 (R n+m ,R n 1 +m 1 ) with n 1 + m 1 e n + m
can simply represent a part of the state and input variables. Let us remember that in the classical LPV form, the parameter vector F explicitly depends on the time variable only. Then, the quasi-LPV representation of a nonlinear system is different from the classical LPV form in that the underlying parameters F are now functions of the state and input variables. [START_REF] Huang | Nonlinear Hinf Control: An enhanced Quasi-LPV Approach[END_REF] The trajectory of the parameters is actually determined by the behavior of the system variables. For example, in the introductory example, 6 is a quasi-LPV form in which the matrix A depends on x ) (x 1 ,x 2 ), so the vector F(x,u) ) x and the matrix B depends only on x 1 , so the vector F(x,u) ) x 1 .

It has to be remarked that the quasi-LPV 16 form represents a state and control affine representation of a general nonlinear system.

The matrix function A(F(x,u)) is in this case given by

The matrices B(F(x,u)), C(F(x,u)), and D(F(x,u)) are given in the same way as A(F(x,u)). Presenting the nonlinear system 15 in a quasi-LPV form 16 amounts to expressing each state component x i (t) and each output component y j (t) in the following way: for all i ) 1, ..., n and j ) 1, ..., l. This explicit quasi-LPV form 16 realizes a repartition of the system nonlinearities (a i,q , b i,s , c j,q , d j,s ) and the different state or input components.

Considering the quasi-LPV form 16, we define the premise Variable set V z in the following way:

where 1,n ) {1, ..., n}.

Subsequently, this set is noted more simply:

where p represents the dimension of the premise variable set V z and is also the number of nonlinearities identified using the quasi-LPV form of the system 15. The quantities z 1 (F(x,u)), ..., z p (F(x,u)) are the premise Variables. Remark 1. (n + l) × (n + m) represents the largest number of premise variables.

Remark 2. The number of submodels that constitute the multiple model is equal to r ) 2 p e 2 (n+l) (n+m) .

The matrices with variable parameters intervening in the quasi-LPV form of system 15 are linear combinations of matrices with constant coefficients (A j ,B j ,C j ,D j ); for example, the matrix A(x,u) may be expressed in the following way:

The set I A contains all the indices corresponding to the premise variables which are contained in the entries of the matrix A(x,u). The matrix A 0 contains the constant entries (a i,j )ofthe

submodel i σ i A i B i µ i (x) 1 (1,1,1) A(z 1,1, z 2,1, z 3,1 ) B(z 1,1, z 2,1, z 3,1 ) F 1,1 (x)F 2,1 (x)F 3,1 (x) 2 (1,1,2) A(z 1,1, z 2,1, z 3,2 ) B(z 1,1, z 2,1, z 3,2 ) F 1,1 (x)F 2,1 (x)F 3,2 (x) 3 (1,2,1) A(z 1,1, z 2,2, z 3,1 ) B(z 1,1, z 2,2, z 3,1 ) F 1,1 (x)F 2,2 (x)F 3,1 (x) 4 (1,2,2) A(z 1,1, z 2,2, z 3,2 ) B(z 1,1, z 2,2, z 3,2 ) F 1,1 (x)F 2,2 (x)F 3,2 (x) 5 (2,1,1) A(z 1,2, z 2,1, z 3,1 ) B(z 1,2, z 2,1, z 3,1 ) F 1,2 (x)F 2,1 (x)F 3,1 (x) 6 (2,1,2) A(z 1,2, z 2,1, z 3,2 ) B(z 1,2, z 2,1, z 3,2 ) F 1,2 (x)F 2,1 (x)F 3,2 (x) 7 (2,2,1) A(z 1,2, z 2,2, z 3,1 ) B(z 1,2, z 2,2, z 3,1 ) F 1,2 (x)F 2,2 (x)F 3,1 (x) 8 (2,2,2) A(z 1,2, z 2,2, z 3,2 ) B(z 1,2, z 2,2, z 3,2 ) F 1,2 (x)F 2,2 (x)F 3,2 (x) A i ) A(z 1,σ i 1 , z 3,σ i 3 ) B i ) B(z 2,σ i 2 ) µ 3 (x) ) F 1,σ 3 1 (x)F 2,σ 3 2 (x)F 3,σ 3 3 (x) (14)
{ x ˙(t) ) f(x(t), u(t)) y(t) ) g(x(t), u(t)) (15) [ x ˙(t) y(t) ] ) [ A(F(x, u)) B(F(x, u)) C(F(x, u)) D(F(x, u)) ][ x(t) u(t) ] (16) A(F(x, u)) ) [ a 1,1 (F(x, u)) ••• a 1,n (F(x, u)) l a n,1 (F(x, u)) ••• a n,n (F(x, u)) ] (17)
x ˙i(t)

) ∑ q)1 n a i,q (F(x, u)) x q (t) + ∑ s)1 m b i,s (F(x, u)) u s (t) y j (t) ) ∑ q)1 n c j,q (F(x, u)) x q (t) + ∑ s)1 m d j,s (F(x, u)) u s (t) (18)
V z ) {a i,q (F(x, u)

)|a i,q * const, i ) 1, n, q ) 1, n } ∪{b i,s (F(x, u))|b i,s * const, i ) 1, n, s ) 1, m } ∪{c j,q (F(x, u))|c j,q * const, j ) 1, l, q ) 1, n } ∪{d j,s (F(x, u))|d j,s * const, j ) 1, l, s ) 1, m } V z ) {z 1 (F(x, u)), ..., z p (F(x, u))|p e (n + l) × (n + m)} A(x, u) ) A 0 + ∑ j∈I A z j (x, u)A j ( 19
)
matrix A(x,u). Each matrix A j gives the linear dependency of A(x,u)t oz j and contains at the position corresponding to the nonlinearity a i,q (identified with the premise variable z j ) the constant term 1, and zero for all the other positions:

The matrices B(x,u), C(x,u), D(x,u) are expressed in the same way as A(x,u) using the sets I B , I C , and I D . In this way, for the previous example (eq 6) we have I A ) {1,3} and I B ) {2}.

By generalizing the presented method in the introductory example, we generate the 2 p submodels of a multiple model characterized by the p premise variables which are partitioned into two, using the convex polytopic transformation: with where the scalars z j,1 and z j,2 are defined in eq 22:

To each submodel i, represented in Table 2 by the ith line, corresponds a p -uplet σ i which codes the partitions of premise variables intervening in the corresponding weighting function. It is thus possible to construct, just like in the previous example, a table gathering the partition sets of premise variables. By multiplying the functions that represent these partitions we obtain the corresponding weighting function µ i (z):

where σ i k represents the index in the kth position in the p-uplet σ i . Then ∑ i)1 r µ i (z) ) 1 and µ i (z) g 0( i ) 1, ..., r), because

We have to note that the matrices A(x,u), B(x,u), C(x,u), and D(x,u) contain the premise variables z j for j ) 1, ..., p. In this way, the matrices A, B, C, and D will be evaluated in the vertices of the polytope defined by the premise variable partitions that occur in these matrices in order to determine the matrices A i , B i , C i , and D i (i ) 1, ..., r). These matrices are the matrices with constant coefficients relative to each submodel: B i , C i , and D i can be written in the same way. This fact is detailed in the demonstration of the following theorem:

Theorem 2. The multiple model eq 1, which represents a convex combination of linear subsystems, {(A i , B i , C i , D i )} (i ) 1, ..., r), is equivalent to the system 16; r ) 2 p is the number of submodels and p is the number of premise variables.

Proof. We first want to prove that the matrices A i (i ) 1, ..., r), representing the r submodels of the multiple model, are calculated by evaluating the matrix function A in the polytope vertex defined by the premise variable partitions.

As already mentioned in eq 19

After replacing z p (like in 20 for j ) p) in the previous equality eq 25 and multiplying the other terms of the sum by F p,1 + F p,2

) 1, we obtain

We then factor the terms that contain F p,1 and F p,2 and we obtain After replacing z p-1 (like in 20 for j ) p -1) and after multiplying the other terms of the sum by F p-1,1 + F p-1,2 ) 1, we obtain:

After factoring again the terms that contain F p-1,1 and F p-1,2 , inside the brackets we obtain

∏ j)1 p F j,σ 2 p j A j ) q [ 0 ••• 0 ••• 0 l 0 ••• 1 ••• 0 l 000 ] i z j (x, u) ) F j,1 (z j )z j,1 + F j,2 (z j )z j,2 (20)
F j,1 (z j ) ) z j (x, u) -z j,2 z j,1 -z j,2 F j,2 (z j ) ) z j,1 -z j (x, u) z j,1 -z j,2 (21)
z j,1 ) max x,u {Zj(x, u)} z j,2 ) min x,u {Zj(x, u)} ∀j ) 1, ..., p (22)
µ i (z) ) ∏ j)1 p F j,σ i j (z j (x, u)) (23) F j,1 (z j (x, u)) + F j,2 (z j (x, u)) ) 1, ∀j ) 1, ..., p A i ) A 0 + ∑ j∈I A z j,σ i j A j (24) A(z) ) A 0 + ∑ j)1 p z j A j (25) A(z) ) (F p,1 + F p,2 )(A 0 + ∑ j)1 p-1 z j A j ) + (z p,1 F p,1 + z p,2 F p,2 ) • A p A(z) ) F p,1 (A 0 + ∑ j)1 p-1 z j A j + z p,1 A p ) + F p,2 (A 0 + ∑ j)1 p-1 z j A j + z p,2 A p ) A(z) ) F p,1 • [(F p-1,1 + F p-1,2 )(A 0 + ∑ j)1 p-2 z j A j ) + (z p-1,1 F p-1,1 + z p-1,2 F p-1,2 ) • A p-1 + z p,1 (F p-1,1 + F p-1,2 )A p ] + F p,2 • [(F p-1,1 + F p-1,2 )(A 0 + ∑ j)1 p-2 z j A j ) + (z p-1,1 F p-1,1 + z p-1,2 F p-1,2 ) • A p-1 + z p,2 (F p-1,1 + F p-1,2 )A p ]
By reiterating this factorization mechanism and replacing at each step z j with the expression 20, until we have replaced all the premise variables, we obtain-

We succeeded in expressing the matrices A i (i ) 1, ..., r)b y using the polytope vertices defined by the premise variable partitions:

After applying the same reasoning for the input and output parts, we proved in this way the equivalence between the multiple model 1 and the system 15.

In the next subsection will be given some general tools concerning the observer/controller design based on MM structure, in order to support the general analytical method and motivate the choice of the final MM structure.

Choice Criteria for Quasi-LPV Form

The existing results concerning the performance analysis and the synthesis of controller/observer for a multiple model [START_REF] Akhenak | Estimation of state and unknown inputs of a nonlinear system represented by a multiple model[END_REF][START_REF] Chadli | On the stability analysis of multiple model systems[END_REF][START_REF] Wang | An approach to fuzzy control of nonlinear systems: Stability and design issues[END_REF][START_REF] Tanaka | Fuzzy Control System Design and analysis[END_REF][START_REF] Ichalal | State estimation of nonlinear systems using multiple model approach[END_REF][START_REF] Guerra | Control law proposition for the stabilization of discrete Takagi-Sugeno models[END_REF][START_REF] Tanaka | A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions[END_REF][START_REF] Bergsten | Luenberger observers for TS fuzzy systems[END_REF][START_REF] Bergsten | Fuzzy Observers[END_REF] are based on the resolution of linear matrix inequalities (LMI), by using the Lyapunov method. Because of the convex sum property of the weighting functions, the LMI are only evaluated at the polytope vertices (A i ,B i ) and the weighting functions do not occur in the resolution of the LMIs. Only the matrices A i and B i have a contribution in the resolution of the LMIs. It is important to mention that the LMI formulation of controller or observer design for MM consists only in sufficient but not necessary conditions. That is the reason why it is essential to propose choice criteria for the MM structure.

In general, the quasi-LPV form 16 for a nonlinear system 15 is not unique; to each quasi-LPV form corresponds a particular premise variables set; choosing a quasi-LPV form is equivalent to choosing the premise variable set. The choice of premise variable set V z is important, because it has an influence on the submodel number and on the global model structure. This is a degree of freedom that can be used to ease the controllability, observability, and stability analysis studies.

First of all, to ensure the controllability/observability of the global system, represented in a multiple model form, that is, to ensure a solution for the LMIs associated to the MM in order to compute the controller/the observer, the controllability/ observability of each submodel is necessary. [START_REF] Guerra | Control law proposition for the stabilization of discrete Takagi-Sugeno models[END_REF] Thus, the quasi-LPV forms producing submodels that are not controllable/ observable must be eliminated. Note that this is not a sufficient condition to ensure the controllability/observability of the multiple model. [START_REF] Murphey | Nonsmooth controllability theory and an example[END_REF] For instance, the different quasi-LPV forms of the system eq 3 in the introductory example do not share the same structure. The form 5 implies a null matrix B(z), thus the multiple model cannot be controllable, whereas in eqs 4 and 6 the matrix B(z) is non-null and therefore the necessary controllability condition of these multiple models is verified.

As said before, in the stability analysis studies of multiple models, the observer stability conditions are given in terms of linear matrix inequalities (LMI). Different techniques are proposed to reduce the number of LMI conditions, which leads to less computational requirements [START_REF] Tanaka | A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions[END_REF] and eases the solution existence. [START_REF] Bergsten | Fuzzy Observers[END_REF] As a consequence, the number of LMI to be satisfied is linear or polynomial in the number of submodels. [START_REF] Tanaka | Fuzzy Control System Design and analysis[END_REF] That is the reason why a multiple model composed by a low number of submodels is searched. This number is related to the number of premise variables, so that a quasi-LPV form which has a small set of premise variables will be preferred. For example, the quasi-LPV forms 4 and 5 implies four premise variables and will thus lead to 16 submodels, whereas the form 6 only requires three premise variables and then eight submodels. In addition to that, the observer/controller design for MM with premise variables depending on the state variables is a lot more complex than if the premise variables are known. [START_REF] Ichalal | Simultaneous state and unknown inputs estimation with PI and PMI observers for Takagi-Sugeno model with unmeasurable premise variables[END_REF] As a consequence, MM form with premise variables depending on a minimal number of state variables is preferable.

To increase the probability that a solution exists, the dimension of the state vector should be made as small as possible. [START_REF] Bergsten | Luenberger observers for TS fuzzy systems[END_REF] Taking into account the previous remarks, the following rules can guide the choice of the quasi-LPV form leading to the most appropriate model:

(i) Eliminate all quasi-LPV forms in which the matrix B has null columns and the matrix C has null rows, since these matrices do not respect the rank observability/controllability conditions. Indeed, retain all quasi-LPV forms in which matrices satisfy the conditions (ii) Among all quasi-LPV forms for which the premise variables z i (F(x,u)) (i ) 1, ..., p) depend on the same number of state variables, choose the quasi-LPV form that contains the lowest number of premise variables z i (F(x,u)) (taking into account the next rule).

(iii) Identify the decompositions that relate on the same premise variables for different state and/or output eqs 18, reducing in this way the number of premise variables.

(iv) Among all quasi-LPV forms that have the same number of premise variables z i (F(x,u)) (i ) 1, ..., p), choose the quasi-LPV form for which the premise variables z i (F(x,u)) depend on the lowest number of state variables.

Application to an Activated Sludge Reactor Model

In this section, the analytical decomposition method developed in the previous section will be applied to an activated sludge reactor model, a reduced form of the ASM1 model, considering only the carbon pollution. 17

A(z) ) F p,1 F p-1,1 [A 0 + ∑ j)1 p-2 z j A j + z p-1,1 A p-1 + z p,1 A p ] + F p,1 F p-1,2 [A 0 + ∑ j)1 p-2 z j A j + z p-1,2 A p-1 + z p,1 A p ] + F p,2 F p-1,1 [A 0 + ∑ j)1 p-2 z j A j + z p-1,1 A p-1 + z p,2 A p ] + F p,2 F p-1,2 [A 0 + ∑ j)1 p-2 z j A j + z p-1,2 A p-1 + z p,2 A p ] A(z) ) ∑ i)1 2 p { ∏ j)1 p F j,σ i j • [A 0 + ∑ j)1 p z j,σ i j • A j ]} A i ) A 0 + ∑ j)1 p z j,σ i j • A j (26) rank(O i ) ) rank [ C i C i A i l C i A i n-1 ] ) n rank(C i ) ) rank[ B i A i B i ... A i n-1 B i ] ) n, ∀i ) 1, ..., r (27)
The wastewater treatment using activated sludge consists in mixing used waters with a rich mixture of bacteria in order to degrade the organic matter. In studies on the activated sludge reactor, several operating modes are considered: regulation from a reference volume, reactor with constant volume, free flow at the output reactor, and controllable input and output flows. Only the first situation using a reference volume is reported here.

Biomass, Carbon and Oxygen Modeling in an Aerobic Environment. Considering only the carbon pollution of an activated sludge reactor, the elimination process of this pollutant is realized in the reactor-clarifier group. The simplified diagram of the studied process is given in Figure 1. The clarifier is supposed to be perfect, that is, with no internal dynamic process and no biomass in the effluent. In this case we can write at each time instant: where q in , q R , q W represent respectively the input, recycled, and wasted flow and where X BH , X BH,R , S S and S S,R are different concentrations whose meaning is pointed in Table 3.

The whole process can be represented by the following nonlinear model: where the variables involved are presented in Table 3 and r H , r S , and r O represent the kinetics of heterotrophic biomass X BH , carbon S S , and oxygen S O reactions, in a heterotroph and aerobic environment and are modeled by

The different coefficients occurring in eq 29 and eq 30 are presented in Table 4.

Under the reactor homogeneity hypothesis, the following assumptions are often made: [START_REF] Olsson | Wastewater Treatment Systems. Modelling, Diagnosis and Control[END_REF] In general, q R and q W represent fractions of input flow q in :

We suppose that the dissolved oxygen concentration at the reactor input (S O,in ) is null.

The system 29 becomes in this case:

Regulation from a Reference Volume. Hereafter, it is assumed that the volume is controlled in order to obtain V equal V ref .

To regulate the output q out the control law combining a priori/a posteriori control is defined by The control variables are the air flow (q a ), the recycled flow (q R ), and the rejected flow (q W ). In this case, the state and input vectors are defined by After replacing r H , r S , and r O given in eq 30, the system 31 becomes:

q flow q in q out q R V reactor volume q a air flow (q in + q R )X BH ) (q R + q W )X BH,R (28a)
S S,R ) S S (28b) { dV dt ) q in + q R -q out d(V • X BH ) dt ) q in X BH,in + q R X BH,R -q out X BH,out + r H V d(V • S S ) dt ) q in S S,in + q R S S,R -q out S S,out + r S V d(V • S O ) dt ) q in S O,in + q R S O,R -q out S O,out + r O V + Kq a V(S O,sat -S O ) (29) { r H ) µ H S S K S + S S S O K OH + S O X BH -b H X BH r S )- 1 Y H µ H S S K S + S S S O K OH + S O X BH + (1 -f)b H X BH r O ) Y H -1 Y H µ H S S K S + S S S O K OH + S O X BH (30)
{ X BH,out ) X BH S S,out ) S S S O,out ) S O q R ) f R q in ,1 e f R e 2 q W ) f W q in ,0 < f W < 1 { V ˙) q in + q R -q out X ˙BH ) q in V X BH,in - q W V q in + q R q W + q R X BH + r H S ˙S ) q in V (S S,in -S S ) + r S S ˙O )- q in V S O + Kq a (S O,sat -S O ) + r O (31) q out ) (f R + 1)q in -K 1 (V ref -V) x ) [VX BH S S S O ] T (32a) u ) [X BH,in S S,in q a V ref ] T (32b)
To put this nonlinear system into a multiple model form, each state equation first has to be presented in a quasi-LPV form like eq 18. Then, taking into account eq 32 and eq 33, the matrices are defined as follows:

where

The system presents three nonlinearities. The premise variables are

The idea of taking the premise variable z 1 as the product of the two Monod laws, without separating them, is to reduce in this way the premise variable number. The chosen premise variables are expressed respectively as the Monod law product (z 1 ), the dilution rate (z 2 ), and the air flow (z 3 ). Concerning the physical meaning in the real process, the first premise variable z 1 represents the growth of the heterotrophic biomass X BH , the second one (z 2 ) illustrates the dwell-time, and the third premise variable z 3 expresses the oxygenation level. Subsequently, by using the convex polytopic transformation, the three premise variables z 1 , z 2 , and z 3 are written like in eq 20, where the two partitions of each premise variable are constructed in the same way as in eq 21 and with the scalars of eq 22. To calculate the maximum and the minimum values that occur in z 1 , z 2 , and z 3 expressions, we use the fact that V, S S , S O , q in , and q a are physically bounded (Table 5).

The number of submodels is 2 3 ) 8 and will be represented by the pairs (A i ,B i )( i ) 1, ..., 8):

To be clear with these notations, let us recall for example, that A 1 is obtained using definition 34, where a 2,2 , a 3,2 , a 3,3 , a 4,2 , a 4,4 are computed by using the minimal value of z 1 , z 2 , z 3 (eq 35).

The multiple model is obtained by an interpolation of the eight previous submodels:

The weighting functions µ i (x,u)( i ) 1, ..., 8) are calculated in the same way as in eq 23, in conformity with definition 35.

For example, we have

In Figure 3 we can see the evolution of the system eq 33 which has been obtained by using formulation 36 obtained for: and under the inputs presented in Figure 2.

K S ) 20 g/m 3 , K OH ) 0.2 g/m 3 , S O,sat ) 10 g/m 3 , K ) 2.3 m -3 , K 1 ) 0.01, b H ) 0.4 (1/24) h -1 , µ H ) 3.733 (1/24) h -1 , Y H ) 0.6, f ) 0.1, f R ) 1.1, f W ) 0.03, with the initial conditions: { V ˙) K 1 (V ref -V) X ˙BH ) q in V X BH,in - q W V q in + q R q W + q R X BH + µ H S S K S + S S S O K OH + S O X BH -b H X BH S ˙S ) q in V (S S,in -S S ) + (1 -f)b H X BH - µ H Y H S S K S + S S S O K OH + S O X BH S ˙O )- q in V S O + Kq a (S O,sat -S O ) + Y H -1 Y H µ H S S K S + S S S O K OH + S O X BH (33) A(z 1 , z 2 , z 3 ) ) ( a 1,1 000 0 a 2,2 00 0 a 3,2 a 3,3 0 0 a 4,2 0 a 4,4 ) B(z 2 ) ) ( 000 b 1,4 b 2,1 000 0 b 3,2 00 00 b 4,3 0 ) (34) a 1,1 )-K 1 a 2,2 ) µ H z 1 - f W (1 + f R ) f W + f R z 2 -b H a 3,2 )- µ H Y H z 1 + (1 -f)b H a 3,3 )-z 2 a 4,2 ) Y H -1 Y H µ H z 1 a 4,4 )-Kz 3 -z 2 b 1,4 ) K 1 b 2,1 ) b 3,2 ) z 2 b 4,3 ) K•S O,sat { z 1 (S S , S O ) ) S S K S + S S • S O K OH + S O z 2 (q in , V) ) q in V z 3 (q a ) ) q a (35)
A 1 ) A(z 1,1 , z 2,1 , z 3,1 ) B 1 ) B(z 2,1 ) A 2 ) A(z 1,1 , z 2,1 , z 3,2 ) A 3 ) A(z 1,1 , z 2,2 , z 3,1 ) A 4 ) A(z 1,1 , z 2,2 , z 3,2 ) A 5 ) A(z 1,2 , z 2,1 , z 3,1 ) A 6 ) A(z 1,2 , z 2,1 , z 3,2 ) A 7 ) A(z 1,2 , z 2,2 , z 3,1 ) A 8 ) A(z 1,2 , z 2,2 , z 3,2 ) B 2 ) B(z 2,1 ) B 3 ) B(z 2,2 ) B 4 ) B(z 2,2 ) B 5 ) B(z 2,1 ) B 6 ) B(z 2,1 ) B 7 ) B(z 2,2 ) B 8 ) B(z 2,2 ) x ˙) ∑ i)1 8 µ i (x, u)(A i x + B i u) (36) µ 1 (x, u) ) F 1,1 (S S , S O )F 2,1 (V, q in )F 3,1 (q a ) ) S S K S + S S S O K OH + S O -0.06 0.16 -0.06 • q in V -0.48 0.92 -0.48 • q a -1.
The q in input flow influences the three concentrations X BH , S S , and S O . An increase of the input flow (q in ) produces an increase of the substrate concentration (S S ) and the heterotrophic biomass concentration (X BH ), as well as a decrease of the dissolved oxygen concentration (S O ). At the same time, S O is directly influenced by the air flow (q a ), X BH , and S S inside the reactor being influenced by the corresponding input reactor concentrations (X BH,in and S S,in ). The range of speed response of the dissolved oxygen, soluble carbon, and heterotrophic biomass is obvious.

For space and clarity reasons, not all of the eight weighting functions of the multiple model 36 are presented in Figure 4 but only four of them. It has to be remarked that q in , q a , V ref , X BH,in and S S,in also have an influence on the weighting functions of the multiple model. For example, the seventh submodel is active if a good oxygenation is ensured (high level of the air flow q a ), if there is no important growth of the soluble components and if the dwell-time (corresponding to important dilution rate z 2 ) is weak (see the periods of time t ) 2.3-3.5 h and t ) 5.5-6.5 h). For a high dwell-time, the fifth submodel plays an important role (see the periods of time t ) 3.5-3.8 h and t ) 6.5-6.7 h). A weak oxygenation (1.7-2.3 h and 6.7-9 h), a weak growth of the soluble components, and weak dwelltime (2-2.3 h) activate the eighth submodel, and so on.

Conclusion

This article proposes a general analytical method to simplify the complexity of a nonlinear system by rewriting it into a multiple model. This substitution consists in finding a set of submodels with a simple linear structure and a set of appropriated weighting functions in order to combine these submodels to constitute the global model.

To obtain a multiple model this article proposes a general method to rewrite the nonlinear model into a multiple model form avoiding the need to choose the operating points or trajectories. Additionally, systematic rules are proposed to choose the premise variables.

From the expression of a nonlinear system, we showed how to realize a systematic decomposition of a system into simpler linear systems by constructing a multiple model. It is important to note that the obtained multiple model is identical to the initial system. This decomposition method can be applied to any order systems. Another important aspect concerning this decomposition method is the given possibility to choose between several quasi-LPV forms of the nonlinear system that lead to specific properties of the multiple model structure. Such properties may be controller or observer design for multiple model structures with regard to the study objectives. This method also gives the possibility to reduce the number of submodels by an adequate choice of premise variables between several sets of candidate premise variables.

From a practical point of view, this method is applied to a simplified activated sludge reactor model and leads to a multiple model structure based on eight local models. Even if in this article we have treated the case of a simplified ASM1 model, considering only the carbon pollution, the proposed method can be applied to the whole ASM1 model, as the difficulties are similar (nonlinearities expressed by the Monod law, the dilution rate, and air flow).

Figure 1 .

 1 Figure 1. The diagram of activated sludge wastewater treatment process.

155 8 .

 8 25 -1.155x(0) ) [ V(0) X BH (0) S S (0) S O (0) ] ) [ 1250 m 3 887 g/m 3 4.2 g/m 3 3 g/m 3 ]

Table 1 . Decomposition Table for Amultiplemodel with Three Premise Variables and Two Partitions for Each Variable

 1

Table 2 . Premise Variable Decomposition Table for a Multiple Model with p Premise Variables and Two Partitions for Each Variable

 2

	submodel i	σ i	A i	B i	µ i (z)
	1 2	(1, 1, ..., 1) (1, 1, ..., 2)	A(z 1,1 , z 2,1 , ..., z p,1 ) A(z 1,1 , z 2,1 , ..., z p,2 )	B(z 1,1 , z 2,1 , ..., z p,1 ) B(z 1,1 , z 2,1 , ..., z p,2 )	j ∏ j)1 p F j,σ 1 ∏ j)1 p F j,σ 2 j
	l	l	lll
	2 p-1	(1, 2, ..., 2)
	2 p-1 + 1	(2, 1, ..., 1) (2, 1, ..., 2)	A(z 1,2 , z 2,1 , ..., z p,1 )	B(z 1,2 , z 2,1 , ..., z p,1 )	j ∏ j)1 p F j,σ (2 p-1 +1)
	l	l	lll
		(2, 2, ..., 1)
	2 p	(2, 2, ..., 2)	A(z 1,2 , z 2,2 , ..., z p,2 )	B(z 1,2 , z 2,2 , ..., z p,2 )

Table 3 . Table of Variables

 3

			reactor	reactor
	variable	definition	input	output	recycled
	X BH	heterotrophic biomass	X BH,in	X BH,out	X BH,R
		concentration
	S S	fast biodegradable	S S,in	S S,out	S S,R
		substrate concentration
	S O	dissolved oxygen	S O,in	S O,out	S O,R
		concentration

Table 4 . Table of Constants

 4

	S O,sat	oxygen saturation concentration
	K S , K OH	half-saturation constants
	Y H	yield coefficient for heterotrophic biomass
	f	fraction of biomass leading to particulate products
	b H , µ H	heterotrophic kinetic coefficients
	K	regulator gain in oxygen

Table 5 . Table of Maximum and Minimum Values

 5

	min	max

Acknowledgment

We acknowledge the financial support received from the "Fonds National de la Recherche Luxembourg". This research is also partially supported by the PHC TASSILI no.07 program under MDU Grant 714.

Literature Cited


O Isnard

V Pop

Magnetic properties of the iron sub-lattice in the YFe 12-x M x compounds (M=Ti, Mo or V; x=1 to 3.5)

Keywords: 75.30.-m, 75.30.Cr, 75.30.Et Spin fluctuation, Paramagnetic susceptibility, Exchange interaction

The magnetic properties of the YFe 12-x M x compounds (M=Ti, Mo or V; x=1 to 3.5) have been determined in the ordered ferromagnetic state as well as in the paramagnetic state. The iron magnetic moment has been determined from 4K up to the Curie temperature whereas the analysis of the paramagnetic region has led to the determination of the effective iron magnetic moment. The number of spins has been calculated below and above the Curie temperature in order to discuss the degree of itinerancy of the Fe magnetic behavior in the YFe 12-x M x compounds. All the YFe 12-x M x compounds (M=Ti, Mo or V; x=1 to 3.5) have very similar crystalline properties: they crystallize in the same crystal structure and all the M elements used here are known to substitute for iron on the same crystal site. On the contrary, they exhibit a wide range of magnetic behavior: the Curie temperature varies from 63 to 539K and the mean magnetic moment per iron atom is also very dependent upon the M element used and its concentration. Furthermore the degree of itinerancy of the iron is not preserved along YFe 12- x M x compounds but is found to depend significantly upon the nature of the substituting element M and its concentration. The results are discussed and compared to earlier published results obtained on binary R-Fe and ternary R-Fe-B compounds.

Introduction

It has been shown already for many decades that combining of rare-earth elements and Mn in a ratio 1:12 leads to intermetallic compounds of the ThMn 12 structure [1]. Corresponding Fe compounds do not form. Later investigations [2,3] have shown that formation of ternary compounds between rareearths and Fe of crystal structure related to ThMn 12 is fairly general. The majority component of these compounds is Fe and their formula is described by RFe 12-x M x . Nowadays, an extensive class of materials of the type RFe 12-x M x (where M=Ti, Cr, V, Mo W, Si…) is known to retain the ThMn 12 structure [2][3][4][5][START_REF] Li | Handbook of Magnetic Materials[END_REF][START_REF] Verhoef | [END_REF][START_REF] Buschow | Concertedd european action on magnets[END_REF]. The stability range of the RFe 12-x M x compounds is highly dependent upon the nature of the substituting element M [2,4,[START_REF] Li | Handbook of Magnetic Materials[END_REF].

These series of RFe 12-x M x intermetallics offer two main advantages as hard magnetic material: (i) a high Fe content that favor a high magnetization, (ii) a relatively high Curie temperature, in particular for the Ti containing compounds. They have attracted much interest as potential candidate for high performance permanent magnet applications. This interest has been increased in the last decade since it has been shown that the magnetic properties of these RFe 12-x M x compounds can significantly been improved by insertion of light elements such as H, C or N within the crystal lattice [START_REF] Coey | Rare-Earth iron Permanent Magnets[END_REF][START_REF] Hurley | [END_REF][START_REF] Tomey | [END_REF][12][START_REF] Fujii | Handbook of magnetic materials[END_REF][START_REF] Coey | [END_REF][15][16][17].

Here we focus our attention on the YFe 12-x M x compounds in order to investigate the influence of the substitution on the magnetic properties of the iron sub-lattice in these phases. In particular the magnetic behavior in the paramagnetic state will be analyzed and compared to the behavior in the ordered region in order to determine the degree of localization of the Fe atoms. The results will be analyzed in the light of the self-consistent renormalization theory of spin fluctuations [18][19][START_REF] Kübler | Theory of itinerant electron magnetism[END_REF].

Experimental

The alloys have been obtained by melting the high purity starting elements (better than 99.95%) in a high frequency induction furnace equipped with a water cooled copper crucible. The homogeneity of the ingot was subsequently achieved by annealing at 950°C for 10 days.

The homogeneity of the sample was checked by conventional X-ray powder diffraction with iron K α radiation and, in all cases, less than ca. 5 volume percent of α-Fe was found in each of the samples.

The indexation of the Bragg peaks was performed with a tetragonal unit cell compatible with the I4/mmm space group.

The magnetic ordering temperatures have been determined with a Faraday type balance at heating and cooling rates of 5 K per minute. A sample of c.a. 50 to 100 mg was sealed under vacuum in a small silica tube in order to prevent oxidization of the sample during heating. The magnetization curves of the YFe 12-x M x compounds were recorded at 5 K by the extraction method [START_REF] Barlet | [END_REF] in a continuous field of up to 9 T. The spontaneous magnetization values have been derived from an extrapolation to zero field of the magnetization obtained in fields higher than 4 T. A detailed description of the experimental set up can be found elsewhere [START_REF] Barlet | [END_REF]. The same cryostat was used with another insert [22] enabling any temperature between 300 and 850 K to be regulated. This set up has been used to investigate the magnetic properties of the YFe 12-x M x compounds above room temperature and in particular in the paramagnetic region.

Results and discussion

According to the X-ray investigation the samples where all found to retain the I4/mmm space group and the ThMn 12 type structure. The lattice parameters determined by a least square refinement of all the observed Bragg reflections are listed in table 1. These parameters are in agreement with that reported earlier for isotype compounds [2,[START_REF] Li | Handbook of Magnetic Materials[END_REF][START_REF] Tomey | [END_REF]23]. It is clearly seen that the unit cell volume of the Mo containing compounds increases of about 5 Å 3 per Mo atom.

In these compounds, the rare earth is known to be located on the 2a crystallographic site, whereas the iron atoms are distributed over the three inequivalent crystal sites : 8i, 8j, and 8f. In the RFe 12-x M x compounds, the Ti, Mo and V atoms are all known to show a strong preference for occupying the 8i site [START_REF] Tomey | [END_REF][23][24][25][26][27]. We can consider that in the YFe 12-x M x compounds the 8i site is randomly occupied by the iron and the M atoms whereas the 8j and 8f sites are fully occupied by iron. According to earlier published neutron diffraction investigation, all the compounds studied here are exhibiting a uniaxial alignment of the Fe magnetic moment along the c axis of the crystal structure [START_REF] Tomey | [END_REF]23,26,27].

The spontaneous magnetization obtained for all the studied YFe 12-x M x compounds are summarized in table 2. As expected the largest magnetization are obtained for the smallest M content. The largest magnetization being 19.8 and 18.6 μ B /formula unit for YFe 11 Mo and YFe 11 Ti respectively. As can be seen from the YFe 12-x Mo x series of compounds a large decrease of the magnetization is observed upon increasing the Mo content. Similar concentration dependence of the Fe magnetic moment has been reported for other M elements such as Ti, W, Cr, V [START_REF] Verhoef | [END_REF][START_REF] Buschow | Concertedd european action on magnets[END_REF]. This magnetization decrease is in agreement with the strong decrease of the Curie temperature observed for these YFe 12-x Mo x compounds when the Mo concentration increases -table 2-. Such magnetization decrease has been reported by Sun et a. [28] for the YFe 12-x Mo x compounds. The spontaneous magnetization obtained here are somewhat smaller than that reported in reference [23]. This most probably originates from the small magnetic fields used (<1.6T), whereas in the present study a much better saturation can be obtained by using a magnetic fields up to 9 T for the magnetization measurements. In this RFe 12-x M x structure the iron magnetic moments have been found to be very sensitive to the local atomic environment, the larger Fe moment being observed on the 8i site [29]. The saturation magnetization of the vanadium containing compounds is only about 10 μ B /formula unit which, assuming no magnetic moment on the M atoms, leads to a mean magnetic moment of 1.17 μ B per Fe atom. This value is much lower than the 1.7 or 1.8 μ B per formula unit obtained for YFe 11 Ti and YFe 11 Mo respectively. This calculation probably underestimates the mean Fe magnetic moment value in YFe 8.5 V 3.5 . Indeed, it is well known that Fe-V hybridization leads to a significant negative polarization on the V atoms. This has been described extensively for the binary Fe-V system [30] but also reported from electronic structure calculation on YFe 10 V 2 [31] where a negative polarization of about -0.5 μ B per V has been obtained. More recently, this has been experimentally shown for the isotype RFe 10.5 V 1.5 ternary compounds [26] where a negative polarization as high as 0.8 μ B per V atoms has been obtained. The relatively low saturation magnetization of YFe 8.5 V 3.5 compared to its large ordering temperature 410K may indicate the contribution of vanadium atoms to the magnetization and the magnetic order. The 4, 300 and 380K isothermal magnetization curves of YFe 8.5 V 3.5 are shown in Fig. 1. The temperature dependence of the spontaneous magnetization for YFe 8.5 V 3.5 , YFe 11 Ti and YFe 11 Mo are shown in Fig. 2. It is noticeable that at 4K, YFe 11 Ti has a smaller magnetization than YFe 11 Mo, whereas the situation is reversed at room temperature. This larger magnetization at room temperature is clearly due to the much higher Curie temperature of YFe 11 Ti. The 4K and 300K saturation magnetization are given in table 3 for all the studied compounds. It is noteworthy that the values obtained here are in good agreement with that reported earlier [START_REF] Li | Handbook of Magnetic Materials[END_REF]23].

The Curie temperatures have been determined by two different methods: i) from the thermomagnetic analysis by plotting the temperature dependence of the square of the magnetization ii) from the Arrott plots [32] figure 3. The two methods are found to give results that are in excellent agreement. The Curie temperature T c of the YFe 12-x M x compounds studied here are summarized in table 3. It is noteworthy that the Curie temperatures obtained here are in good agreement with that earlier reported in [2,4,33]. The Curie temperature value of 471±5K obtained by thermomagnetic analysis for YFe 11 Mo agrees with the value of 475±4K obtained from the Arrott plots given in Fig. 3. The T c values of the YFe 12-x Mo x compounds are in excellent agreement with that reported by Sun et al [28].

The molecular field coefficient n Fe-Fe and exchange field B ex have been derived from the Curie temperature of the YFe 12-x M x compounds assuming that the T c results only from Fe-Fe exchange interactions. The equations used are given below where M and C correspond to the magnetization and the Curie Weiss constant respectively. The corresponding values of the molecular field coefficient n Fe- Fe and exchange field B ex are given in table 3.

C T M B c ex = ( 1 ) C T n c Fe Fe = - ( 2 )
The composition dependence of the exchange field in the YFe 12-x Mo x compounds exhibits a dramatic decrease versus the Mo content. It is clear that the Mo for Fe substitution leads to a dramatic decrease of the exchange field. For the same composition, the Ti containing compound exhibit a similar but slightly higher exchange interactions than the Mo isotype compound. The n Fe-Fe exchange interactions determined here are of the same magnitude than that reported recently in ThFe 11 C x [34] but much larger than what is usually found for the R 2 Fe 17 compounds [35].

Magnetic measurements have been undertaken at high temperature for all the studied compounds in order to investigate the magnetic behavior of Fe in the paramagnetic state. Since the iron impurity orders at a much higher temperature than the YFe 12-x M x compounds its magnetic influence most likely will affect the determination of the intrinsic susceptibility of the YFe 12-x M x compounds. In order to avoid such an influence of the magnetic ordered iron impurities on the susceptibility values χ, the latter were obtained from field-dependent measurements according to Honda-Owen plots [START_REF] Bates | Modern Magnetism[END_REF]:

H cM H M s + = χ ( 3 )
by extrapolating to H -1 0. The impurity content is represented by c whereas M s corresponds to the saturation magnetization of the impurity phase.

The thermal variation of the reciprocal susceptibility has been measured for all YFe 12-x M x compounds.

At high temperature, the YFe 12-x M x compounds are exhibiting a Curie-Weiss behavior. The thermal evolution of the reciprocal susceptibilities is plotted in Fig. 4 for all the studied compounds. A linear fit of the data according to a Curie-Weiss law,

p T C θ χ - = ( 4 )
leads to the values reported in table 2 and 3.

The determined paramagnetic temperature θ p are slightly larger than the Curie temperatures deduced from the thermomagnetic measurements, which is reminiscent of short-range magnetic interactions above T C . As can be seen from table 3, θ p is about 30 to 40 K larger than T C for the YFe 11 Mo, YFe 11 Ti, YFe 11 Mo 2 compounds . The case of YFe 8.5 V 3.5 is peculiar since θ p value is slightly lower than T c . This may be indicative of the presence of antiferromagnetic interactions, thus confirming the negative polarization on the vanadium as deduced above from the magnetization analysis.

The magnetic behavior of the YFe 12-x M x compounds may be analyzed using the self-consistent renormalization theory of spin fluctuations [18][19][START_REF] Kübler | Theory of itinerant electron magnetism[END_REF]. In this model two extremes cases are described : i) local moment and ii) exchange enhanced paramagnetism This two extreme cases are characterized by different value of the ratio r =S p /S 0 between the number of spins determined from Curie constants (S p ) and the saturation magnetization (S 0 ). For a local moment we have r = 1 while for longitudinal spin fluctuations r = ∞. The ratio r has been calculated for all the YFe 12-x M x compounds and is reported in table 2. A significant increase of the ratio r is observed upon increasing the M content. This bears witness to a progressive delocalization of the Fe magnetic moment [START_REF] Burzo | [END_REF]. According to the renormalisation theory of spin fluctuations, when the amplitude of the local spin fluctuation is large and fixed there is the local moment limit where only the transverse component of local spin fluctuations are important. On the contrary, for a nearly ferromagnetic alloy, the longitudinal spin fluctuations dominate. The substitution induced increase of the r ratio observed for the YFe 12-x M x compounds suggests that the contribution of the longitudinal spin fluctuations increases as the M content increases. The r values obtained for the YFe 12-x M x compounds -see table 2-reveals that the iron magnetic moment is in the similar degree of localization in YFe 12-x M x than in Th 2 Fe 17 C x compounds [35] and slightly less localized than in the ThFe 11 C x phases [START_REF] Bates | Modern Magnetism[END_REF]. It is worth to mention that none of these phases are as localized than elemental α-Fe (r = 1.05).

In the RFe 2 compounds the effective magnetic moment per Fe atom is found to be about 3.0 to 3.4 μ B /Fe atom in YFe 2 and GdFe 2 respectively. For these compounds the ratio r between the number of spins obtained from the Curie constants and that determined from saturation magnetization measurements has been reported to be constant in the Gd x Y 1-x Fe 2 system with a value of 1.5 [START_REF] Burzo | [END_REF]. This value is very similar to that obtained for YFe 11 Mo showing that the Fe sublattice is in the same degree of localization in both compounds. Burzo [38] reported a r value of 1.30 and 1.24 for the Gd x Y 1-x Fe 3

and R 2 Fe 14 B compounds respectively which are known to have exchange interaction similar to the one obtained for the YFe 12-x M x compounds.

It has been shown that for many families of compounds, the r ratio varies linearly versus the exchange fields. Such plot is given in Fig. 5 evolution taking into account the YFe 12-x M x compounds studied here. Keeping in mind that the contribution of the M atoms to the exchange interactions have been neglected the agreement is reasonable [38]. This confirms that the major contribution to the magnetic properties arise from the Fe atoms.

Recent investigation of the ternary Th-Fe-C interstitial compounds has shown that a pronounced reduction of the r value upon carbon insertion in the lattice [34,35]. This change of the ratio between the ordered and the effective spin numbers (r = S p / S 0 ) is slightly modified upon interstitial element insertion showing that the spin fluctuations change with the interstitial concentration. This reduction of the r ratio observed for the ThFe 11 C x and Th 2 Fe 17 C x series of compounds, shows that the carbon addition induces an evolution towards a local Fe magnetic moment.

Similar behavior can be expected for the YFe 12-x M x X y (X=H, C, N) compounds whose Curie temperature is known to increase significantly upon interstitial element insertion [START_REF] Li | Handbook of Magnetic Materials[END_REF][START_REF] Tomey | [END_REF]25,28].

According to the Rhodes and Wolfarth curve [39], one can expect that the magnetic properties of the YFe 12-x M x X y ( X=H, C, N) compounds evolved towards a more localized like magnetism. Such interpretation can be proposed to account for the magnetic properties of the carbides and nitrides.

Further study on the YFe 12-x M x X y compounds would be needed to confirm this expectation and to quantify this evolution of the r ratio. This is rather difficult to confirm experimentally since the YFe 12-

x M x X y compounds are metastable. They can partially decompose upon heating while performing the magnetic investigation in the high temperature paramagnetic state. This is particularly true for the nitrides which form rare-earth nitride and Fe 4 N [24] or in the case of the hydrides for which hydrogen is released upon heating [29] thus hampering to analyze the magnetic properties in a wide temperature range in the paramagnetic state.

Conclusion

The magnetic properties of the YFe 12-x M x compounds are found to dramatically dependent upon the M element nature and concentration. The iron magnetic moment as well as the Curie temperature and the exchange interactions are significantly reduced upon M for Fe substitution (M=Ti, Mo, V). The present study has shown that the spin fluctuations can vary from a rather localized to a strongly delocalized type for YFe 12-x M x compounds. In particular, the degree of itinerancy is dramatically increased upon increasing the Mo concentration. The ratio r between the number of spins obtained from the Curie constants and that determined from saturation magnetization is found to vary from 1.5 to 2.2 for YFe 11 Mo and YFe 8.5 V 3.5 respectively. This shows a much more delocalized magnetism than in iron even more than in the binary YFe 2 compounds. The YFe 12-x M x compounds are thus offering the unique opportunity to cover a wide range of magnetic behavior over the Rhodes and Wolfarth curves for compounds keeping the same crystal structure. T (K)

Figure 1 .

 1 Figure 1. The 4, 300 and 380 K isothermal magnetization curves temperature recorded for the YFe 8.5 V 3.5 sample.

Figure 2 .

 2 Figure 2. The temperature dependence of the spontaneous magnetization of YFe 11 Ti , YFe 11 Mo and YFe 8.5 V 3.5 compounds.

Figure 3 .Figure 4 .

 34 Figure 3. Arrott plots at the indicated temperature around the Curie temperature of YFe 11 Mo

Figure 5 .

 5 Figure 5. The evolution of the r = S p / S 0 values as function of the exchange fields. The YFe 12-x M xcompounds are drawn in black circles, the others points are taken from reference[START_REF] Burzo | [END_REF].

  Figure 1

  Figure 5

Table 1 .

 1 Lattice parameters of the YFe 12-x M x compounds

	Compound	a	c	V
		(Å)	(Å)	(Å 3 )
	YFe 11 Ti	8.503 (1)	4.795 (1)	346.7
	YFe 11 Mo	8.519 (1)	4.780 (1)	346.9
	YFe 10 Mo 2	8.553 (1)	4.792 (1)	350.6
	YFe 8.5 V 3.5	8.508 (1)	4.779 (1)	345.9

Table 2 .

 2 Saturation magnetization M s , Curie Weiss constant C, paramagnetic effective moment μ eff , mean number of spins in the ordered S 0 or paramagnetic states S p and corresponding ratio r=S p /S 0 for the YFe 12-x M x compounds compared to that of the pure Fe.

	Compound	M s 300K (μ B /f.u.)	M s 4K	M Fe 4K	S 0	μ eff (μ B /Fe atom)	S p	r	C (μ B *K/f.u.*T)
			(μ B /f.u.)	(μ B /atm)
	YFe 11 Ti	16.3	18.6	1.69	0.845	3.745	1.439	1.70	34.59
	YFe 11 Mo	15.6	19.8	1.8	0.90	3.649	1.392	1.55	32.80
	YFe 10 Mo 2	5.5	13.1	1.31	0.655	3.412	1.278	1.95	26.07
	YFe 8.5 V 3.5	6.5	9.96	1.17	0.586	3.409	1.277	2.18	22.12
	Fe α	2.17	2.22	2.22	1.11	3.18	1.17	1.05	2.26

Table 3 .

 3 Curie temperature T c , paramagnetic Curie temperature θ p , molecular field coefficient n Fe-Fe and exchange field B ex in the YFe 12-x M x compounds

	Compound	Number of	T c	θ p	n Fe-Fe	B ex
		Fe atom/f.u.	(K)	(K)	(μ 0 )	(T)
	YFe 11 Ti	11	539	565	232	290
	YFe 11 Mo	11	469	500	213	283
	YFe 10 Mo 2	10	340	384	275	171
	YFe 8.5 V 3.5	8.5	410	372	275	185
	Fe α	1	1043	1093	466	1023

ACKNOWLEDGMENTS

V. P. thanks the 'University Joseph Fourier, Grenoble, France' for the grant of 'Maître de conférence invité' at the Faculty of physics. O. I. would like to thank E. Burzo for interesting discussions.