Anca Maria Nagy

Benoît Marx

Gilles Mourot

Georges Schutz

José Ragot

State estimation of the three-tank system using a multiple model

This paper addresses the exact transformation of nonlinear systems into a multiple model form with unmeasurable premise variables. The multiple model structure serves to treat the observability and the state estimation problem of nonlinear systems. Using a method with no information loss, a nonlinear system is transformed into a multiple model, depending on the choice of premise variables. It is a key point, since it allows to choose, between different multiple model forms, the one that has suitable structure and properties, in order to design an observer. The convergence conditions of the state estimation error are expressed in LMI formulation using the Lyapunov method. These proposals are investigated and applied to the three-tank system.

I. INTRODUCTION

The fault detection and diagnosis (FDD) methods for linear systems had already reached a certain development and maturity. Nevertheless, nonlinear models are often needed to represent real system behaviors. As a consequence, there is a need to extend FDD methods to nonlinear systems, even if it is a difficult point. It is the reason why the concept of multiple models has received attention in the last two decades. The multiple model structure gives the possibility to reduce the complexity of nonlinear systems, by constructing linear submodels aggregated using weighting functions [START_REF] Takagi | Fuzzy identification of systems and its application to modeling and control[END_REF], [START_REF] Tanaka | Fuzzy Control System Design and analysis. A Linear Matrix Inequality Approach[END_REF]. The interests to use this structure is already well known, the main advantage is the possibility to extend many tools and results of the linear theory to nonlinear systems, such as the observer synthesis. In the linear framework, the fault detection can be realized using observer based methods. There already exists several works on observer design for multiple model systems. A multi-observer based on Luenberger observers was proposed and applied for diagnosis in [START_REF] Patton | Fuzzy observers for nonlinear dynamic systems Fault diagnosis[END_REF]. In [START_REF] Akhenak | Estimation of state and unknown inputs of a nonlinear system represented by a multiple model[END_REF], sliding mode observers for linear systems, were transposed to the multiple model case, and in [START_REF] Akhenak | Estimation of state and unknown inputs of a nonlinear system represented by a multiple model[END_REF], [START_REF] Akhenak | Unknown input multiple observer based approach. Application to secure communications[END_REF] unknown input multi-observers were implemented. Moreover, one of the main interests to use this type of observers is its robustness with regard to modeling uncertainties. However, these papers assume that the weighting functions depend on measurable premise variables, which is not always the case. In the literature, few works are devoted to the case of unmeasurable decision variables [START_REF] Bergsten | Observers for Takagi-Sugeno Fuzzy Systems[END_REF], [START_REF] Ichalal | Design of Observers for Takagi-Sugeno Systems with Immeasurable Premise Variables: an L2 Approach[END_REF], [START_REF] Palm | Sliding mode observers for Takagi-Sugeno Fuzzy Systems[END_REF]. As discussed in [START_REF] Ichalal | Design of Observers for Takagi-Sugeno Systems with Immeasurable Premise Variables: an L2 Approach[END_REF], multiple model with unmeasurable decision Anca Maria Nagy, Benoît Marx, Gilles Mourot, José Ragot are with Centre de Recherche en Automatique de Nancy, Nancy Université, 2, avenue de la Forêt de Haye, 54516 Vandoeuvre-les-Nancy Cedex France anca-maria.nagy, gilles.mourot, benoit.marx, jose.ragot@ensem.inpl-nancy.fr Georges Schutz is with CRP "Henri Tudor", LTI Unité Business and Systems Analytics, 29, Avenue John F. Kennedy, L-1855 Luxembourg -Kirchberg georges.schutz@tudor.lu variables is more adequate to fault diagnosis since a single model can be used, both for sensor and actuator fault diagnosis. [START_REF] Ichalal | Design of Observers for Takagi-Sugeno Systems with Immeasurable Premise Variables: an L2 Approach[END_REF] proposes a method which reduces the conservatism of the precedent works by reducing the number of LMIs to be solved and relaxing the conditions under which the method is applicable. In addition, the number of LMIs does not depend on the number of submodels of the multiple model. Different techniques to obtain a multiple model exist in the literature. Firstly, model identification techniques or parameter estimation using experimental data [START_REF] Abonyi | Modified Gath-Geva fuzzy clustering for identification of Takagi-Sugeno fuzzy models[END_REF] can be considered. Secondly, the techniques based on existent mathematical nonlinear models, where a linearization around one (or several) operating point(s) as well as a dynamic linearization near an arbitrary trajectory are proposed [START_REF] Johansen | On the interpretation and identification of dynamic Takagi-Sugeno fuzzy models[END_REF]. The main drawbacks of this technique are: the loss of information, the choice of the premise variables expressing different nonlinearities of the system is not systematically realized, the choice of different operating points still remains very delicate. The method used here avoids this particular loss and gives an equivalent multiple model form of the initial nonlinear system [START_REF] Nagy | Model structure simplification of a biological reactor[END_REF]. In addition to that, the method offers several multiple models with different structures, depending on the choice of premise variables. So, it is possible to choose from different multiple model structures the one that ensures the convergence of the multi-observer, with respect to given conditions. The method is here improved and adapted in order to facilitate the choice of the suitable MM structure, by introducing a pondered sum of the state variables when constructing the MM form. The convergence conditions of the state estimation error are expressed in LMI formulation using the Lyapunov method. The method used here for the observer design [START_REF] Ichalal | Design of Observers for Takagi-Sugeno Systems with Immeasurable Premise Variables: an L2 Approach[END_REF] requires the observability of the multiple model. It is the reason why an observable multiple model structure is sought. These proposals are investigated and applied in this paper to the three-tank system. This system was largely studied in the literature [START_REF] Rincon-Pasaye | Fault diagnosis in nonlinear systems: An application to a three-tank system[END_REF], [START_REF] Rato | Multimodel based fault tolerant control of the 3-Tank System[END_REF], [START_REF] Theilliol | Fault diagnosis and accommodation of a three-tank system based on analytical redundancy[END_REF] and is considered as a benchmark, in different frameworks. The controller switching approach used for fault tolerant control in [START_REF] Rato | Multimodel based fault tolerant control of the 3-Tank System[END_REF], implies a period of time between two switchings (dwell time). In [START_REF] Rincon-Pasaye | Fault diagnosis in nonlinear systems: An application to a three-tank system[END_REF] a sliding mode observer for the three-tank system is proposed, where a particular region of operation is considered. The method proposed in this paper, avoids the need to separate different operating modes and give a multiple model which encompasses all possible modes, without any transition management. In [START_REF] Theilliol | Fault diagnosis and accommodation of a three-tank system based on analytical redundancy[END_REF] a fault diagnosis and accommodation method is applied to the three-tank system, that is linearized around an operating point using Taylor expansion. It is well known that the choice of the different operating points still remains a very delicate subject. The method proposed in this paper consists in rewriting the nonlinear models. As a consequence no operating points have to be chosen. In order to ensure the possibility to design an observer, a preliminary study of observability for nonlinear systems has to be realized. The uniform observability notion is closely related to observer design. There are different approaches to verify this condition. In [START_REF] Hou | Observing a Three-Tank System[END_REF] a systematic observability analysis for the three-tank system is realized. As a consequence, the observer design for the three-tank system is possible. The paper is organized as follows. Section II starts with the presentation of the three-tank system in II-A. In section II-B the main steps of the method used to obtain an equivalent multiple model form of a nonlinear system (here, the threetank system) are given. This method proposes a choice criterion of multiple models with suitable structure and properties, in order to design a multi-observer (section II-C). The design of the proposed observer based on multiple model form is described in section II-D. In section III some conclusions and work perspectives are given.

II. THE THREE-TANK SYSTEM A. Process description

The three-tank system is a MIMO system with two inputs and two outputs, presented in Fig. 1. The system consists in three cylindrical tanks of the same section S. They are linked to each other through connecting cylindrical pipes having the section S n . The nominal outlet is located at tank 3. The incoming flows Q 1 and Q 2 are considered as the inputs of the system. The control objective is to regulate the water levels in tanks 1 and 2 by adjusting Q 1 and Q 2 . α i (i = 1, ..., 3) are the outflow coefficients, g = 9.82[m/s 2 ] the gravitational acceleration. The objectives here are to analyze observability of the system based on single or double measurements and to design observers for estimating all the tank levels. The

ẋ1 (t) = Q 1 (t) S -α 1 S n S sgn[x 1 (t) -x 3 (t)] 2g |x 1 (t) -x 3 (t)| ẋ2 (t) = Q 2 (t) S + α 3 S n S sgn[x 3 (t) -x 2 (t)]• 2g |x 3 (t) -x 2 (t)| -Q 3 (t) S ẋ3 (t) = α 1 S n S sgn[x 1 (t) -x 3 (t)] 2g |x 1 (t) -x 3 (t)| -α 3 S n S sgn[x 3 (t) -x 2 (t)] 2g |x 3 (t) -x 2 (t)| (1
) where Q 3 (t) = α 2 S n 2gx 2 (t) and sgn(•) denotes the sign function. For the sake of the simplicity, let us note:

W i,k = α i S n S sgn[x i (t) -x k (t)], i = k (2) Define x =   x 1 x 2 x 3   , u = Q 1 Q 2 (3)
Let us consider H the maximal liquid level in the tanks and suppose that the tanks are never simultaneously empty. This assumption can be made because the levels x 1 , x 2 and x 3 are controllable using the flows Q 1 and Q 2 . In this case, the three-tank levels x 1 , x 2 and x 3 are bounded:

x ∈ R 3 + .

B. The multiple model representation

Let us consider the multiple model representation of a nonlinear system:

   ẋ(t) = r ∑ i=1 µ i (ξ (t)) [A i x(t) + B i u(t)] y(t) = C x(t) (4)
A i , B i and C are real known matrices with appropriate dimensions. The weighting functions µ i (•) are generally nonlinear in ξ (t) and satisfy the convexity property:

r ∑ i=1 µ i (ξ (t)) = 1, ∀t 0 ≤ µ i (ξ (t)) ≤ 1, i ∈ {1, ..., r}, ∀t (5)
The premise variable ξ (t) can depend on measurable variables, as the input u(t) or the output y(t) of the system, and/or on unmeasurable variables like the state x(t). In order to obtain a multiple model structure from a given nonlinear system different methods exists. The majority of them are linearization based techniques, which come with an information loss of the initial nonlinear system. The method used here avoids this particular loss and gives an equivalent multiple model form of the initial nonlinear system. Only a brief description of the important points of this method will be given as follows, the technical details can be found in [START_REF] Nagy | Model structure simplification of a biological reactor[END_REF]. Considering a nonlinear system with bounded nonlinearities, it can be written in a quasi linear-parameter-varying form:

ẋ(t) = A(κ(t)) x(t) + B(κ(t)) u(t) y(t) = C x(t) (6)
where κ(t) can represent a subset of the state variables.

To each quasi-LPV form corresponds a particular set of premise variables and a particular multiple model structure.

As a consequence, it is possible to choose between different multiple models the one with convenient properties. This represents a degree of freedom that will be used to make easier the construction of an observer. 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 requirement [START_REF] Tanaka | A Descriptor System Approach to Fuzzy Control System Design via Fuzzy Lyapunov Functions[END_REF] and ensures the solution existence [START_REF] Bergsten | Observers for Takagi-Sugeno Fuzzy Systems[END_REF]. Due to the convexity of the interpolation made between the submodels, 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. A Linear Matrix Inequality Approach[END_REF]. Therefore, it is tried to find a multiple model composed by a minimal number of submodels. This number is related to the number of premise variables, so that a quasi-LPV form which has a minimal set of premise variables will be preferred. Besides, the complexity of the LMI conditions increases with the number of state variables involved in the premise variables. In order to ensure the solution existence, the dimension of the state vector involved in the premise variables should be made as small as possible [START_REF] Bergsten | Thau-Luenberger Observers for TS Fuzzy systems[END_REF].

Taking into account the previous remarks, the choice of the quasi-LPV form leading to the simplest observable multiple model will be realized, using the following rules:

1) Eliminate all quasi-LPV forms for which the matrices A and/or B have null rows and/or columns. 2) Identify the decompositions containing common premise variables for different state equations, reducing in this way the number of premise variables. 3) Chose the quasi-LPV form for which the premise variables depend on a minimal number of state variables.

The premise variables associated to the quasi-LPV form will be partitioned into two parts, using the convex polytopic transformation. The two partitions of each bounded nonlinearity will contribute to the construction of the submodels and to the corresponding weighting functions. The multiple model will thus be a convex combination of linear submodels, the nonlinearity being transferred into weighting functions related to each submodel.

In the following, the previous essential points of these method will be considered in order to transform the threetank system into a multiple model form. Nevertheless, a new technique will be implemented concerning the quasi-LPV forms. These forms will be highlighted through a weighted sum of the state variables. The three-tank system is rewritten in an equivalent way, using a weighted sum of the state variables, via the real scalars λ i and γ i (i = 1, ..., n). The choice of real scalars λ i and γ i gives the possibility to obtain different multiple model structures. In fact, the use of these scalars allows to highlight the existence of different quasi-LPV forms of the system. Taking into account the notations (2) the three-tank system (1) is then rewritten:

ẋ1 (t) = 1 S Q 1 (t) - W 1,3 (t) 3 ∑ i=1 λ i x i (t) 2g |x 1 (t) -x 3 (t)| 3 ∑ i=1 λ i x i (t) ẋ2 (t) = 1 S Q 2 (t) -α 2 S n S 2gx 2 (t) + W 3,2 (t) 3 ∑ i=1 γ i x i (t) 2g |x 3 (t) -x 2 (t)| 3 ∑ i=1 γ i x i (t) ẋ3 (t) = W 1,3 (t) 3 ∑ i=1 λ i x i (t) 2g |x 1 (t) -x 3 (t)| 3 ∑ i=1 λ i x i (t) - W 3,2 (t) 3 ∑ i=1 γ i x i (t) 2g |x 3 (t) -x 2 (t)| 3 ∑ i=1 γ i x i (t) (7)
for non null λ i and γ i , and since x(t) is non null, then 3 ∑ i=1 λ i x i (t) and 3 ∑ i=1 γ i x i (t) are not null.

Let us define the premise variables:

z 1 (x) = W 1,3 (t) 3 ∑ i=1 λ i x i 2g |x 1 -x 3 | z 2 (x) = α 2 S n S 2g x 2 z 3 (x) = W 3,2 (t) 3 ∑ i=1 γ i x i 2g |x 3 -x 2 | (8)
where

x i = x i (t), i = 1, ..., 3.
The system (7) takes the following quasi-LPV form:

ẋ(t) = A(z 1 , z 2 , z 3 ) x(t) + B u(t) (9)
where

A(z 1 , z 2 , z 3 )=   -λ 1 z 1 -λ 2 z 1 -λ 3 z 1 γ 1 z 3 γ 2 z 3 -z 2 γ 3 z 3 λ 1 z 1 -γ 1 z 3 λ 2 z 1 -γ 2 z 3 λ 3 z 1 -γ 3 z 3   (10) B =   1 S 0 0 1 S 0 0   (11)
Using the convex polytopic transformation [START_REF] Tanaka | Fuzzy Control System Design and analysis. A Linear Matrix Inequality Approach[END_REF], the three premise variables z j are divided into two parts:

z j (x) = z j (x) -z j,2 z j,1 -z j,2 z j,1 + z j,1 -z j (x) z j,1 -z j,2 z j,2 = F j,1 (z j (x)) z j,1 + F j,2 (z j (x)) z j,2 (12)
where the scalars z j,1 and z j,2 are:

z j,1 = max x z j (x) z j,2 = min x z j (x) ∀ j = 1, ..., 3 (13)
The multiple model is composed of 8 = 2 3 submodels. The corresponding weighting functions are calculated as the products of the partition functions F j,σ j i (•) from (12) (where σ j i can take values 1 or 2) in order to obtain all 8 possible combinations between the functions:

µ i (x) = F 1,σ 1 i (z 1 (x)) F 2,σ 2 i (z 2 (x)) F 3,σ 3 i (z 3 (x)) (14)
with (σ 1 i , σ 2 i , σ 3 i ) = σ i corresponding to the submodel i is given in the table I and indicates which of the partition functions F j,σ j i (•) will be used in the computation of the weights µ i (x). Similarly, the matrices A i (i = 1, ..., 8) are obtained using the scalars z j,σ j i ( j = 1, ..., 3) expressed in [START_REF] Rato | Multimodel based fault tolerant control of the 3-Tank System[END_REF] order to obtain the 8 possible combinations:

A i = A(z 1,σ 1 i , z 2,σ 2 i , z 3,σ 3 i ) i = 1, ..., 8 (15)
where the matrix A(z 1 , z 2 , z 3 ) is given in [START_REF] Palm | Sliding mode observers for Takagi-Sugeno Fuzzy Systems[END_REF]. Considering [START_REF] Rincon-Pasaye | Fault diagnosis in nonlinear systems: An application to a three-tank system[END_REF], all the matrices B i are identical and thus the matrices

B i = B (i = 1, ..., 8
).

σ i i F 1,1 F 1,2 F 2,1 F 2,2 F 3,1 F 3,2 1 1 0 1 0 1 0 (1,1,1) 2 1 0 1 0 0 1 (1,1,2) 3 1 0 0 1 1 0 (1,2,1) 4 1 0 0 1 0 1 (1,2,2) 5 0 1 1 0 1 0 (2,1,1) 6 0 1 1 0 0 1 (2,1,2) 7 0 1 0 1 1 0 (2,2,1) 8 0 1 0 1 0 1 (2,2,2)
Then, the three-tank model ( 1) is transformed into the multiple model:

ẋ(t) = 8 ∑ i=1 µ i (x(t)) [A i x(t) + Bu(t)] y(t) = Cx(t) (16)
It has to be remarked that in conformity with ( 8) and [START_REF] Palm | Sliding mode observers for Takagi-Sugeno Fuzzy Systems[END_REF] the weighting functions µ i (x) and the matrices A i depend on the real scalars λ i and γ i (i = 1, 2, 3). So, for different choices of these scalars, different structures of the multiple model will be obtained. All these multiple model structures (excluding the degenerated cases λ 1 = λ 2 = λ 3 = 0 and/or γ 1 = γ 2 = γ 3 = 0) are analytically equivalent with the initial nonlinear form of the three-tank model (1).

In the next section will be discussed the conditions to be respected when choosing the real scalars λ i and γ i (i = 1, 2, 3) (and so the multiple model form) in order to design an observer for the three-tank system using the observer proposed in [START_REF] Ichalal | Design of Observers for Takagi-Sugeno Systems with Immeasurable Premise Variables: an L2 Approach[END_REF].

C. Observability criterion choice of the multiple model

In order to design an observer for a multiple model, the observability of the system is required to solve the LMI, given in [START_REF] Ichalal | Design of Observers for Takagi-Sugeno Systems with Immeasurable Premise Variables: an L2 Approach[END_REF], and obtain the gains of the observer. It is well known that the observability of each submodel is a requirement for the observability of the global multiple model. The next geometric conditions are used for the multiple model [START_REF] Takagi | Fuzzy identification of systems and its application to modeling and control[END_REF]:

rank(O i ) = rank      C CA i . . . CA n-1 i      = n, ∀i = 1, ..., 8 (17)
In conformity with (15) the matrices A i (i = 1, ..., 8) have the same structure as that of the matrix A given in [START_REF] Palm | Sliding mode observers for Takagi-Sugeno Fuzzy Systems[END_REF], with the difference that z 1 , z 2 and z 3 are replaced with constants representing the maximum or the minimum of the three premise variables (z ). Taking into account that the degenerate cases (x 1 = x 3 , x 2 = x 3 and x 1 = x 2 = x 3 = 0) are excluded, the maximum and the minimum of the premise variables are always non null. As a consequence, the eight conditions [START_REF] Theilliol | Fault diagnosis and accommodation of a three-tank system based on analytical redundancy[END_REF] are verified if the same condition is verified for the matrix A. Firstly, for the sake of simplicity, only one measure will be considered, for example x 1 , which defines the matrix C = [1 0 0]. It has to be remarked that the observer design based on a single measure may be interesting for fault diagnosis studies. The observability matrix O 1 for the matrix A has the following form:

O 1 =   C CA CA 2   =   1 0 0 -λ 1 z 1 -λ 2 z 1 -λ 3 z 1 ( * ) 1 ( * ) 2 ( * ) 3  
where * represent non null terms:

( * ) 1 = λ 2 1 z 2 1 -λ 3 z 1 (λ 1 z 1 -γ 1 z 3 ) -λ 2 z 1 γ 1 z 3 ( * ) 2 = λ 1 z 2 1 λ 2 -λ 2 z 1 (γ 2 z 3 -z 2 ) -λ 3 z 1 (λ 2 z 1 -γ 2 z 3 ) ( * ) 3 = λ 1 λ 3 z 2 1 -λ 3 z 1 (λ 3 z 1 -γ 3 z 3 ) -λ 2 z 1 γ 3 z 3 The determinant of the observability matrix O 1 is in this case det(O 1 ) = λ 2 γ 3 z 2 1 z 3 (λ 2 -λ 3 ) -λ 3 λ 2 z 2 1 (γ 2 z 3 + z 2 ) + λ 2 3 γ 2 z 2 1 z 3 .
It can be deduced that the rank of the observability matrix O 1 is equal to n = 3 if and only if λ 3 = 0 and λ 2 = 0. For λ 3 , λ 2 = 0 each submodel is observable, which is necessary for observer design. Secondly, it can be considered the case of two measured outputs. If the classical case of the three-tank system is considered, where x 1 and x 2 are the measured outputs, the matrix C has the form:

C = 1 0 0 0 1 0 (18)
The observability matrix O 1,2 (i = 1, ..., 8) has the following form:

O 1,2 =            1 0 0 0 1 0 -λ 1 z 1 -λ 2 z 2 -λ 3 z 1 γ 1 z 3 γ 2 z 3 -z 2 γ 3 z 3 λ 1 z 2 1 (λ 1 -λ 3 ) + γ 1 λ 3 z 1 z 3 ( * ) 1 ( * ) 2 ( * ) 3 ( * ) 4 ( * ) 5           
where * represent the following non null terms:

( * ) 1 = z 1 [λ 2 z 1 (λ 1 -λ 3 ) -γ 2 z 3 (λ 2 -λ 3 ) + λ 2 z 2 ] ( * ) 2 = λ 3 z 2 1 (λ 1 -λ 3 ) -γ 3 z 1 z 3 (λ 2 -λ 3 ) ( * ) 3 = λ 1 z 1 z 3 (γ 3 -γ 1 ) -γ 1 z 3 [z 2 -z 3 (γ 3 -γ 2 )] ( * ) 4 = (γ 2 z 3 -z 2 ) 2 -γ 2 γ 3 z 2 3 ( * ) 5 = γ 3 z 3 [(γ 2 -γ 3 ) z 3 -z 2 + λ 3 z 1 ] -λ 3 γ 1 z 1 z 3
In this case the multiple model structures with λ 3 = γ 3 = 0 should be avoided in order to ensure the observability condition (17) of the submodels. Similar studies can be realized for the other possible cases, with one or two measures. In the next section the observer design for the three-tank system is realized, taking into account the criterion choices of the multiple model presented in this section.

D. Observer design

The method used here to design observers for nonlinear systems is based on the multiple model form [START_REF] Takagi | Fuzzy identification of systems and its application to modeling and control[END_REF]. The case where the weighting functions µ i depend on unmeasurable premise variables (state variables) is considered. Let us consider the matrices A 0 and A i defined by:

A 0 = 1 r r ∑ i=1 A i A i = A i -A 0 (19)
By substituting A 0 and A i in the multiple model state equation [START_REF] Takagi | Fuzzy identification of systems and its application to modeling and control[END_REF], the proposed observer for the three-tank system has the following form:

ẋ(t) = A 0 x(t) + 8 ∑ i=1 µ i ( x) A i x(t) + Bu(t) + L(y(t) -ŷ(t)) ŷ(t) = C x(t) (20)
where L is a matrix to be determined. The state estimation error is given by:

e(t) = x(t) -x(t) (21)
Its dynamic is obtained using ( 21) and (20):

ė(t) = ẋ(t) -ẋ(t) = (A 0 -LC)e(t) + Δ(x, x) (22)
where:

Δ(x, x) = r ∑ i=1
A i (µ i (x(t)) x(t) -µ i ( x(t)) x(t))

+B i (µ i (x(t)) -µ i ( x(t)))u(t)]

(23) can be considered as a small perturbation if x tends toward x. In conformity with [START_REF] Ichalal | Design of Observers for Takagi-Sugeno Systems with Immeasurable Premise Variables: an L2 Approach[END_REF] it is assumed that the following conditions hold:

|µ i (x(t)) -µ i ( x(t))| < τ i |x(t) -x(t)| |µ i (x(t)) x(t) -µ i ( x(t)) x(t)| < ω i |x(t) -x(t)| |u(t)| < ϕ (24)
Since the weighting functions µ i (x) take values in the in- terval [0, 1], are continuous and derivable, and the state x is bounded, the conditions (24) are easy to check. The first condition in (24) is verified, since the weighting functions µ i (x) are globally Lipschitz. Given a function f of class C 1 , we have:

f (x) -f ( x) = x x ḟ (t)dt, (25)
Applying this property for all eight weighting functions µ i we obtain:

|µ i (x) -µ i ( x)| ≤ x x | μi (t)| dt ≤ τ i |x -x| , i = 1
, ..., 8 τ i can be easily founded, knowing the extrema of functions µ i (i = 1, ..., 8). Since µ i (x) are globally Lipschitz and x is bounded and assumed to be continuous, the second condition in (24) is verified too. Applying the same principles for f (x) = µ i (x)x, the second one is checked similarly. In this case, the term Δ(x, x) may be bounded by:

Δ(x, x) < ψ |x(t) -x(t)| (26)
where

ψ = r ∑ i=1 ( σ (A i )ω i + σ (B i )τ i ϕ) (27)
where σ (M) denotes the maximum singular value of the matrix M. In conformity with the convergence theorem presented in [START_REF] Ichalal | Design of Observers for Takagi-Sugeno Systems with Immeasurable Premise Variables: an L2 Approach[END_REF] (Lemma 1), the state estimation error between the multiple model ( 4) and the multiple observer (20) converges asymptotically towards zero, if there exist matrices P = P T > 0, Q = Q T > 0 and K such that the following condition holds:

A T 0 P + PA 0 -C T K T -KC + ψ 2 Q P P -Q < 0 (28)
The gain of the observer (20) is then computed by:

L = P -1 K
It has to be remarked that the number of LMIs does not depend on the number of submodels of the multiple model. It depend only on A 0 , the mean of these submodels (A i ). Nevertheless, A 0 and A i have the same structure, so the choice of the multiple model is important in order to find L. The simulation results are given in figure 2 for λ 1 = -0.7, λ 2 = 5, λ 3 = 17, γ 1 = 0, γ 2 = -3, γ 3 = -1 and ψ = 0.506. These parameters have to be chosen in order to respect the observability conditions, as discussed previously. Considering different initial conditions for the multiple model (x 0 = [ 1.8; 1.7; 5.5 ]) and for the observer ( x0 = [ 4.8; 5; 2.1 ]), a fast convergence of the state estimation error towards zero is obtained. The estimation results are correct (see figure 2), although a noise measurement has been added to the two outputs, as can be seen in figure 3. The LMI condition (28) holds and the observer gain is:

L =   1

III. CONCLUSIONS AND FUTURE WORKS

This paper proposes an equivalent rewriting of nonlinear systems into a multiple model form with unmeasurable premise variables in order to design a multi-observer and reconstruct the state of this system. The transformation method is applied to the three-tank system and gives very good results concerning the state reconstruction. The interest to use the multiple model structures is already well known, the main advantage is the ability to extend the tools of the linear theory to nonlinear systems. Other advantages and good properties of the multiple model are, firstly, the possibility to reduce the complexity of nonlinear systems, by constructing linear models that will be aggregated using weighting functions and, secondly, the fact that these aggregating functions possess the convexity property. The multiple model form developed here uses unmeasurable premise variables, the states of the system. In this way it covers a larger class of nonlinear systems than those using measurable premise variables and allows to develop only one model of the system behavior to detect and isolate actuator and sensor faults. In order to obtain the multiple model structure, a method with no information loss is used. The initial nonlinear system is rewritten into an equivalent multiple model form. This method gives the possibility to determine between the different possible multiple models the one that is observable and for which the observer design is possible. These conditions are less restrictive than those proposed in the literature with regard to the Lipschitz constants. In addition, the number of LMIs to solve is restrained and does not depend on the number of submodels of the multiple model. Using the method presented in this paper, it is envisaged to extend the study to fault detection and diagnosis.

Fig. 1 .

 1 Fig. 1. The three-tank system dynamic mathematical model is based on Bernoulli's law for liquids and on mass conservation law:

3 Fig. 2 .

 32 Fig. 2. State estimation for the three-tank system

Fig. 3 .

 3 Fig. 3. Measured and reconstructed outputs

IV. ACKNOWLEDGMENTS

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


Anca Maria Nagy

Gilles Mourot

Benoît Marx

Georges Schutz

José Ragot

Model structure simplification of a biological reactor

Keywords: dynamic modeling, reduction, weighting functions, subsystems, wastewater treatment

This article proposes an analytical method for decomposing a dynamic nonlinear system into a multiple model form in order to reduce its complexity and to study more easily identification, stability analysis and controller design problems. The majority of existing methods are order reduction based techniques, which come with an information loss of the initial system, whereas the method proposed here avoids this particular loss. The multiple model constitutes an efficient tool to represent nonlinear systems. These are decomposed into several linear time invariant systems (LTI) which are weighted and aggregated that allows to benefit from important analysis tools. This method is applied to a simplified activated sludge reactor model.

INTRODUCTION

The complexity problem of dynamic nonlinear systems appears in a great number of scientific and engineering domains. Some decomposition and simplification techniques were developed in the last years, in order to realize a complexity reduction, according to objectives like identification, controller design, stability analysis. In most studies, the complexity reduction consists in the reduction of the system order that comes with an information loss [START_REF] Dolgin | Model reduction of uncertain systems retaining the uncertainty structure[END_REF], [START_REF] Petzold | Model reduction for chemical kinetics: an optimization ppproach[END_REF], [START_REF] Sayesel | Model simplification and validation with indirect structure validity tests[END_REF]). Another way to reduce the complexity of dynamic nonlinear systems is to rewrite the nonlinear system in a much easier way, by decomposing it into much simpler unities, without loosing information. The multiple model (MM) representation (Murray-Smith and Johansen [1997]) constitutes actually a commonly used tool for nonlinear system modeling. The MM formalism is based on a decomposition of the system dynamic behavior into several operating zones, each zone being characterized by a submodel. Depending on the system evolution in each zone, each submodel contributes more or less to the approximation of the global system behavior. The contribution of each submodel in the global model, which is a convex combination of submodels, is defined by a weighting function. In the literature several equivalent terminologies are used to define this class of models: the MM (Murray-Smith and Johansen [1997]), 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, Modelling and Control with Polytopic Linear Models[END_REF]). The convenience of using this type of models is that important properties like stability, controllability, observability, extensively used in the case of linear time-invariant sys-tems (LTI) can be generalized, at least partially, to MM if the constitutive submodels are linear and for particular weighting functions [START_REF] Akhenak | Estimation of state and unknown inputs of a nonlinear system represented by a multiple model[END_REF]). The MM structure is then used here to reduce the complexity of a simplified ASM1 model (Activate Sludge Model no.1) [START_REF] Olsson | Wastewater Treatment Systems. Modelling, Diagnosis and Control[END_REF]) which describes a biological degradation process of an activate sludge reactor. Different linearization techniques were proposed, like a linearization around one or several operating points [START_REF] Smets | A linear ASM1 based multi-model for activated sludge systems[END_REF]) in order to obtain a multiple model. The loss of information constitutes the first drawback of this technique. Secondly, the choice of premise variables expressing the nonlinearities of the system, just as the choice of different operating points, still remain very delicate. Afterwards, we will present a systematic procedure to transform a nonlinear system by rewriting it into a MM form, avoiding the major drawbacks: the transformation is realized without loss of information, the choice of different operating points is no longer necessary and the choice of premise variables is realized in a more precise way. Starting with a general form of nonlinear system with bounded nonlinearities, a MM state representation is realized. This MM representation constitutes a quasi linear parameter varying (quasi-LPV) system because the convex combinations of constant matrices calculated from the polytope vertices give use to matrices with variable parameters. The vertices are obtained using the convex polytopic transformation (CPT) [START_REF] Wang | An approach to fuzzy control of nonlinear systems: stability and design issues[END_REF]). The constant matrices define the submodels and the nonlinearities are rejected into the submodel weighting functions. The MM obtained with this method is not unique, it depends on the choice of the scalars defining the CPT and on the quasi-LPV model structure. This method and its characteristics will be pointed out by accentuating on biological reactor model.

METHODOLOGY

This section is dedicated to a general methodology to transform a nonlinear system presented in a general form into a multiple model. The proposed method is totally analytical and the obtained MM is equivalent to the initial nonlinear system i.e. the two representations give the same responses. This will be illustrated on a simplified activated sludge reactor model. Considering a nonlinear system with bounded nonlinearities, the MM form is written. In general, for a nonlinear dynamic system, this representation is not unique, that is the reason why different criteria of choice for a MM form have to be defined to match the study objectives. To each MM form corresponds a premise variable set. Each premise variable will be partitioned in two parts using the convex polytopic transformation [START_REF] Morère | Mise en oeuvre de lois de commandes pour les modèles flous de type Takagi-Sugeno[END_REF]). The two partitions of each premise variable will contribute to the construction of the corresponding weighting functions of the MM. The multiple model will thus be a convex combination of linear submodels [START_REF] Murray-Smith | Multiple model approaches to modelling and control[END_REF]), the nonlinearity being transferred into weighting functions related to each submodel.

Analytical method of structure simplification

The activated sludge reactor model, like a large category of nonlinear dynamic systems, can be represented by:

ẋ(t) = f (x(t), u(t)) y(t) = g(x(t), u(t)) (1)
where

f (x (•) , u (•)) ∈ R n and g (x (•) , u (•)) ∈ R l .
To reduce the complexity of nonlinear dynamic systems (1), a system decomposition into a multiple model form is realized. This type of model allows us to represent nonlinear dynamic systems into a convex combination of linear submodels:

         ẋ(t) = r i=1 µ i (x, u) [A i x(t) + B i u(t)] y(t) = r i=1 µ i (x, u) [C i x(t) + D i u(t)] (2)
x ∈ R n is the state vector, u ∈ R m is the input vector and y ∈ R l the output vector. A i , B i , C i , D i are constant matrices of appropriate dimensions. The function µ i (x, u) represents the weight of the i th submodel

{(A i , B i , C i , D i }.
The weighting functions µ i (x, u) have the following property:

r i=1 µ i (x, u) = 1 and µ i (x, u) ≥ 0, ∀(x, u) ∈ R n × R m
In order to obtain the multiple model form, first the system will be transformed in a quasi-LPV form. This is an intermediate form which allows to obtain the multiple model structure. 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 (1) can be represented in a quasi-LPV form:

ẋ(t) = A(x(t), u(t)) x(t) + B(x(t), u(t)) u(t) y(t) = C(x(t), u(t)) x(t) + D(x(t), u(t)) u(t) (3)
where

A (x, u) ∈ R n×n , B (x, u) ∈ R n×m , C (x, u) ∈ R l×n , D (x, u) ∈ R l×m .
Let us denote a i,j (x, u) (resp. b i,j (x, u), c i,j (x, u) and d i,j (x, u)) the entry of A(x, u) (respectively B(x, u), C(x, u) and D(x, u)) lying on the i th row and the j th column.

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

A(x, u) =    a 1,1 (x, u) • • • a 1,n (x, u) . . . a n,1 (x, u) • • • a n,n (x, u)    (4)
B(x, u), C(x, u) and D(x, u) are constructed in the same way with appropriate dimensions.

Considering the quasi-LPV form resulting from system (1) and given in an explicit way in (3), we define the premise variable set V z in the following way:

V z = a i,q (x, u) a i,q = const, i = 1, n, q = 1, n ∪ b i,s (x, u) b i,s = const, i = 1, n, s = 1, m ∪ c j,q (x, u) c j,q = const, j = 1, l, q = 1, n ∪ d j,s (x, u) d j,s = const, j = 1, l, s = 1, m
where 1, n = {1, ..., n} and when a •,• = const. means that a •,• does not reduce to a constant value. In fact every time varying entry of the matrices A(x, u), B(x, u), C(x, u) and D(x, u) will be considered as a decision variable, denoted z i (x, u). Subsequently, this set (V z ) is noted more simply:

V z = {z 1 (x, u), . . . , z p (x, u)}
where p (verifying p ≤ (n + l)(n + m)) denotes the number of premise variables and is the number of nonlinearities identified using the quasi-LPV form of the system (1). z 1 (x, u), . . ., z p (x, u) are the premise variables. Remark 1. (n+l)×(n+m) represents the largest number of premise variables. However, for practical applications the number of premise variables is less that this bound.

In general, the quasi-LPV form (3) for a nonlinear system of the form (1) is not unique; for each quasi-LPV form exist a particular premise variable set; choosing a quasi-LPV form is equivalent to choose the premise variable set. The choice of the premise variable set V z is important, because it has an influence on the submodel numbers and on the global model structure. This is a degree of freedom that should be used to ease the controllability, the observability and the stability analysis studies. First of all, the controllability/observability of each submodel is necessary to ensure the controllability/observability of the global system, represented in a multiple model form [START_REF] Guerra | Control law proposition for the stabilisation of discrete takagisugeno models[END_REF]). Thus, the quasi-LPV forms producing submodels which are not controllable/observable will not be considered. However it should be pointed out 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]). Secondly, in the stability analysis study of multiple model, different techniques are proposed to reduce the number of LMI (Linear Matrix Inequality) conditions, which extend the solution existence [START_REF] Bergsten | Observers for Takagi-Sugeno fuzzy systems[END_REF]) and leads to less computational requirement [START_REF] Tanaka | A descriptor system approach to fuzzy control system design via fuzzy lyapunov functions[END_REF]). Due to the convexity of the interpolation made between the submodels, the LMI are verified for all the vertices. 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 it is tried to find a multiple model composed by a minimal number of submodels. This number is related to the number of premise variables, so that a quasi-LPV form which has a set of minimal number of premise variables will be preferred. Besides that, the complexity of the LMI conditions increases with the number of state variables involved. In order to ensure the solution existence, the dimension of the state vector should be made as small as possible [START_REF] Bergsten | Observers for Takagi-Sugeno 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 simplest model:

• Eliminate all quasi-LPV forms for which the matrices have null columns (B) and/or rows (C). • Identify the decompositions that contain common premise variables for different state and/or output equations (3), reducing in this way the number of premise variables. • Among quasi-LPV forms for which the premise variables z i (x, u) (i = 1, ..., p) depend on the same number of state variables, choose the quasi-LPV form that contains the minimal number of premise variables z i (x, u) (taking into account the next rule). • Among quasi-LPV forms that have the same number of premise variables z i (x, u) (i = 1, ..., p), choose the quasi-LPV form for which the premise variables z i (x, u) depend on a minimal number of state variables.

Let us introduce a notion used here after: the 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

F i : [x 0 , x 1 ] × [u 0 , u 1 ] -→ [0, 1] (i = 1, 2) (x(t), u(t)) -→ F i (x(t), u(t))
with F 1 (x(t), u(t)) + F 2 (x(t), u(t)) = 1 such as:

h(x(t), u(t)) = F 1 (x(t), u(t)) • h 1 + F 2 (x(t), u(t)) • h 2 for all h 1 ≥ max x,u {h(x, u)} and h 2 ≤ min x,u {h(x, u)}
The two functions F 1 and F 2 are defined as:

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
Let us note that this decomposition is not unique.

After the choice of a quasi-LPV form and of the premise variables (using the rules mentioned above), a convex polytopic transformation for the p premise variables is performed.

z j (x, u) = F j,1 (z j ) • z j,1 + F j,2 (z j ) • z j,2 (5)
F j,1 (z 1 ) = z j (x, u) -z j,2 z j,1 -z j,2 F j,2 (z 1 ) = z j,1 -z j (x, u) z j,1 -z j,2 (6)
z j,1 = max x,u {z j (x, u)} z j,2 = min x,u {z j (x, u)} (7)
Using the convex polytopic transformation (Lemma 1), we generate the 2 p submodels of a MM characterized by p premise variables. Each premise variable is thus described like in (5) with the partitions (6), and the scalars z j,1 and z j,2 defined in ( 7), for all j = 1, ..., p.

To each submodel i corresponds a p-uplet σ i which codes the partitions of the premise variables occuring in the corresponding weighting function, just like in table 1, where the case of three premise variables is illustrated.

Considering the i th row, the entries equal to 1 indicate the partition of each premise variable, intervening in the i th weighting function.. Multiplying the functions describing these partitions, we obtain the corresponding weighting function µ i (z):

µ i (z) = µ i (x, u) = p k=1 F k,σ k i (z k (x, u)) (8)
where σ k i represents the index in the k th position in the p-uplet σ i . Then r i=1 µ i (z) = 1 and µ i (z) ≥ 0(i = 1, ..., r), since:

F j,1 (z j (x, u)) + F j,2 (z j (x, u)) = 1, ∀j = 1, ..., p
Remark 2.The number of submodels that constitute the multiple model is r = 2 p ≤ 2 (n+l)(n+m) .

The matrices with variable parameters intervening in the quasi-LPV form of system (1) are linear combinations of constant matrices (A j , B j , C j , D j ); for example, the matrix A(x, u) may be expressed in the following way:

A(x, u) = A 0 + j∈IA z j (x, u) • A j (9)
where the set I A contains all the indexes 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 ) of the matrix A(x, u). The matrix A j contains at the position corresponding to z j the constant term 1, and zero for all the other positions.

In the matrices A(x, u), B(x, u), C(x, u) and D(x, u), written like in (9), call upon the premise variables z j for j = 1, ..., p; in this way, the matrices A, B, C and D will be evaluated at the vertices of the polytope defined by the premise variable partitions that intervene in these matrices for determining the matrices A i , B i , C i and D i (i = 1, ..., r), that are constant coefficients matrices relatives to each submodel:

A i = A 0 + j∈IA z j,σ j i • A j (10)
B i , C i and D i can be written in the same way.

We can give the following result:

The multiple model (2) which represents a convex combination of linear submodels {(A i , B i , C i , D i )} (i = 1, ..., r) is equivalent to the system (1).

For reasons of space the proof of this theorem can not be developed here. We can notice that obtaining a MM with this method generally needs neither to make some approximation nor to choose linearization points. This method is illustrated in the next session with a biological reactor model.

THE ACTIVATED SLUDGE REACTOR MODELING

The biological reactor model

In this section, some general considerations are given concerning an activated sludge reactor model, the reduced ASM1 model: the carbonated pollution part of ASM1 model will be considered.

The activated sludge wastewater treatment consists in mixing used waters with a rich mixture of bacteria in order to degrade the organic matter. In the activated sludge reactor study, several operating modes can be considered: reactor with constant volume, regulation from a reference volume, free flow at the output reactor, controllable input and output flows. The first situation is reported here.

The elimination process of carbonated pollution is realized in the bioreactor-clarifier group. The simplified diagram is given in figure 1. The reactor-clarifier group can be represented by the following nonlinear system:

                                         dV (t) dt = q in (t) + q R (t) -q out (t) d(V (t) • X BH (t)) dt = q in (t)X BH,in (t) + q R (t)X BH,R (t) -q out (t)X BH,out (t) + r H (t)V (t) d(V (t) • S S (t)) dt = q in (t)S S,in (t) + q R (t)S S,R (t) -q out (t)S S,out (t) + r S (t)V (t) d(V (t) • S O (t)) dt = q in (t)S O,in (t) + q R (t)S O,R (t) -q out (t)S O,out (t) + r O (t)V (t) + Kq a (t)V (t)(S O,sat -S O (t)) (11)
where the variables involved are: V the reactor volume, X BH the heterotrophic biomass concentration, X BH,R the recycled one, S S the fast biodegradable substrate concentration, S O the dissolved oxygen concentration, q the flow and q a the air flow. The in and out indexes correspond respectively to the reactor input and output, q R , q W are respectively the recycled and the rejected flow. r H , r S and r O representing the kinetics of heterotrophic biomass X BH , carbon S S and oxygen S O reactions in a heterotroph and aerobic environment are modeled by:

                   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 (12)
The time dependence of different variables will not be explicitly mentioned in the future equations, in order to facilitate the writing.

The different coefficients intervening in ( 11) and ( 12) are: S O,sat = 10 [g/m 3 ] (oxygen saturation concentration), K S = 20 [g/m 3 ] and K OH = 0.2 [g/m 3 ] (half-saturation constants), Y H = 0.6 (g cell COD formed) and f = 0.1 (conversion rate coefficients for the products in different reactional processes (increase or mortality)), b H = 0.4 [1/24h] and µ H = 3.733 [1/24h] (heterotrophic kinetic coefficients), K = 2.3 [1/m 3 ] (regulator gain in oxygen).

The clarifier is supposed to be perfect with no internal dynamic process and no biomass in the effluent. In this case, we can write:

(q in + q R )X BH = (q R + q W )X BH,R S S,R = S S (13)
Under the reactor homogeneity hypothesis, the following assumptions are made:

X BH,out = X BH S S,out = S S S O,out = S O
It is supposed that the dissolved oxygen concentration at the reactor input (S O,in ) is null.

In general, q R and q W represent fractions of input flow q in :

q R = f R q in , 1 ≤ f R ≤ 2 q W = f W q in , 0 < f W < 1
The reactor volume is assumed to be constant and thus: q out = q in + q R . Replacing r H , r S and r O given in ( 12) and taking into account the above assumptions (by eliminating the first equation which corresponds to the volume variation) then the system (11) becomes:

                                     ẊBH = 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 ṠS = 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 ṠO = - 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 (14)

The multiple model for the biological reactor

To obtain the multiple model form, in a first step, the system (14) has to be transformed into a quasi-LPV form, just as in (3). Taking into account the rules mentioned in the previous section, the chosen quasi-LPV form is represented by:

A(z 1 , z 2 , z 3 ) = a 1,1 0 0 a 2,1 a 2,2 0 a 3,1 0 a 3,3 x = X BH S S S O B(z 2 ) = b 1,1 0 0 0 b 2,2 0 0 0 b 3,3 u = X BH,in S S,in q a ( 15
)
where

a 1,1 = µ H • z 1 - f W (1 + f R ) f W + f R • z 2 -b H a 2,1 = - 1 Y H µ H • z 1 + (1 -f )b H a 2,2 = -z 2 a 3,1 = Y H -1 Y H µ H • z 1 a 3,3 = -K • z 3 -z 2 b 1,1 = b 2,2 = z 2 b 3,3 = K • S O,sat (16)
The system presents three nonlinearities and we have chosen as premise variables:

         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 (17)
Remark. The flow q in is an input for the biological reactor and is not integrated in the quasi-LPV form (15) because this will generate a null column in the matrix B, which is not a desired structure for controllability studies. In a second step, using the transformation presented in Lemma 1, the three premise variables z 1 , z 2 and z 3 are written like in (5), where each premise variable is constructed as a convex sum (6) of scalars (7)( j = 1, ..., 3). The maxima and the minima in (7) are calculated using the fact that V , S S , S O , q a and q in are bounded. In table 1, are generated all the submodels of a multiple model characterized by three premise variables z j (x, u) (j = 1, ..., 3) which are partitioned into two parts, the number of submodels being equal to 2 3 . To each submodel i corresponds a triplet σ i which codes the partitions of the premise variables z j (x, u) (j = 1, ..., 3) intervening in the corresponding weighting function. In this way, in the table 1, considering the i th row, the entries equal to 1 indicate the partition of each premise variable, intervening in the weighting function µ i (x):

µ 1 (x) = F 1,1 (z 1 ) F 2,1 (z 2 ) F 3,1 (z 3 ) µ 2 (x) = F 1,1 (z 1 ) F 2,1 (z 2 ) F 3,2 (z 3 ) µ 3 (x) = F 1,1 (z 1 ) F 2,2 (z 2 ) F 3,1 (z 3 ) µ 4 (x) = F 1,1 (z 1 ) F 2,2 (z 2 ) F 3,2 (z 3 ) µ 5 (x) = F 1,2 (z 1 ) F 2,1 (z 2 ) F 3,1 (z 3 ) µ 6 (x) = F 1,2 (z 1 ) F 2,1 (z 2 ) F 3,2 (z 3 ) µ 7 (x) = F 1,2 (z 1 ) F 2,2 (z 2 ) F 3,1 (z 3 ) µ 8 (x) = F 1,2 (z 1 ) F 2,2 (z 2 ) F 3,2 (z 3 )
To express the constant coefficient matrices A i and B i of (2), characterizing each submodel i (i = 1, ..., 8) we use the matrices defined in (15). Let us note that the premise variables z 1 , z 2 and z 3 (resp. z 2 ) are involved in A(x, u) (resp. B(x, u)). In this way, we will evaluate the matrices A and B at the vertices of the polytope defined by the partitions of premise variables which occur in these matrices. For example, to determine the matrices A 3 and B 3 , we proceed in the following way. The triplet corresponding to the third line is retained σ 3 = (1, 2, 1), which has as components σ k 3 (k = 1, ..., 3) that can take the values 1 or 2. Taking into account ( 15), ( 16) and ( 17),

A 3 = A(z 1,σ 1 3 , z 2,σ 2 3 , z 3,σ 3 3 ) and B 3 = B(z 2,σ 2
3 ). In a more general way A i and B i (i = 1, ..., 8) are noted:

A i = A(z 1,σ 1 i , z 2,σ 2 i , z 3,σ 3 i ) B i = B(z 2,σ 2
i ) The multiple model is obtained by an interpolation of the eight previous submodels, just as in (2) (r = 8). In figure 2 we can see the evolution of the system (14) (represented in the form (2)) and the weighting functions of the multiple model obtained for f R = 1.0, f W = 0.037, with the initial conditions: (V (0), X BH (0), S S (0), S O (0)) T = (1250, 887, 4.1, 1.0) T . The volume is expressed in m 3 . For space and clarity reasons only four weighting functions are presented in the figure (µ 5 , µ 6 , µ 7 , µ 8 ). It has to be pointed out that the evolution of the weighting functions depends on the control input evolution.

CONCLUSION

This article proposes a general analytical method to simplify the complexity of a nonlinear system. 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. Let us insist to the fact that in order to obtain a multiple model, different techniques of linearization applied around different well chosen operating points were proposed in the literature. The major drawback is that these techniques imply a loss of information. Besides, the difficulty of choosing these different operating points is the reason why this article proposes a more general method to rewrite the nonlinear model into a multiple model form avoiding the need to choose these operating points and avoiding loss of information.

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 has the same response that those of the initial system. Another important aspect concerning this decomposition method is the given possibility to reduce the number of submodels by an adequate choice of premise variables between several sets of candidate premise variables. This method is applied to a simplified activated sludge reactor model and leads to a multiple model structure based on eight submodels. Even if in this article we have treated the case of a reduced ASM1 model, considering only the carbon pollution, the proposed method can be expanded to the complete ASM1 model, the difficulties being similar.

z 2 z 3 σ i i F 1,1 F 1,2 F 2,1 F 2,2 F 3,1 F 3,2 1 1 0 1 0 1 0 (1,

Fig. 1 .

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

Fig. 2 .

 2 Fig. 2. The evolution of system (14) presented into a multiple model form and some weighting functions of the multiple model

Table 1 .

 1 Decomposition table for a multiple model with three premise variables and two partitions for each variable

	1,1)

ACKNOWLEDGMENT

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