Soliman El Kabir

Frédéric Dubois

Rostand Moutou Pitti

Naman Recho

Yuri Lapusta

A new path-independent integral generalized for three-dimensional crack problem: analytical and numerical results

Keywords: Crack problem, Energy release rate, Path-independent integral, J integral, Three dimensional

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

Abstract :

The complex mechanical loading and high climatic variations on structures implies having a better understanding of their fracture mechanical behavior by considering humidity heterogeneity. In this case, it is necessary to consider three-dimensional case in the study of reel crack growth problems.

For it the most common approach is the determination of the energy release rate by energetic method. Most of the studies carried out deal with two-dimensional case. The studies of complex structures request the development of specific tools for three-dimensional configurations. This paper present a new analytical formulation and its numerical application for three-dimensional crack growth problem. This work is based on a generalization of the Rice's integral for threedimensional crack problem. A new integral parameter in real three-dimensional case, which computes the energy release rate combining an arbitrary crack front, is developed. A physical interpretation allows to evaluate the efficiency of the proposed integral. The new integral is compared with the three-dimensional Bui's integral. Applying the theta method a integral, generalized to a volume domain, is implemented into a finite element software. The non-path dependence is proved with the use of numerical application. The energy release rate distribution along the crack front line is obtained, and compared to Bui's integral.

A numerical validation, in terms of energy release rate, is carried out on a DCB (Double Cantilever Beam) specimen under opening mode loading for wood material. Various visions are also proposed to evaluate integral parameter and the energy release rate distribution along the crack front line.

Introduction

The complex mechanical loading and high climatic variations on structures implies having a better understanding of their fracture mechanical behavior. According a no heterogeneity of mechanical fields in the cross section, it is necessary to consider three-dimensional case in the study of reel crack growth problem. For it the most common approach is the determination of the energy release rate by energetic approach. Most of the studies carried out deal with two-dimensional case. The studies of massive elements request the development of specific tools for three-dimensional configurations.

In this context, two approaches can be developed [START_REF] Li | A comparison of methods for calculating energy release rates[END_REF]. The first one is based on the subdivision of a three-dimensional problem by some two-dimensional ones [START_REF] Li | A comparison of methods for calculating energy release rates[END_REF][START_REF] Bakker | On the numerical evaluation of J in three dimensions[END_REF][START_REF] Amestoy | On the definition of local path independent integrals in three dimensional[END_REF]. This approach doesn't need much more effort because all tools developed for two-dimensional problems can be easily reused. However, even if this technique can include a non-linear crack front, it cannot allow considering an external out-of-plane loading inducing, for instance, torsion effects.

The second approach is to rewrite the J-integral formalism, starting from the very beginning with a three-dimensional analysis. The topic of this paper deals with the generalisation of the Jintegral formalism to a three-dimensional problem and its adaptability of the G-theta method for a finite element implementation.

-integral description

For plan problems and for static cracks, Rice (1968) [START_REF] Rice | A path independent integral and the approximate analysis of strain concentrations by notches and cracks[END_REF] has defined a path independent integral which allows to computes energy release rate around the crack tip. For cracked linear elastic material, Rice [START_REF] Rice | A path independent integral and the approximate analysis of strain concentrations by notches and cracks[END_REF] have used J-integral to compute energy release rate for curvilinear contour. -integral takes the following definition :

(1)

Where denotes the strain energy density, is arbitrary curvilinear contour oriented by its normal vector, is the displacement component and is the stress component.

Fig. 1: Description integral domain of J and For three-dimensional problem, the crack front line is defined as the intersection of two surface (figure 2).

Fig. 2: Crack surface with crack system coordinate

To take into account in global formalism the most general case in which the crack front line is arbitrary, we consider some volumes and surfaces integration domains as described in Fig. 3.

Fig. 3: Closed volume and surfaces integration domains

The new -integral formulation is based on an energetic approach using consideration described in figure 3. After some mathematical developments, we obtain:

(2) the J 3D -integral is composed by three separated terms. The first one designates the classical part used for the determination of the crack growth initiation. It can be completed by the effects of a crack lips pressure introduced by the second term. The last part allows the generalization for the crack propagation ensuring the non-path dependence when the crack tip moves inside the integral domain. The is interpreted as the integration of the -integral along the crack front line:

(3) denotes the Bui's integral for three-dimensional crack problem. The -s s line:

(4)

To implement this integral in a finite element software, it is easier to consider a volume domain integral [START_REF] El Kabir | Numerical study of crack path by MMCG specimen using M integral[END_REF][START_REF]M w s s m θ_integral[END_REF].

(

The allows to compute the distribution of energy release rate along the crack front line per slim thickness. We can establish a relation between and as follows: (6) The average energy release rate per every slim thickness can be equal to energy release rate per each plan in studied solid.

Finite element analysis

The numerical implementation is based on a Double Cantilever Beam (DCB) loaded in an open mode.

The DCB specimen was adapted by Dubois et al to wood material. In this part, we recall the wood specimen dimensions of DCB device. Fig. 4 presents the dimensions in millimetres of the initial wood specimen. In this wood specimen, two holes are machined in order to fixe the Arcan device. This allows a loading fixations in tensile mode. The geometry of the DCB specimen has been optimized by using a finite element computation. This specimen is adapted to obtain a stable crack growth rate during propagation for tensile mode. The finite element computation is realized for an elastic isotropic behavior. Wood material used is Douglas fire and has the following elastic characteristics E=14100 MPa, Poisson ratio NU=0.3. The initial crack length is fixed to 60mm.

In the follows, the results of numerical study are exposed. In order to observe the effect of thicknesses on the DCB specimen we plot the evolution of the energy release rate as function of the crack front line.

The description of field around the crack front line is shown in Fig. 4. The field is equal to zero on outside surface, and 1 on inside surface. The average energy release rate is calculated along (Fig. 6). Fig. 7 shows us the energy release rate distribution along the crack front line versus the thickness.

Conclusions and outlooks

This paper deals with a new formulation of the J-integral for the study of fracture process in element by taking into account three dimensional effects. A theoretical and numerical approach are established. At this stapes, more numerical investigations are necessary. In this case, the integral is generalized to three-dimensional approach using the Lagrangian variation. Also, it will be necessary to extend the integral to a mixed mode loading case [START_REF] Pitti | A new integral parameter for mixed-mode crack growth in viscoelastic orthotropic media[END_REF] in order to introduce hydrological and thermal effects for three dimensional problems [8].

Fig. 4

 4 Fig. 4: DCB specimen

Fig. 5 :Fig. 6 :

 56 Fig. 5: Description of field around the crack front line

Fig. 7 :

 7 Fig. 7: Energy release rate distribution along the crack front line

  . Université de Poitiers, Institut PPRIME, Bd Marie et Pierre Curie, 86962 Chasseneuil Cedex, France + soliman.elkabir@yahoo.fr b. Université de Limoges, GC2D, 19300 Egletons, France + frederic.dubois@unilim.fr c. Université Clermont Auvergne, CNRS, Institut Pascal, BP 10448, 63000 Clermont-Ferrand, France + rostand.moutou_pitti@uca.fr d. Université Clermont Auvergne, CNRS, Institut Pascal, BP 10448, 63000 Clermont-Ferrand, France + naman.recho@moniut.univ-bpclermont.fr e. French Institute of Advanced Mechanics, Institut Pascal, 63171 Aubière, France + yuri.lapusta@sigma-clermont.fr

	A NEW PATH-INDEPENDENT INTEGRAL
	GENERALIZED FOR THREE-DIMENSIONAL
	CRACK PROBLEM: ANALYTICAL AND
	NUMERICAL RESULTS
	S. EL KABIR a , F. DUBOIS b , R. MOUTOU PITTI c , N. RECHO d , Y.
	LAPUSTA e ,
	Résumé :
	est développée grâce
	à la méthode du champ et implémenté dans un logiciel d'éléments finis. Pour cela l'éprouvette appelée DCB (Double Cantilever Beam) est utilisée. Cette éprouvette permet d'observer une plage importante de stabilité le long de la fissure en mode d'ouverture (mode I). Le travail réalisé est une étude numérique appliquée à l'éprouvette DCB afin de valider cette nouvelle intégrale et de vérifier son indépendance vis-à-vis du domaine d'intégration. Le taux de restitution d'énergie est calculé le long du front de la fissure. L'intégrale J tridimensionnelle est comparé à l'intégrale de Bui.

aPrédire le comportement des structures face à la fissuration dans le cas des chargements mécanique sous environnements variables, implique de considérer une répartition tridimensionnelle des champs hydrique et nécessite donc de généraliser une approche tridimensionnelle de la mécanique de la rupture pour évaluer les risques de fissuration dans les sections transversales. L'objectif principal de ce travail est de présenter une nouvelle intégrale invariante adaptée aux problèmes de fissurations dans le cas tridimensionnel. Ce travail est basé sur le développement numérique de l'intégrale J pour des problèmes tridimensionnel. L'intégrale


Emna Sghaier
email: emna.sghaier@insa-lyon.fr

Adeline Bourdon

Didier Rémond

Jean-Luc Dion

Nicolas Peyret

D Remond

J-L Dion

A novel identification tool for rotors working under non-stationary conditions based on the IAS

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

Introduction

Rotor dynamics are widely present in different engineering fields. In the automotive field for example, with the growing use of the electric motors rotating at very high speeds, it has become crucial to understand the dynamic behavior of rotors spinning at very high velocity especially when going through critical speeds. This critical working condition, as well as the presence of bearings or gears in the rotor structure, leads to a fluctuation of the instantaneous angular speed (IAS). However, conventional vibration-monitoring techniques are based on some assumptions on the IAS. The latter is often considered either constant or following a perfect linear or exponential law of variation [START_REF] Lalanne | Rotordynamics prediction in engineering[END_REF]. Those assumptions lead to models which are not able to simulate rotating machinery under fluctuating load conditions. Even though the major part of the scientific works on rotor dynamics focus on the study of the stationary working conditions, the transient response analysis is considered as the most inclusive analysis since it allows the study of different working conditions of the rotating machinery as the start up and shut down processes, stability, and impact [START_REF] Kirk | Transient response of rotor-bearing systems[END_REF]. Since many rotors operate above their critical speeds, a main interest was particularly given to the study of the behavior of rotating machinary when going through one or more critical speeds. Al-Bedoor [START_REF] Al-Bedoor | Transient torsional and lateral vibrations of unbalanced rotors with rotor-to-stator rubbing[END_REF] developed a model for the coupled torsional and lateral vibrations of unbalanced rotors that accounts for the rotor-to-stator rubbing. He showed that a split in resonance is observed due to the rubbing condition when the rotor torsional flexibility is considered. Zhou et Shi [START_REF] Zhou | The analytical imbalance response of jeffcott rotor during acceleration[END_REF] gave an analytical solution for the unbalance response of the Jeffcott rotor during acceleration.They showed that, quantitatively, the motion consists of three parts: a transient vibration at damped natural frequency, a synchronous vibration with the frequency of instantaneous 'synchronous' frequency, and a suddenly occurring vibration at damped natural frequency. Li and Singh [START_REF] Li | Analysis of transient amplification for a torsional system passing through resonance[END_REF] studied also the analytical solutions of the transient response but focused on the envelopes of the lateral displacement, velocity and acceleration of a linear torsional oscillator excited by an instantaneous sinusoidal torque. They also gave a new analytical approximation to find the maximum amplifications and corresponding peak frequencies.

When the dynamic model takes as an input the induced driving torque and not the speed of rotation as a function of time, one must add an equation that describes how the energy source supplies the energy to the equations that govern the corresponding dynamical system. Mutual interaction between the driving source and the rotor motion is observed and we deal in this case with non-ideal problem of passage through resonance or what is also called sommerfeld effect [START_REF] Sommerfeld | Beiträge zum dynamischen ausbau der festigkeitslehe[END_REF]. This interaction manifests itself as a modification of the motor frequency or regime of operation near the resonance and changes in the stable-unstable portions of the dynamical system response [START_REF] Manoel Balthazar | An overview on non-ideal vibrations[END_REF].

In the following study, since we consider as a given of the dynamic problem the induced torque and as an unknown the speed of rotation (IAS), we will be dealing with a non-ideal system. A new dynamic model is then built to simulate the non-stationnary working conditions especially when going through one or more critical speeds. This model is developeed in order to serve as a tool for identification, based on the IAS, for certain parameters of the system as the mass unbalance and damping.

New model describing the rotordynamics

We consider a simple idealized rotor made of a flexible shaft, a rigid disk and supported by flexible damped bearings. The main excitation of the rotor is the mass unbalance (fig. 3). In our approach we consider that a torque is induced to the rotating machinary and we make no preliminary assumption on the rotating speed. The angular displacement about the X and Y axis as well as the spinning angle about the Z axis are calculated using Euler angles as follows:

θ x = θ.cos(φ) (1a) θ y = θ.sin(φ) (1b)
θ z = φ + ψ (2)
It is very important to notice here that θ z accounts for both the free body rotating motion as well as for the torsional deformation.

T = T D + T s + T u (3)
The finite element method is used and the dsiplacement fields are approximated using shape functions. We denote by {δ} the nodal displacement vector. Then the Lagrange equations are applied, which leads to the following matrix form:

d dt ∂T ∂ δ - ∂T ∂δ = ([M Dconst ] + [M Dvar ({δ c })]) δ + C D ( θz ) δ + [M u (θ z )] δ + {F nlu (θ z )} + [M s ] { δ} + [S gyr ({δ})]{ δ} + {F nl gyr ( {δ})} (4)
where:

• [M Dconst ] and [M s ] are the classical mass matrices of the disk and the shaft respectively.

• C D ( θzc ) is the classical skew-matrix related to the gyroscopic effects of the disk.

• {F nlu (θ zc )} is the centrifugal force resulting from the mass unbalance and proportional to the square of the spinning speed.

The two matrices [M Dvar ] and [M u ] respectively related to the disk and the mass unbalance result from the assumption of non-stationnary working condition where:

[M Dvar ({δ})] =         0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I Dw 2 θ yc 0 0 0 0 0 -I Dw 2 θ xc 0 0 0 I Dw 2 θ yc -I Dw 2 θ xc 0         (5) [M u (θ z )] = d m u         0 0 0 0 0 cos(θ zc ) 0 0 0 0 0 -sin(θ zc ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 cos(θ zc ) -sin(θ zc ) 0 0 0 d         (6)
We notice that, by considering non-stationnary working conditions, matrices with extra-diagonal terms appear in the equation of dynamics of the rotor. Those terms clearly include coupling between the flexural and torsional degrees of freedom. Finally, the matrix [S gyre ] and the force vector {F nl gyre } derive from the new formulation of the gyroscopic effect of the shaft under non-stationnary regime such as:

[S gyre ({δ})] = -N 3 (l) t {δ} t [M g 67 (l)] t -[M g 67 (l)] {δ} N 3 (l) + ∂N 3 ∂z t {δ} t [M g * 67 (l)] t + [M g * 67 (l)] {δ} ∂N 3 ∂z (7)
and,

{F nl gyr ( {δ}, {δ})} = -N 3 (l) t {δ} t [M g 67 (l)] {δ} -2 N 3 (l) {δ} [M g 67 (l)] {δ} + ∂N 3 ∂z t {δ} t [M g * 67 (l)] {δ} + 2 ∂N 3 ∂z {δ} [M g * 67 (l)] {δ} (8)
where N 3 is a vector with polynomial shape functions of first order. [M g 67 ] and [M g * 67 ] are skew-symmetric matrices with constant terms obtained by the multiplication of shape function vectors.

Once the strain energy of the shaft and the virtual works of the bearings, assumed to be with linear stiffness and damping are calculated, the classical stiffness matrix is obtained and we finally write the equation of the dynamics of rotating shaft element such as:

([M s ] + [M Dconst ]) { δ} + [C s ] {δ} + [K c ]{δ} = -([S gyr ({δ})] + [M Dvar ({δ})] + [M u (θ zc )]) { δ} -C D ( θzc ) {δ} -{F nl gyr ( {δ})} + {F nlu (θ zc )} + {F ext } (9)

Results and discussions

We consider an example of an idealized rotor with the properties detailed in table 1. The rotor is made of a flexible shaft, a rigid disk and is excited by the presence of a mass unbalance. Table 1: Proprieties of the studied rotor

In this exemple we simulate the start up of a rotor. Therefore a driving torque following the law shown in figure 4 is induced to the system. The aim of applying a constant torque before the linear time-varying one is to better intialize the simulation for the non-stationary regime. It is important to mention that, for this exemple of rotor with the mentioned properties, the first critical speed is at 2527 rpm and the second one is at 26364 rpm. During the accelerarion period the rotor will go through the two mentioned critical speeds.

The simulation results are used for time-frequency analysis for both lateral displacement and instantaneous angular speed.

Figure 5 shows the results for the lateral displacement. As we can see, both the excitation frequency f e and the first natural frequency f 1 contributes to the motion of the rotor. The results are qualitatively coherent with the analytical solution proposed by Zhou et Shi [START_REF] Zhou | The analytical imbalance response of jeffcott rotor during acceleration[END_REF] saying that, during acceleration, the motion consists of three parts: a transient vibration at damped natural frequency, a synchronous vibration with the frequency of instantaneous 'synchronous' frequency, and a suddenly occurring vibration at damped natural frequency. We can see that the rotor has more difficulties crossing the second natural frequency than the first one.

The spectrogram of the instantaneous angular speed ( fig. 6) shows a coupling between the excitation frequency f e and the first eigen frequency f 1 . This simulation result proofs that the IAS, whose analyses is

Conclusion

The big majority of previous works on the non-stationary rotordynamics were conducted under the assumption of a given law of the rotational speed. In our study, a new dynamic model is built considering no assumption on the IAS. It was shown, for a simple exemple of rotors, that when the IAS is considered as an unknown of the dynamic problem,it is highly influenced by the rotor dynmic properties. Based on the obtained results, we assume that the IAS is a physical quantity which is rich enough with informations about the studied system and that a better analysis using kalman filters is of big intesrest in order to identify the system parameters using the new proposed dynamic model for non-stationnary rotor dynamics.

Figure 1 :

 1 Figure 1: Rotor simplified model

Figure 2 :

 2 Figure 2: Euler angles

Figure 3

 3 Figure 3: rotor exemple

Figure 4 :Figure 5 :

 45 Figure 4: Torque law as a function of time

Figure 6 :

 6 Figure 6: Spectrogram of the instantaneous angular speed


Souad Bezzaoucha

Benoît Marx

Didier Maquin

José Ragot

State and multiplicative sensor fault estimation for nonlinear systems

Keywords: Sensor faults estimation, Takagi-Sugeno models, sector nonlinearity approach, convex polytopic transformation, state and fault observer, linear matrix inequality

This paper addresses the state and sensor fault estimation for nonlinear systems represented by Takagi-Sugeno (T-S) models. The considered faults are time-varying and with multiplicative effect on the sensor output signals. The proposed estimation procedure is based firstly on the sector nonlinearity approach using the convex polytopic transformation where the original system is equivalently rewritten as a Takagi-Sugeno system with unmeasurable premise variables and, secondly on the design of an observer allowing the fault estimation by solving an LMI optimization problem. An application of the proposed approach to a simplified model of an activated sludge reactor model is proposed.

I. INTRODUCTION

Due to an increasing demand for higher performances, industrial processes tend to be, more and more safety critical, robust and reliable. For this reason, state and fault estimation, and more generaly diagnosis, especially the modelbased ones, receive considerable interest from the control community. In the literature, the available works either address the problems of both sensor fault tolerance and estimation accuracy (see [START_REF] Edelmayer | Federated filtering for fault tolerant estimation and sensor redundancy management in coupled dynamics distributed systems[END_REF] for example), where the Extended Kalman Filter is used as state estimator, or sensor redundancy management logic to maintain estimation functionality in nonlinear systems. In this kind of approach, only the state estimation is considered without any detection or estimation of the sensor fault. However, in the Fault Detection and Diagnosis (FDD) design, both fault detection and isolation are considered, usually by characterizing the normal behaviour of sensor readings, and identify significant deviation from this normal situation as faults [START_REF] Sharma | Sensor faults: Detection methods and prevalence in real-world datasets[END_REF]. In [START_REF] Astorga-Zaragoza | Sensor fault diagnosis for a class of LPV descriptor systems[END_REF] for example, a strategy for detecting and isolating sensor faults using a bank of residuals within a generalized observer scheme (GOS) is presented. This scheme provides an estimator dedicated to a particular sensor, this estimator being driven by all outputs except that of the considered sensor. In [START_REF] Ichalal | Fault diagnosis for Takagi-Sugeno nonlinear systems[END_REF], the authors proposed methods for state estimation and fault diagnosis for nonlinear systems represented by Takagi-Sugeno (T-S) models. In [START_REF] Bezzaoucha | On the unknown input observer design : a decoupling class approach with application to sensor fault diagnosis[END_REF] and [START_REF] Aguilera-Gonzalez | Sensor fault and unknown input estimation based on proportional integral observer applied to LPV descriptor systems[END_REF], both sensor fault and unknown input estimation were considered. The main aim of the bank of observers is to generate an incidence matrix or a coupling graph between the fault and the residuals. In fact, a signal that is obtained from the generated residuals defines the effects associated with the fault. This strategy proved its efficiency for the detection and isolation part, but, it requires the same number of observers as outputs. Moreover, it should be highlighted that most of the considered sensor faults are additive ones, i.e. the measured outputs are linear in respect to the faults. The few available works on multiplicative faults use a sliding mode observer and are based on their re-modelling into the framework of additive faults and then apply previous works on reconstructing additive faults (see [START_REF] Tan | Multiplicative fault reconstruction using sliding mode observers[END_REF]). In the present work, is tackled a simultaneous state and multiplicative sensor fault observer design for nonlinear systems represented by T-S models with unmeasurable premise variables. Based on the sector nonlinearity transformation (SNT) [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF], [START_REF] Nagy | Systematic multimodeling methodology applied to an activated sludge reactor model[END_REF], both state observer with sensor fault tolerance and estimation accuracy and time-varying fault observer (for detection, isolation and estimation at once) are designed. The proposed method has the advantage to be analytical and systematic without any loss of information since it consists in rewriting the time-varying sensor fault and the system nonlinearities in a polytopic form allowing to transform the original nonlinear system into a T-S model based on the sector nonlinearity approach and the convex polytopic transformation. The T-S model obtained with this transformation has the major interest to exactly represent the system without any loss of informations or any approximation. The proposed method is an extention to the sensor fault estimation of the author's previous work presented in [START_REF] Bezzaoucha | State and parameter estimation for nonlinear systems: a Takagi-Sugeno approach[END_REF] and [START_REF] Bezzaoucha | Nonlinear joint state and parameter estimation : Application to a wastewater treatment plant[END_REF]. This procedure allows to design a state and fault observer by minimizing the L 2 gain from the fault to the state and fault estimation errors. Using the Lyapunov theory, the L 2 gain minimization is expressed as a LMI constraints. The paper is organized as follows. Section II introduces the T-S representation of the nonlinear system with multiplicative sensor fault. In section III, joint state and unknowm time-varying fault estimation is proposed for T-S systems with unmeasurable premise variables. An application of the proposed approach to a simplified model of an activated sludge reactor model with some simulation results are given in section IV. Conclusions are summarized in section V.

II. PROBLEM STATEMENT: T-S MODELLING OF MULTIPLICATIVE TIME-VARYING SENSOR FAULT

A first contribution of this work is to model nonlinear time-varying systems using the T-S representation. An efficient way consists in rewriting the original nonlinear system in a simplier form, like the T-S one. Originaly introduced by [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF], the T-S representation is based on time-varying interpolation between time invariant linear submodels and allows to describe the exact nonlinear behaviour of a system, under the condition that its nonlinearities are bounded. This is reasonable as variables of physical systems are real and always bounded. See for example [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF], [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF] and the references therein. The T-S representation is very interesting in the sense that it simplifies the mathematical developments for observer and controller design compared to the original nonlinear models Moreover, it allows to extend the use of some tools developed in the linear framework to the nonlinear systems, for the stability study, the control design and the observer synthesis. In the present paper, the T-S models with multiplicative timevarying sensor fault are considered. In order to design a joint state and fault observer, each time-varying sensor fault is rewritten under a particular form. Let us consider the nonlinear T-S system with multiplicative time-varying sensor fault represented by equation ( 1)

   ẋ(t) = r ∑ i=1 µ i (x(t))(A i x(t) + B i u(t)) y(t) = C(t)x(t) (1)
The weighting functions µ i (ξ (t)) of the r submodels satisfy the convex sum property

   r ∑ i=1 µ i (ξ (t)) = 1 0 ≤ µ i (ξ (t)) ≤ 1, i = 1, . . . , r (2)
It is importnt to note that in the remaining of the paper, the weighting functions depend on the state variable x(t). This case is more difficult to deal with and then less studied than the one with measurable premise variables, but naturally appears when the T-S system is obtained with the SNT. The matrice C(t) is defined as follows

C(t) = (I m + F(t))C (3)
s.t. F(t) ∈ R m×m defined by:

F(t) = diag( f (t)) (4)
where diag( f (t)) refers to a diagonal matrix with the terms f j (t) (sensor faults) on its diagonal. F(t) is also expressed as

F(t) = m ∑ j=1 f j (t)F j (5)
with F j matrices of dimension R m×m and where the element of coordinate (i, i) is equal to 1 and 0 elsewhere. The terms f j (t) are time-varying unknown parameters and represent multiplicative sensor faults. Each fault component is unknown but bounded where the lower bound f 2 j and the upper one f 1 j are known. Thus f j (t) can be written as: 6) in [START_REF] Bezzaoucha | State and parameter estimation for nonlinear systems: a Takagi-Sugeno approach[END_REF], it becomes:

f j (t) = µ 1 j ( f j (t)) f 1 j + µ 2 j ( f j (t)) f 2 j , f j (t) ∈ [ f 2 j , f 1 j ] (6) with µ 1 j ( f j (t)) = f j (t) -f 2 j f 1 j -f 2 j , µ 2 j ( f j (t)) = f 1 j -f j (t) f 1 j -f 2 j (7) µ 1 j ( f j (t)) + µ 2 j ( f j (t)) = 1, ∀t Replacing (
F(t) = m ∑ j=1 2 ∑ k=1 µ k j ( f j (t)) f k j F j (8)
In order to write F(t) as a simple polytopic matrix, the convex sum property of the functions µ k j ( f j (t)) can be exploited (see [START_REF] Bezzaoucha | State and parameter estimation for nonlinear systems: a Takagi-Sugeno approach[END_REF] for calculation details):

F(t) = 2 m ∑ j=1 µ j ( f (t))F j (9) with        µ j ( f (t)) = m ∏ k=1 µ σ k j k ( f k (t)) F j = m ∑ k=1 f σ k j k F j ( 10
)
where the µ j ( f (t)) satisfy the convex sum property.

In the following, f (t) is the fault vector of components f j (t), j = 1, . . . , m. The indices σ k j equal to 1 or 2, indicate which partition of the k th parameter ( µ k 1 or µ k 2 ) is involved in the j th submodel. The relation between the j th submodel and the σ k j indices are given by the following equation

j = 2 m-1 σ 1 j + 2 m-2 σ 2 j + . . . + 2 0 σ m j -(2 1 + 2 2 + . . . + 2 m-1 ) (11
) Finally, using equations ( 10) and ( 3), (1) becomes:

         ẋ(t) = r ∑ i=1 µ i (x(t))(A i x(t) + B i u(t)) y(t) = 2 m ∑ j=1 µ j ( f (t)) C j x(t) ( 12
)
C j = C + F j C (13)

III. STATE AND SENSOR FAULT OBSERVER

Based on the obtained T-S model [START_REF] Tan | Multiplicative fault reconstruction using sliding mode observers[END_REF], a simultaneous state and sensor fault observer may be designed and implemented. An L 2 attenuation approach will be proposed to minimize the effect of the time-varying fault on the state and fault error estimation. The state and sensor fault observer is defined by

                 ẋ(t) = r ∑ i=1 µ i ( x(t)) (A i x(t) + B i u(t) + L i (y(t) -ŷ(t))) ḟ (t) = r ∑ i=1 µ i ( x(t))(K i (y(t) -ŷ(t)) -α i f (t)) ŷ(t) = 2 m ∑ j=1 µ j ( f (t)) C j x(t) (14)
where L i ∈ R n x ×m , K i ∈ R m×m and α i ∈ R m×m are the gains to be determined such that the estimated state and fault converge to the system state and fault. Let us define the state and fault estimation error e x (t) and e f (t) as

e x (t) = x(t) -x(t) e f (t) = f (t) -f (t) (15)
Their dynamics cannot be easily computed directly from (15) since in [START_REF] Tan | Multiplicative fault reconstruction using sliding mode observers[END_REF] the weighting functions depend on the unmeasurable variables ( f (t) and x(t)). Because of that, based on the convex sum property of the weighting functions, ( 12) is rewritten as follow

             ẋ(t) = r ∑ i=1 [µ i ( x(t))(A i x(t) + B i u(t))+ (µ i (x(t)) -µ i ( x(t))) (A i x(t) + B i u(t))] y(t) = 2 m ∑ j=1 µ j ( f (t)) C j x(t) + ( µ j ( f (t)) -µ j ( f (t))) C j x(t)
(16) This form allows a better comparison of x(t) with x(t), since µ i ( x(t)) and µ j ( f (t)) not only appears in [START_REF] Zhou | Robust stabilization of linear systems with norm-bounded time-varying uncertainty[END_REF], but also in (16). Let us define:

∆A(t) = r ∑ i=1 [µ i (x(t)) -µ i ( x(t))]A i = A Σ A (t)E A (17) ∆B(t) = r ∑ i=1 [µ i (x(t)) -µ i ( x(t))]B i = BΣ B (t)E B ( 18
)
∆C(t) = 2 m ∑ j=1 ( µ j ( f (t)) -µ j ( f (t))) C j = C Σ C (t)E C (19) with A = A 1 . . . A r , Σ A (t) = diag(δ 1 (t), . . . , δ r (t)), B = B 1 . . . B r , Σ B (t) = diag(δ 1 (t), . . . , δ r (t)), C = C 1 . . . C 2 m , Σ C (t) = diag( δ 1 (t), . . . , δ 2 m (t)), E A = I n x . . . I n x T , E B = I n u . . . I n u T E C = I 2 m . . . I 2 m T δ i (t) = µ i (x(t)) -µ i ( x(t)), δ j (t) = µ j ( f (t)) -µ j ( f (t)) (20) Thanks to property (2), it follows -1 ≤ δ i (t) ≤ 1, -1 ≤ δ j (t) ≤ 1 (21)
which implies from definition (20) 17), ( 18) and (19), the system (16) is then written as an uncertain system given by:

Σ T A (t)Σ A (t) ≤ I, Σ T B (t)Σ B (t) ≤ I, Σ T C (t)Σ C (t) ≤ I (22) Using (
         ẋ(t) = r ∑ i=1 µ i ( x(t))((A i + ∆A(t))x(t) + (B i + ∆B(t))u(t)) y(t) = 2 m ∑ j=1 µ j ( f (t))( C j + ∆C(t))x(t) (23)
From equations ( 23), ( 14) and (15), the dynamics of the state estimation error is given by ėx

(t) = r ∑ i=1 µ i ( x(t)) (A i e x (t) + ∆A(t)x(t) -L i (y(t) -ŷ(t)) + ∆B(t)u(t)) (24)
The output error y(t)ŷ(t) is then calculated as follows

y(t) -ŷ(t) = 2 m ∑ j=1 µ j ( f (t))( C j e x (t) + ∆C(t)x(t)) (25)
Replacing ( 25) in ( 24), the dynamics of the state estimation error is given by

ėx (t) = r ∑ i=1 2 m ∑ j=1 µ i ( x(t)) µ j ( f (t))((A i -L i C j )e x (t)+ (∆A(t) -L i ∆C(t))x(t) + ∆B(t)u(t)) (26)
From equations ( 25) and ( 14), the dynamics of the fault estimation is given by

ė f (t) = r ∑ i=1 2 m ∑ j=1 µ i ( x(t)) µ j ( f (t))(-K i C j e x (t) -α i e f (t) ḟ (t) -K i ∆C(t)x(t) + α i f (t)) (27
) Due to the coupling between the errors e f (t) and e x (t), it is convenient to consider the augmented vectors e a (t) and ω(t)

e a (t) = e x (t) e f (t) , ω(t) =     x(t) f (t) ḟ (t) u(t)     (28)
From ( 26), ( 27) and ( 28), it follows

ėa (t) = r ∑ i=1 2 m ∑ j=1 µ i ( x(t)) µ j ( f (t)) (Φ i j e a (t) + Ψ i (t)ω(t)) (29)
with

Φ i j = A i -L i C j 0 -K i C j -α i Ψ i (t) = ∆A(t) -L i ∆C(t) 0 0 ∆B(t) -K i ∆C(t) α i I 0 (30)
Considering (29), the objective is to design a joint state and fault observer with a minimal L 2 gain of the transfer from ω(t) to e a (t). The computation of the observer gains is detailed in the next theorem. Theorem 1: There exists a joint robust state and multiplicative sensor fault observer (14) for a nonlinear system (1) with an L 2 gain from ω(t) to e a (t) bounded by β (β > 0) if there exists matrices

P 1 = P T 1 > 0, P 2 = P T 2 > 0, Γ 1 , Γ 2 , Γ 3 , Γ 4 > 0, α i , K i , R i and scalars β , λ 1 , λ 1C > 0, λ 2C > 0
and λ B > 0 solutions of the optimization problem (31) under LMI constraints (32) and (33) (see next page) min

P 1 ,P 2 ,R i ,K i ,α i ,λ 1 ,λ 1C ,λ 2C ,λ B β ( 31
) for i = 1, . . . , r and j = 1, 2 m Γ k < β I for k = 1, 2, 3, 4 (32)
with

Q 11 i j = P 1 A i + A T i P 1 -R i C j -C T j R T i + I n x Q 33 = -Γ 1 + λ 1 E T A E A + λ 1C E T C E C + λ 2C E T C E C (34)
The observer gains are given by  



L i = P -1 1 R i K i = P -1 2 K i α i = P -1 2 α i (35)                Q 11 i j -C T j K T i 0 0 0 0 P 1 A P 1 B R i C 0 * -α i -α T i + I m 0 α i P 2 0 0 0 0 K i C * * Q 33 0 0 0 0 0 0 0 * * * -Γ 2 0 0 0 0 0 0 * * * * -Γ 3 0 0 0 0 0 * * * * * -Γ 4 + λ B E T B E B 0 0 0 0 * * * * * * -λ 1 I 0 0 0 * * * * * * 0 -λ B I 0 0 * * * * * * 0 0 -λ 1C I 0 * * * * * * 0 0 0 -λ 2C I                < 0 (33)
Proof: In the remaining of the paper, the following lemma is used:

Lemma 1: [START_REF] Zhou | Robust stabilization of linear systems with norm-bounded time-varying uncertainty[END_REF] Consider two matrices X and Y with appropriate dimensions, a time-varying matrice ∆(t) and a positive scalar ε. The following property is verified

X T ∆ T (t)Y +Y T ∆(t)X ≤ εX T X + ε -1 Y T Y (36) for ∆ T (t)∆(t) ≤ I.
Let us consider the following quadratic Lyapunov function

V (e a (t)) = e T a (t)Pe a (t), P = P T > 0 (37) Using (29), its time derivative is given by

V (e a (t)) = r ∑ i=1 2 m ∑ j=1 µ i ( x(t)) µ j ( f (t)) e T a (t)((Φ i j ) T P +PΦ i j )e a (t) + e T a (t)PΨ i (t)ω(t) + ω T (t)Ψ T i (t)Pe a (t) (38)
It is known that e a (t) asymptotically converges toward zero when ω(t) = 0 and that the L 2 gain from ω(t) to e a (t) is bounded by β if the following inequality holds V (e a (t)) + e T a (t)e a (t) -ω T (t)Γω(t) < 0 (39)

with Γ = diag(Γ k ), Γ k < β I, for k = 1, 2, 3, 4 (40)
An adequate choice of Γ allows to attenuate the transfer from some components of ω(t) to e a (t). From (38), (39) becomes:

r ∑ i=1 2 m ∑ j=1 µ i ( x(t))µ j ( f (t)) e a (t) ω(t) T Φ T i j P + PΦ i j + I 2n x PΨ i (t) Ψ T i (t)P -Γ e a (t) ω(t) < 0
(41) The Lyapunov matrix P is chosen as

P = diag(P 1 , P 2 ) (42)
From ( 15), (30), ( 40) and ( 42), (41) holds if

µ i ( x(t)) µ j ( f (t)) Q i j + Q(t) + Q T (t) < 0 (43)
with:

Q i j =         Q 11 i j -C T j K T i 0 0 0 0 * -α i -α T i + I m 0 α i P 2 0 * * Q 33 0 0 0 * * * -Γ 2 0 0 * * * * -Γ 3 0 * * * * * -Γ 4         (44)
Q 11 i j = P 1 A i + A T i P 1 -R i C j -C T j R T i + I n x (45)
As ( 43) is time depending, it is convenient to look for bound to the matrix Q(t). For that purpose, based on (17) and ( 18), the time-varying term of (43) can be expressed as:

Q(t) =         P 1 A 0 0 0 0 0         Σ A (t) 0 0 E A 0 0 0 +         P 1 B 0 0 0 0 0         Σ B (t) 0 0 0 0 0 E B +         P 1 L i C P 2 K i C 0 0 0 0         Σ C (t) 0 0 -E c 0 0 0 (46)
Using lemma 1 and property (21), there exists positive scalars λ 1 , λ B , λ 1C and λ 2C , such that

Q(t) + Q T (t) <         Q 1 0 0 0 0 0 0 Q 2 0 0 0 0 0 0 Q 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 λ B E T B E B         (47)
with:

Q 1 = λ -1 1 P 1 A A T P 1 + λ -1 B P 1 BB T P 1 + (λ 1C ) -1 P 1 L i C C T L T i P 1 Q 2 = λ -1 2C P 2 K i C C T K T i P 2 Q 3 = λ 1 E T A E A + (λ 1C + λ 2C )E T C E C ( 48
)
for i = 1, . . . , r and j = 1, . . . , 2 m . From inequality (47), since µ i ( x(t)) and µ j ( f (t)) satisfy the convex sum property, with the variable changes (35), the LMI (33) implies ( 43) and (39). As a consequence, the L 2 -gain of the transfer from ω(t) to e a (t) is bounded by β , which achieves the proof. T-S model of the system is proposed. The proposed state and fault observer is then designed by solving an LMI optimization problem, i.e. by minimizing the L 2 gain from the augmented input to the estimation errors. The chosen application example is an activated sludge reactor with multiplicative sensor fault on the output. From the nonlinear equations of the system, a T-S model of the system is derived. The proposed observer is synthetized and the obtained results illustrate its performance.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Output with and without f 1 (t)

Fig. 3 .Fig. 4 .

 34 Fig. 3. System states and their estimates

IV. NUMERICAL EXAMPLE In this section, the proposed approach is applied to a biological wastewater treatment plant. A reduced form of an activated sludge reactor model is considered with only the carbon pollution and two state variables. Starting from the nonlinear equations of the system, a T-S representation is given. Multiplicative sensor fault are considered. The objective is to synthesize an observer in order to simultaneously estimate the system states and the fault. The process consists in mixing used waters with a rich mixture of bacteria in order to degrade the organic matter [START_REF] Nagy Kiss | State estimation of two time-scale multiple models with unmeasurable premise variables. Application to biological reactors[END_REF]. Under specific assumptions, some simplifications can be made and the nonlinear system can be represented with the following equations [START_REF] Bezzaoucha | State and parameter estimation for nonlinear systems: a Takagi-Sugeno approach[END_REF]:

with x 1 (t) and x 2 (t), the biomass and substrat concentration respectively, and where a, b and d are known parameters. The input u(t) represents the dwell-time in the treatment plant.

The measured output is the biomass concentration (y(t) = x 1 (t)).

It is assumed that a bounded multiplicative sensor fault f 1 (t) affects the output y(t) such that:

As previously explained, f 1 (t) can also be written as:

) are defined by [START_REF] Ichalal | Fault diagnosis for Takagi-Sugeno nonlinear systems[END_REF]. Parameters b, c, d have been identified and set to b = 0.07, c = 0.7 and d = 2.5. From the system nonlinearities, let us consider the following premise variables:

From ( 49) and (52), the following quasi-LPV form is obtained:

Since a T-S model is obtained in a compact set of the state space, maximum and minimum values that occur in z 1 (t) and z 2 (t) may be calculated using the knowledge of the domain of variation of u(t): z

Using the convex polytopic transformation, two partitions for each premise variable are constructed as follows:

where the scalars z 1 1 , z 2 1 , z 1 2 and z 2 2 are defined as

The submodels are defined by the pairs (A i , B i ) with i = 1, . . . , 4. Due to the choice of premise variables, all the B i matrices are equal to B T = 0 d . The matrices A i are given by:

The T-S model of the system with the multiplicative sensor fault is obtained by an interpolation of the four previous submodels for the state and two submodels for the output equations:

The weighting functions µ j ( f (t)) are calculated from (7) and µ i (z(t)) as the following:

As mentioned at the begining of this section, the objective is to synthesize a robust state and fault observer applying the proposed approach. To illustrate the time varying fault effect on the system, figure 1 depicts the output with and without the sensor fault. The system input, the state variables and their estimates, the time-varying fault and its estimate are depicted in the figures 2, 3 and 4 respectively. The initial conditions are taken as x(0) = 0.1 1.5 for the system and x(0) = 0.09 2.3 , f1 (0) = 0 for the state and fault observer respectively. From the depicted figures, one can conclude on the efficiency of the synthetized state observer, since the two states are perfectly estimated as well as the time-varying multiplicative sensor fault f 1 (t).

V. CONCLUSION

In the present paper, a new systematic procedure is presented to deal with the state and multiplicative sensor fault estimation for nonlinear systems. It consists in transforming the original system into a Takagi-Sugeno model, based on the sector nonlinearity approach and the convex polytopic transformation. This transformation has the major interest to exactly represent the system without any loss of informations. The considered procedure is the following: from the nonlinear time-varying equations of the process, a global


Souad Bezzaoucha
email: souad.bezzaoucha@ims-bordeaux.fr

Benoît Marx
email: benoit.marx@univ-lorraine.fr

Didier Maquin
email: didier.maquin@univ-lorraine.fr

José Ragot
email: jose.ragot@univ-lorraine.fr

José Ragot State

State tracking control for Takagi-Sugeno models

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction and paper outline

The main objective of this work is to deal with reference model tracking for nonlinear systems described by Takagi-Sugeno (T-S) models. The T-S models are known to be an efficient way to deal with the problems of estimation and control of nonlinear systems by writing them in a polytopic form. Originally introduced by [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF], the T-S representation allows to exactly describe nonlinear systems, provided that the nonlinearities are bounded. This is reasonable since state variables as well as parameters of physical systems are bounded, and so is the input of the system which may be considered naturally stable, or equipped with a stabilizing control (see for example [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF] and the references therein). Despite an abundant literature on stability conditions of T-S models, few authors have dealt with the tracking problem for all the state of the system; indeed most of the works deal with the tracking of the system output, which generally reduces to some combinations of these variables. One can refer to some works concerned with state or output feedback with H ∞ performances [START_REF] Taniguchi | A new PDC fuzzy reference models[END_REF], [START_REF] Tseng | Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model[END_REF] and [START_REF] Mansouri | Output feedback LMI tracking control conditions with H ∞ criterion for uncertain and disturbed models[END_REF]. The nonlinear tracking control problem is expressed in terms of Linear Matrix Inequality (LMI) and is based on the T-S and Parallel Distributed Compensation (PDC) structures (an L 2 tracking performance related to the tracking error is formulated and a PDC state feedback control (or output feedback control) is designed. See [START_REF] Mansouri | Output feedback LMI tracking control conditions with H ∞ criterion for uncertain and disturbed models[END_REF], [START_REF] Tseng | Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model[END_REF] for examples). However, in the cited references, a referred "suitable" choice for the reference model is made without any explanations nor details.

The last remark motivated the present study. In fact, either for the linear case or the nonlinear one, few works detail the influence of the reference model choice, which is not a trivial task. In [START_REF] Nie | Fault estimation and MRC-based active FTC[END_REF] for example, the authors referred to the Erzberger's conditions, but with no further explanations. For these reasons, in the proposed work, a focus is made not only on the design control procedure, but also on the tracking (matching) conditions. The paper is organized as follows. In section II, the structural conditions to achieve state tracking are introduced in the T-S case. These conditions are an extension of the well known Erzberger's conditions. Two procedures for state tracking are considered: for the first one, the controller structure is firstly fixed and then the structural conditions and the appropriate gains of the controller are deduced. For the second one, no prerequisited control law is considered, with the objective to achieve a null tracking error for both strategies. In section III, the quadratic optimal control with an extension to the Model Predictive Control for T-S models are presented. Simulation examples are given in section IV. Finally, section V summarizes the obtained results.

Structural conditions for exact state tracking for T-S systems

Model and objective

Let us consider the following T-S model [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF]:

x k+1 = A k x k + B k u k (1) s.t. x k ∈ R n x and u k ∈ R n u with: A k = r ∑ i=1 µ i,k (ξ k )A i , B k = r ∑ i=1 µ i,k (ξ k )B i (2)
where B k is supposed to be a full column rank matrix and where the weighting functions µ i,k (ξ k ) depend on the so-called premise variable ξ k which may be a state, input, or output combination. These weighting functions satisfy the following convex sum property:

0 ≤ µ i,k (ξ k ) ≤ 1, r ∑ i=1 µ i,k (ξ k ) = 1 (3)
The considered linear reference model is the following:

x r,k+1 = A r x r,k + B r u r,k (4)
s.t. x r,k ∈ R n x , u r,k ∈ R n u
and where the desired performances are defined by the choice of the matrices A r and B r .

The ideal tracking objective is to adjust, at each instant k, the control u k in such a way that the system state x k follows the reference model state x r,k with a null tracking error. For this purpose, two procedures may be considered. The first strategy consists in setting the controller structure and then deduce the appropriate structural conditions that must satisfy the reference model and controller gains. The second strategy is not based on a prerequisited control law structure. The idea is to find, as for the first strategy, the appropriate structural conditions, but also an analytical expression for the control law. If the ideal tracking is not reachable, some compromises need to be defined such as, for example, tracking of a subset of the states, instead of all of them.

Prerequisited control law

In order to achieve the tracking objective, the following control law is considered:

u k = K k x k + K r,k u r,k (5)
Substituting ( 5) into (1), the closed-loop system is:

x k+1 = (A k + B k K k ) x k + B k K r,k u r,k (6)
The matching conditions for the reference model and the system are then obtained by comparing the closedloop system (6) and the reference model [START_REF] Tseng | Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model[END_REF]. They are given by:

A k + B k K k = A r B k K r,k = B r (7)
From [START_REF] Allgöwer | Simplicial and continuation methods for approximating fixed points and solutions to systems of equations[END_REF], in order to have a solution in respect to the gain K k and K r,k , the following rank conditions have to be fullfiled:

rank[B k ] = rank[B k |A r -A k ] rank[B k ] = rank[B k |B r ] (8)
where B k and A k are defined in [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF]. If conditions (8) are fulfilled, then at each sampling time, the gains K k and K r,k are given by:

K r,k = B + k B r , K k = B + k (A r -A k ) (9)
with B + k a suitable pseudo-inverse matrix of the full column rank B k matrix. Note that in order to satisfy the matching conditions (8), from definitions (2), since the system matrices A k and B k depend on the time, one sufficient, but not unique, condition is to consider the matrices A i , B i and A r , B r in the following canonical form:

A i =   A 0 A i   , A r =   A 0 A r   B i = 0 n x -n u b i , B r = 0 n x -n u b r (10) with A 0 a matrix of dimension (n x -n u ) × n x , A i and A r matrices of dimensions n u × n x . b i and b r are of dimen- sion n u × n u .
The structure (10) means that:

1. the (n xn u ) first rows of the matrices A i are equal to the (n xn u ) first rows of the matrix A r 2. the (n xn u ) first rows of the matrices B i are null 3. the (n xn u ) first rows of the matrix B r are null allowing to fully satisfy the rank conditions [START_REF] Axelsson | Iterative Solution Methods[END_REF]. It is important to note that the matching conditions (8) between the reference model and the system depend on the choice of the control law, u k given by [START_REF] Mansouri | Output feedback LMI tracking control conditions with H ∞ criterion for uncertain and disturbed models[END_REF]. It means that these conditions have to be adapted when changing the structure of the control law.

Numerical example

To illustrate the above conditions, let us consider the following academic example:

A r =    0.2 0.5 0 -0.2 0.99 -0.1 0 0 0.2    , B r =    0 0 1    A 1 =    0.2 0.5 0 -0.2 0.99 -0.1 0 0 0.1    , B 1 =    0 0 1.5    A 2 =    0.2 0.5 0 -0.2 0.99 -0.1 0 0 1.1    , B 2 =    0 0 -0.5    (11)
The weighting functions are taken as:

µ 1,k = 2 -sin(x 1,k ) -tanh(x 2,k ) 4 , µ 2,k = 1 -µ 1,k (12)
Applying the tracking control law ( 5) with ( 9), the system and model reference states are depicted in figure 1 (respectively noted x i and x ir , i = 1, . . . , 3). In figure 2, the control inputs u r,k and u k are represented. From the depicted figures, one can see that the control tracking is efficient for all the three states under the specified structural conditions.

No prerequisited control law

In this subsection, no prerequisited structure of the control law is considered. Then, at time k, in order to achieve the tracking objective at time k + 1 with a null state tracking error between ( 4) and (1), meaning:

x r,k+1 = A k x k + B k u k (13)
the control law u k has to verify:

B k u k = x r,k+1 -A k x k (14)
Note that the exact state tracking is ensured, i.e. (13) has a solution in respect to u k , only if the following matching condition is fulfilled at each sampling time:

rank [B k ] = rank B k |x r,k+1 -A k x k (15)
The tracking control is deduced and given by the following equation:

u k = (B T k B k ) -1 B T k (x r,k+1 -A k x k ) (16)
and the reader is invited to compare the two structures ( 16) and ( 5) with the help of ( 9).

Remark 1 If the premise variables ξ k of the weighting functions µ i,k depend on the input u k , the control law (16) will be implicit (i.e. u k = F(u k )) since A k and B k are input depending [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF]. A solution may be given by an iterative algorithm with the following recurrence:

u ( j+1) k = (B ( j) k ) T B ( j) k -1 B ( j) k T (x r,k+1 -A ( j) k x k ) (17) with B ( j) k = r ∑ i=1 µ i,k (u ( j) k )B i , A ( j) k = r ∑ i=1 µ i,k (u ( j)
k )A i , j = 0, . . . , N with N the number of iterations and u 0 k the input initialization (may be taken as u r,k for example). The convergence of this algorithm may be proved locally, but it will not be considered in the present work since it is not the purpose of the study (see [START_REF] Allgöwer | Simplicial and continuation methods for approximating fixed points and solutions to systems of equations[END_REF], [START_REF] Axelsson | Iterative Solution Methods[END_REF], [START_REF] Kelley | Iterative Methods for Linear and Nonlinear Equations[END_REF] for more details).

As mentioned in the beginning of this section, to achieve the exact tracking, the system and model reference must be in a certain canonical form (structural conditions). Depending on the reference and system models, these conditions may be impossible to meet and thus must be relaxed. One way is to apply the tracking to some states only, but it can affect the functioning of the system since other states are left free. A better way is to use an approximate tracking of all the states.

Approximate state tracking for T-S systems

In this section, the optimized tracking using the norm of the tracking error is addressed. This part concerns the optimal control with the introduction of the MPC for T-S models. As it was mentioned above, due to the restrictiveness of the exact tracking structural conditions, one can consider another approach to relax these structural conditions. This approach is the so-called quadratic optimal control for T-S models and aims at minimizing the tracking error. This is a natural way to deal with the problem by optimizing the tracking error. Indeed, it is less difficult and conservative to deal with a minimization problem than an equality constraint such as expressed by the structural conditions of the previous section.

Control law design

At each time instant k, the objective is to minimize the following criterion which is the norm of the tracking error:

Φ k (u k ) = B k u k -x r,k+1 + A k x k 2 W (18)
where W is a positive definite weighting matrix chosen accordingly to the state components for which some specific tracking is desired. The control tracking law is then given by:

u k = (B T k W B k ) -1 B T k W (x r,k+1 -A k x k ) (19)
where the matrices A k and B k have been already defined in (2).

Remark 2 When the premise variables ξ k depend on the control as explained previously (remark 1), the same procedure goes for the control law (19) with the iterative resolution, it becomes for j = 0, . . . , N:

u ( j+1) k = ((B ( j) k ) T W B ( j) k ) -1 (B ( j) k ) T W (x r,k+1 -A ( j) k x k ) (20) with B ( j) k = r ∑ i=1 µ i,k (u ( j) k )B i , A ( j) k = r ∑ i=1 µ i,k (u ( j) k )A i .
The proposed control tracking law u k (19) (i.e. (20) for input dependent premise variables) aims to ensure the tracking of x r,k+1 by x k+1 at each time instant k.

Since the controller has no ability to anticipate future events, it can not take control actions accordingly and the tracking performances may degrade. Based on this statement, the MPC tracking is introduced for the T-S models. In fact, the main advantage of MPC is to allow the current time slot to be optimized, while keeping future time slot in account. This is achieved by optimizing a finite time horizon, but only implementing the current time slot.

As for linear and nonlinear model reference tracking control, the MPC requires iterative solution and aims to ensure that the tracking error is minimized on a finite sliding horizon. Roughly speaking, the procedure for T-S models is the same as for the conventional MPC. However, some difficulties occur when the premise variables depend on the control.

Premise variables independent of the input

Considering a finite horizon of p + 1 steps, using the state equation

x k+1 = A k x k + B k u k (21)
it follows the state expression at time k + p + 1

x k+p+1 = A k+p x k+p + B k+p u k+p = Φ k+p,k x k + k+p-1 ∑ l=k Φ k+p,l+1 B l u l ( 22
)
where

Φ k,l = A k-1 . . . A l if k > l ≥ 0 I if k = l (23)
with A k and B k defined by [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF]. Gathering the states on the time horizon

[k + 1 : k + p + 1], let us note: xk+1,p =       x k+1 x k+2 . . . x k+p+1       , u k,p =       u k u k+1 . . . u k+p       , A k,p =          A k A k+1 A k . . . p ∏ i=0 A k+p-i          (24) B k,p =          B k 0 . . . 0 A k+1 B k B k+1 . . . 0 . . . . . . . . . . . . p-1 ∏ i=0 A k+p-i B k p-2 ∏ i=0 A k+p-i B k+1 . . . B k+p         
Using ( 22) and (24), the state x is written as:

xk+1,p = A k,p x k + B k,p u k,p , x k,p ∈ R n(p+1) (25)
To ensure the reference model tracking on the time horizon [k + 1 : k + p + 1], the control u k,p is ajusted in order to minimize the criterion:

Φ k,p ( ūk,p ) = x r,k,p -A k,p x k -B k,p u k,p 2 W (26) with x r,k,p = x r,k+1 . . . x r,k+p+1 T ∈ R n(p+1) .
This leads to:

u k,p = (B T k,p W B k,p ) -1 B T k,p W (x r,k,p -A k,p x k ) (27)
where the input at the step k defined by:

u k = I n u 0 . . . 0 ūk,p (28)
is then applied to the system. For the next step, the horizon is moved and the criterion Φ k+1,p is optimized in order to obtain and apply the control u k+1 . As explained before, the control u k,p (in particular u k ) is calculated while keeping future time slot in account, which explains the anticipative character of the MPC control.

Extension to premise variables dependent of the input

Since the weighting functions of the matrices A k,p and B k,p (24) may depend on the control ūk,p , instead of the analytical solution (27), the following iterative algorithm is proposed:

1. define a threshold δ 2. for j = 0, define u ( j) k,p and u

( j-1) k,p 3. compute A ( j) k,p and B ( j) k,p 4. while ||u ( j) k,p -u ( j-1) k,p || > δ u ( j+1) k,p = (B ( j) k,p ) T W B ( j) k,p -1 (B ( j) k,p ) T W (x r -A ( j) k,p x k ) (29) j ← j + 1, compute A ( j+1) k,p and B ( j+1) k,p . with:                                                                  A ( j) k,p =           A ( j) k A ( j) k+1 A ( j) k . . . p ∏ i=0 A ( j) k+p-i           , A ( j) k = r ∑ i=1 µ i,k (u ( j) (k))A i B ( j) k,p =           B ( j) k 0 . . . 0 A ( j) k+1 B ( j) k B ( j) k+1 . . . 0 . . . . . . . . . . . . p-1 ∏ i=0 A ( j) k+p-i B ( j) k p-2 ∏ i=0 A ( j) k+p-i B ( j) k+1 . . . B ( j) k+p           B ( j) k = r ∑ i=1 µ i,k (u ( j) (k))B i
(30) After the algorithm convergence, the control input at the step k is defined by:

u k = I n u 0 . . . 0 u k,p (31)
and is applied to the system. For the next step, the horizon is moved and the criterion Φ k+1,p is optimized in order to obtain u k+1 .

Numerical example

order to illustrate the proposed approach, let us consider the following numerical example:

A r =         0.2 0.5 0 0 0 -0.2 0.19 -0.1 0 0 0 -1 0.2 0 0 0 0.2 -0.2 0.9 -0.2 0.1 0.3 0 0 0.7         A 1 =         0.3 0.5 0 0 0.1 -0.2 0.69 -0.1 0 0 0 -1.1 0.5 0 0.1 -0.51 -0.1 0 0.9 -0.2 0.1 0.3 0 0 1         A 2 =         0.6 0.5 0 0 0 -0.2 0.39 -0.1 0 0 0 -1 0.5 0 0 0 0.2 -0.2 1 -0.2 0.1 0.3 0 0 0.8         B 1 =         -0.2 0.3 0.4 0.5 0.3 1.3 0.55 0.45 -0.7 -0.3         , B 2 =         -0.8 -0.3 -0.2 -0.1 -0.3 0.7 -0.05 -0.15 -1.3 -0.9         B r =         -0.5 0 0.1 0.2 0 1 0.25 0.15 -1 -0.6        
The weighting functions are input dependent and given by:

   µ 1,k = 1 + 2 tanh(u 1,k ) 2 µ 2,k = 1 -µ 1,k (32)
One can verify that the exact tracking conditions [START_REF] Axelsson | Iterative Solution Methods[END_REF] are not fulfilled. For this reason, the MPC is performed for three steps forward (p = 2) with iterative resolution (30). First, the objective is to ensure a good tracking of the fourth and fifth state components. Consequently, the weighting matrix is chosen as W = diag(1, 1, 1, 2, 2), implying a relaxation of the tracking of the first three state components. The system and reference model states are depicted in figure 3 (respectively noted x i and x ir , i = 1, . . . , 5). In figure 4, the control inputs u r,k and u k are represented. From the depicted figures, one can see that the control tracking is efficient (especially for the fourth and fifth states) although the structural conditions are not fulfilled. Secondly, if the main objective is an accurate tracking of x 3r by x 3 , one should set the weighting matrix as W = diag(1, 1, 2, 1, 1). The obtained results are displayed in figure 5 for the system and model reference states and in figure 6 for the control inputs u r,k and u k .

One can observe that for this case, the third state tracking has been improved when the fourth and fifth states tracking have been slightly deteriorated.

In order to quantify the improvement due to the predictive control, let us consider the following criterion: where i the component number of a vector (i = 1, . . . , 5), N is the simulation horizon and x k,i is obtained from non predictive control (16). φ ip is analogously defined with x k,i obtained with MPC:

φ i = N ∑ k=0 |x r,k,i -x k,i |
φ ip = N ∑ k=0 |x r,k,i -x p k,i |
Finally, the performance gain τ i due to MPC is obtained from τ i = 100 φ i -φ ip φ i

For the considered example (W = diag(1, 1, 1, 2, 2)), we obtain the following improvement (for each state): τ 1 = 2.54%, τ 2 = 12.99%, τ 3 = 5.21%, τ 4 = 64.72%, and τ 5 = 14.56%. In order to highlight the influence of the time horizon length p + 1 on the tracking performances, the improvement of φ i , namely τ i , is computed for time horizon characterized by p ∈ {1, 2, 3, 4}. It is also important to highlight that depending on the dynamic characteristic of the reference on a time horizon, a too short, as well as a too long horizon may not give the best expected results. A compromise is then needed. To quantify the best horizon length, a comparative study as the one presented may be a good solution.

Conclusion

In this paper, the tracking objective for nonlinear T-S model was considered. Structural conditions for perfect tracking were established, as well as the quadratic optimal control. A Model Predictive Control for the T-S case with finite time horizon was also developed and the prediction influence was highlighted via a numerical example. A first perspective for the present work is to generalize the matching conditions and the structure proposed in (10) for a general control structure law and establish the relation between the reference model (A r , B r ) and the system matrices (A i , B i ). During the study, a strong correlation between the time horizon length and the model reference dynamics was pointed, in fact, a second interesting perspective will be to present a choice criterion that optimizes the time horizon length according to the model reference dynamics.

Figure 1 .Figure 2 .

 12 Figure 1. System and model reference states

5 , x r, 5 Figure 3 .Figure 4 .

 5534 Figure 3. System and reference model states for W = diag(1, 1, 1, 2, 2)

x 5 , x r, 5 Figure 5 .Figure 6 .

 5556 Figure 5. System and reference model states for W = diag(1, 1, 2, 1, 1)

Table 1 .

 1 The results are gathered in table I. One can conclude, for the presented example, that a horizon of length p = 4 gives the best results.

		p = 1	p = 2	p = 3	p = 4
	τ 1	1.76%	2.54%	8.99% 14.02%
	τ 2	4.80% 12.99% 22.28% 28.85%
	τ 3	3.81%	5.21%	6.21%	7.67%
	τ 4 46.90% 64.72% 77.52% 81.1%
	τ 5 10.62% 14.56% 13.64% 14.81%


Souad Bezzaoucha
email: souad.bezzaoucha@ims-bordeaux.fr

Benoît Marx
email: benoit.marx@univ-lorraine.fr

Didier Maquin
email: didier.maquin@univ-lorraine.fr

José Ragot
email: jose.ragot@univ-lorraine.fr

Finite Memory State Observer Design for Polytopic Systems. Application to Actuator Fault Diagnosis

This paper addresses the Finite Memory Observer (FMO) design applied to polytopic models. After a brief introduction on FMO for linear systems, the nonlinear models represented in a Takagi-Sugeno (T-S) or Polytopic form are then considered. The considered observer design will be applied to investigate the fault diagnosis for nonlinear discrete-time systems subject to unknown input where joint system states and unknown inputs estimation is proposed.

Introduction

In order to detect and isolate a sensor fault through the estimation of system outputs using measurable signals and the model of the system, fault detection and isolation (FDI) techniques based on the time-evolution of the residual signals obtained by the comparison between the measured outputs and the estimated outputs [START_REF] Patton | Issues of Fault Diagnosis for Dynamic systems[END_REF], [START_REF] Ding | Model-Based Fault Diagnosis Techniques Design Schemes, Algorithms and Tools[END_REF] are commonly considered. The procedure is performed by defining and generating some residual signal in order to detect the occurring fault(s). The residual signals often consist in output estimation error, provided by classical or unknown input observers. Then the residual analysis and / or structuration may lead to fault isolation. A way to do so is to establish the theoretical influence of each fault on each residual, namely the signature table. Then, a decision logic is used to generate fault indicators based on these residuals. System states or outputs estimation is the basis for the FDI methods. Among estimation techniques, those using Kalman filters or a Luenberger observer are widely used. These estimators are said to be infinite memory and hence the state estimation error converges to zero in infinite time. In contrast, the Finite Memory Observer (FMO) has the advantage to ensure the convergence of the state estimation in a finite time, at least in the absence of disturbances. Despite the interest mentioned above, few studies have been published on the FMO compared to those on infinite memory observers. The pioneering works are due to Jazwinski [START_REF] Jazwinski | Stochastic processes and filtering theory[END_REF], [START_REF] Buxbaum | Fixed-memory recursive filters[END_REF] and [START_REF] Wonham | Linear multivariable control. Lectures notes in economics and mathematical systems[END_REF] where a state estimation formulation from a discrete or continuous integral form of the inputs-outputs has been proposed. This form was also considered by Medvedev [START_REF] Medvedev | A continuous finitememory deadbeat observer[END_REF], [START_REF]Investigation of a finite memory observer structure[END_REF] and Byrski [8] [9]. This filtering technique applied for the continuous case [START_REF] Engel | A Continuous-Time Observer which converges in finite time[END_REF], [START_REF] Raff | An impulsive observer that estimates the exact state of a linear continuous-time system in predetermined finite time[END_REF] as well as for the discrete one [START_REF] Park | LMS finite memory estimators for discrete-time state space models[END_REF], [START_REF] Soeda | Finite-time settling adaptive observer for linear discrete-time systems[END_REF], offers a generic aspect in the sense that it is applied for state estimation, parameter estimation and control with sliding horizon. Note that the nonlinear case has been less discussed, however one can refer to the following works [START_REF] Alessandri | Neural approximators for nonlinear finite-memory state estimation[END_REF], [START_REF] Li | A high-gain-based global finite-time nonlinear observer[END_REF], [START_REF] Du | Recursive design of finite-time convergent observers for a class of timevarying nonlinear systems[END_REF]. Several works based on sliding horizon for an exact state reconstruction in a finite time (without measurement noise nor model uncertainties) may be found in the literature with different terminologies like exact observers, FMO, integral observers and ideal observers. Most of these studies are academic, nevertheless some of them are applied to electric power transmission networks [START_REF] Dziwinski | The application of the exact state estimation method in electric power systems[END_REF], diesel engines diagnosis [START_REF] Graton | Diagnosis of diesel injection system using finite memobservers[END_REF], fuel cell estimation [START_REF] Rakhtala | Design of finite-time high-order sliding mode state observer: A practical insight to PEM fuel cell system[END_REF], state converters estimation [START_REF] Defoort | Robust finite time observer design for multicellular converters[END_REF] or general applications in the diagnosis framework [START_REF] Nuninger | Finite memory generalized state observer approach for failure detection in dynamic systems[END_REF], [START_REF] Medvedev | State estimation and fault detection by a bank of continuous finite-memory filters[END_REF]. Given the advantage of the FMO, it seems interesting to extend its scope to nonlinear systems. As it was mentioned previously, few works deal with the nonlinear case. This is why in the present paper a particular attention is given to nonlinear systems represented in a polytopic or Takagi-Sugeno (T-S) form. The polytopic model may have different names, such as fuzzy model (Takagi-Sugeno model), multi-model, local model networks, etc. It allows the representation of nonlinear behaviors by the interpolation of a set of linear submodels. Each submodel contributes to the global behavior of the nonlinear system through a weighting function [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF].

The T-S structure may be obtained by transforming the original system into a polytopic linear model based on the sector nonlinearity approach and the convex polytopic transformation. This transformation has the major interest to exactly represent the system without any loss of informations since the considered nonlinearities are bounded (each parameter varies between two known values). In the present work, finite memory observer for nonlinear systems represented in a T-S form are proposed. The paper is organized as follows. Section II introduces the state and unknown input estimation with finite memory observer for linear systems. In section III the T-S systems are considered for both measurable and unmeasurable premise variables. Illustrative examples are presented in Section IV and conclusion results are detailed in section V.

Preliminaries: Finite Memory Observer for linear systems

The FMO is designed on a finite sliding horizon of length r + 1. From available measurements at time k + r in a time interval [k : k +r], the system states are then estimated in finite time. The horizon is moved by one step forward [k + 1 : k + r + 1] which allows to estimate the state at the instant k + r + 1. The next section details the above procedure and expands it to the unknown inputs estimation.

State estimation

Let us consider the following system:

x k+1 = Ax k + Bu k , x ∈ R n x y k = Cx k (1)
where x k is the system state at the instant k, u k ∈ R n u the input and y k the output. A, B and C are the system matrices with appropriate dimensions.

Using the system equation ( 1), the output expression at time k + r is given by:

y k+r = CA r x k +CA r-1 Bu k + • • • +CBu k+r-1 (2)
Gathering the outputs on the time horizon [k : k + r], let us note:

ỹk = M x x k + M u ũk (3)
with: 1) is given by the following structure:

                                             ỹk =      y k y k+1 . . . y k+r      ũk =      u k u k+1 . . . u k+r-1      M x =      C CA . . . CA r      M u =            0 0 . . . . . . 0 CB 0 . . . . . . 0 CAB CB . . . . . . . . . . . . CA r-1 B CA r-2 B . . . . . . CB            Proposition 1 A FMO for system (
     xk+r = A r xk + T ũk xk = (M T x W M x ) -1 M T x W ( ỹk -M u ũk ) T = A r-1 B A r-2 B . . . B ( 4
)
where W is a positive definite weighting matrix of appropriate dimension chosen accordingly to the state components for which some specific importance is given.

One can easily verify that xk+r = x k+r by replacing the expression of xk and (3) in xk+r (4):

               xk+r = A r (M T x W M x ) -1 M T x W ( ỹk -M u ũk ) + T ũk =A r (M T x W M x ) -1 M T x W (M x x k + M u ũk -M u ũk ) +T ũk = A r x k + T ũk = x k+r

Remark 1 Expression (4) shows that the state estimate xk+r at the instant k + r results from the input and outputs filtering on the time horizon

[k : k + r]. Since the same procedure is applied one step forward [k + 1 : k + r + 1]
, it is possible to establish a recurrence relation between the state estimation xk+r and xk+r+1 .

In the next subsection, an extension for joint state and unknown input FMO is considered. The proposed structure is based on the same one given in (4).

State and unknown input finite memory observer

Let us consider the following system subject to non measurable unknown input p:

x k+1 = Ax k + Bu k + Pp k , x ∈ R n x , p ∈ R n p y k = Cx k (5)
The unknown input dynamic is given by:

p k+1 = p k + δ k (6)
where δ k is the unknown input variation at the instant k.

An augmented state x a k =

x k p k is defined with the concatenation of the system state and the unknown input:

x a k+1 = A a x a k + B a u k + P a δ k y k = C a x a k ( 7
)
with:

               x a k = x k p k A a = A P 0 I B a = B 0 P a = 0 I C a = C 0 (8)
Equation ( 3) may be extended in the form:

ỹk = M a x x a k + M a u ũk + M a δ δk (9)
with:

                                   ỹk =      y k y k+1 . . . y k+r      ũk =      u k u k+1 . . . u k+r-1      δk =      δ k δ k+1 . . . δ k+r-1      M a x =       C a C a A a . . . C a (A a ) r                                                            M a u =            0 0 . . . . . . 0 C a B a 0 . . . . . . 0 C a A a B a C a B a . . . . . . . . . . . . C a A a r-1 B a C a A r-2 B a . . . . . . C a B a            M a δ =            0 0 . . . . . . 0
C a P a 0 . . . . . . 0 C a A a P a C a P a . . . . . . . . . . . . C a A a r-1 P a C a A r-2 P a . . . . . . C a P a           
By analogy with the previous subsection results, the augmented state estimate xa k is given by a similar expression than the one given in [START_REF] Buxbaum | Fixed-memory recursive filters[END_REF]. Then from xa k we directly deduce the estimation of xk of the state and pk of the unknown input. In the next section an extension of the above finite memory observer is given for nonlinear T-S systems.

Finite memory observers for T-S systems

The T-S representation of a nonlinear system consists in a time-varying interpolation of a set of linear submodels. Each submodel contributes to the global behavior of the nonlinear system through a weighting function µ i (ξ k ) [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF]. Let us consider the following T-S model [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF]: [START_REF] Engel | A Continuous-Time Observer which converges in finite time[END_REF] with:

x k+1 = A k x k + B k u k
A k = r ∑ i=1 µ i (ξ k )A i , B k = r ∑ i=1 µ i (ξ k )B i ( 11
)
where the weighting functions µ i (ξ k ) depend on the so- called premise variable ξ k which may be a state, input, or output combination. These weighting functions satisfy the following convex sum property:

0 ≤ µ i (ξ k ) ≤ 1, r ∑ i=1 µ i (ξ k ) = 1 (12) s.t. x k ∈ R n x and u k ∈ R n u .
Roughly speaking, the FMO design for T-S models is the same as for the conventional linear case. However, some difficulties occur when the premise variables are not known.

Known premise variables

In the present subsection, the case of known premise variables is considered. Based on the same structure as for the linear case, for the time horizon [k : k + r], the output vector is given by:

ỹk = M x (ξ k )x k + M u (ξ k ) ũk ( 13
)
with the following definitions:

ỹT k = y T k . . . y T k+r , ũT k = u T k . . . u T k+r-1 M x (ξ k ) =         C CA k CA k+1 A k . . . CA k+r-1 . . . A k         , M u (ξ k ) =       (M 1 u (ξ k )) T (M 2 u (ξ k )) T . . . (M r u (ξ k )) T       T (M 1 u (ξ k )) T = 0 (CB k ) T (CA k+1 B k ) T . . . (CA k+r-1 . . . A k+1 B k ) T (M 2 u (ξ k )) T = 0 0 (CB k+1 ) T . . . (CA k+r-1 . . . A k+2 B k+1 ) T (M r u (ξ k )) T = 0 0 . . . (CB k+r-1 ) T (14)
Note that the matrices A k and B k [START_REF] Raff | An impulsive observer that estimates the exact state of a linear continuous-time system in predetermined finite time[END_REF] are time dependent (depend on the premise variables ξ k ) which implies that the matrices M x (ξ k ) and M u (ξ k ) are also time dependent.

At time k, let us consider the following criterion:

Φ(x k ) = ỹk -M x (ξ k )x k -M u (ξ k ) ũk 2 W ( 15
)
where W is a positive definite weighting matrix of appropriate dimension chosen accordingly to the state components for which some specific importance is given.

Supposing that M x is full column rank, the state estimator may be given in the following form:

           xk = M T x (ξ k )W M x (ξ k ) -1 M T x (ξ k )W ỹk -M u (ξ k ) ũk xk+r = A k+r-1 . . . A k xk + T ũk T= A k+r-1 . . . A k+1 B k A k+r-1 . . . B k+1 . . . B k+r-1 (16)
The state estimation at time k + r is then deduced using the data collected on the interval [k : k + r]. Then, the horizon is moved by one step forward [k + 1 : k + r + 1] which allows to estimate the state at the instant k +r +1.

State filtering

The model (1) does not take into account the noise that frequently corrupts the outputs. Without a precise modeling of that noise, it is however possible to reduce its influence on the estimates by introducing, in the optimization criterion, a regularization term:

Φ(x r k ) = ỹk -M x (ξ k )x r k -M u (ξ k ) ũk 2 W + x r k -x k-1 2 F
where F is a positive definite weighting matrix of appropriate dimension which plays the same role as W (chosen accordingly to accentuate the filtering on a considered state component). The derivative of the above criterion with regard to the state is given by:

∂ Φ ∂ x r k = -2M T x (ξ k )W ( ỹk -M x (ξ k )x r k -M u (ξ k ) ũk ) +2 F (x r k -x k-1 ) (17)
The solution is given for:

xr k = (M T x (ξ k )W M x (ξ k ) + F) -1 M T x (ξ k )W ( ỹ -M u ũ) + F xk-1 (18)
Remark 2 By developing the inverse of the matrix M T x (ξ k )W M x (ξ k ) + F, the solution ( 18) may be expressed in terms of the non regularized solution given in [START_REF] Du | Recursive design of finite-time convergent observers for a class of timevarying nonlinear systems[END_REF].

xr k = (M T x (ξ k )W M x (ξ k )) -1 F + I -1 xk + (M T x (ξ k )W M x (ξ k )) -1 F xk-1 ( 19
)
where xk correspond to the state estimation obtained without filtering. Unsurprisingly, from [START_REF] Rakhtala | Design of finite-time high-order sliding mode state observer: A practical insight to PEM fuel cell system[END_REF], we notice that if F → 0, then

xr k → xk and if F → ∞ then xr k → xk-1 .

Unknown premise variables

Let us now consider the case where the the weighting functions of the matrices A k and B k depend on the (unknown) state of the system. Instead of the analytical solution [START_REF] Graton | Diagnosis of diesel injection system using finite memobservers[END_REF], an iterative solution is proposed. For the time horizon [k : k + r], let us note x0 k the initial state estimation, which may be set equal to the previous horizon state estimate xk-1 . The state estimate is then given by:

                     x(1) k = M T x ( x(0) k )W M x ( x(0) k ) -1 M T x ( x(0) k ) W ỹk -M u ( x(0) k ) ũk M x ( x(0) k ) = M x (ξ k ) | ξ k = x(0) k M u ( x(0) k ) = M u (ξ k ) | ξ k = x(0) k (20)
More generally, at iteration q + 1 we get:

                             x(q+1) k = M T x ( x(q) k )W M x ( x(q) k ) -1 M T x ( x(q) k ) W ỹk -M u ( x(q) k ) ũk x(q+1) k+r = A k+r-1 . . . A k x(q+1) k + T ( x(q) k ) ũk M x ( x(q) k ) = M x (ξ k ) | xk = x(q) k M u ( x(q) k ) = M u (ξ k ) | xk = x(q) k (21)
The same idea is applied when considering the noise filtering as explained in section III.B.

State and unknown input estimation

The extension of the proposed observer in section II.B to the T-S case is deduced straightforwardly by replacing A b , B k and C in ( 14) by A a k , B a k and C a defined by [START_REF] Byrski | The survey for the exact and optimal state observers in Hilbert spaces[END_REF] in which A, B and P are replaced by A k , B k and P k respectively.

Illustrative examples

In the following section, numerical examples are given in order to illustrate the effectiveness of the proposed observers. Let us consider the T-S model with two submodels and state dependent premise variables:

                                     A 1 =    0.
   C = 1 0 0 µ 1 (x k ) = 1 2 (1 + tanh(x 1 (k)/0.5)), µ 2 (x k ) = 1 -µ 1 (k) (22)

State estimation

In the considered example, the weighting functions are state dependent (x 1 (k)). The formulation given by ( 20) and ( 21) of section III.C is then applied.

In figure 1 are depicted the system states x i , i = 1, 2, 3 as well as their estimates. Only one state is measured and the considered time horizon length is equal to 3. Figure 2 depicts the system input u(k), output y(k) and its estimate ŷ(k) as well as the weighting function µ 1 (x k ) which covers the two modes (submodels). As seen on these figures, the states are well estimated (but no measurement noise was considered).

Estimation with noise and unknown inputs

In this subsection, unknown inputs are also considered. In order to illustrate the efficiency of proposed filtering algorithm, two simulations results are presented. The first case is about state and unknown input estimation, the measurement are subject to an additive measurement noise but the filtering algorithm is not applied. In the second case, the state filtering given in section III.B is applied. The unknown input matrix P is defined as:

P = 1 1 1 T ( 23
)
In this second example, the matrix C is defined by:

C = 1 0 0 0 1 0 (24)
In figure 3 are depicted the system states and their estimates. The joint estimation state/unknown input was done with measurement noise but without any filtering.

As the figure shows, the third state estimate x3 (k) is greatly affected by the noise. This result may be explained by the fact that since only the states x 1 and x 2 are measured, the estimation is made to the detriment of the third one.

In order to improve the estimation, the filtering proposed in section III.B is then applied. The figure 4 shows the unknown input δ k and its estimate δk as well as the input u k . The filtering effect is clearly illustrated in figures 5 and 6 where the improvement is clearly shown for the third state.

Conclusion

In this paper, a Finite Memory Observer design for nonlinear T-S model was considered. A joint state and unknown input reconstruction algorithm was proposed for both measurable and unmeasurable premise variables. The case of measurement noise was also studied with the proposition of a filtering algorithm. Numerical examples were presented in order to highlight the approach efficiency. As futur work, it turns possible to to extend the proposed approach to sensor fault detection and isolation with the help of a bank of finite memory observers.

Figure 1 .Figure 2 .

 12 Figure 1. System states x k and their estimates xk

Figure 3 .

 3 Figure 3. System states x k and their estimates xk : with noise measurement and without filtering

Figure 4 .

 4 Figure 4. The input u k and unknown input δ k and its estimate: with noise measurement and without filtering

Figure 5 .Figure 6 .

 56 Figure 5. System states x k and their estimates xk : with noise measurement and filtering