Didier Theilliol Mickaël Rodrigues

Dominique Sauter

D Theilliol

Design of an Active Fault Tolerant Control for Nonlinear Systems described by a Multi-Model Representation

In this paper, an new active Fault Tolerant Control (FTC) strategy is developed to nonlinear systems described by multiple linear models to prevent the system deterioration by the synthesis of adapted controllers. When a fault is detected by the fault detection and diagnosis scheme, the reconfigurable controller is designed automatically using a robust gain scheduling strategy. The main contribution concerns the design of state feedback gains through LMI both in fault-free and faulty cases in order to preserve the system performances over a wide operating range. For each separate actuator with which the system is robustly stabilizable, a robust pole placement is designed by pole clustering. The effectiveness and performances of the method have been illustrated in simulation considering a classical nonlinear benchmark.

I. INTRODUCTION

The increasing demands for higher system performance, product quality, productivity and cost efficiency lead to a continuous growth of the complexity and automation degree of technical processes. Typical instances of such kind of complex and distributed systems are sophisticated vehicles, civil aircraft, environment treatment processes, large fleets and infrastructures [START_REF] Noura | Fault-tolerant control in dynamic systems: Application to a winding machine[END_REF]. Associated with these development trends, high reliability, availability and safety become an important system requirement which is included in many international standards and regulations. The objective of Fault Tolerant Control system (FTC) is to maintain current performances closed to desirable performances and preserve stability conditions in the presence of component and/or instrument faults; in some circumstances reduced performance could be accepted as a trade-off. Accommodation capability of a control system depends on many factors such as the severity of the failure, the robustness of the nominal system, and the actuators redundancy. FTC can be motivated by different goals depending on the application under consideration, for instance, safety in flight control or reliability or quality improvements in industrial processes. Various approaches for FTC have been suggested in the literature [START_REF] Patton | Fault-tolerant control: the 1997 situation[END_REF], [START_REF] Zhang | Bibliographical review on reconfigurable Fault-Tolerant Control systems[END_REF], [START_REF] Blanke | Diagnosis and Fault-Tolerant Control[END_REF] and [START_REF] Zhang | Fault Tolerant Control system design with explicit consideration of performance degradation[END_REF] but often deal with linear systems. For nonlinear systems, the design of Fault Tolerant controller is far more complicated. Nonlinear systems based on multiple linear models, represents an attractive solution to deal with the control of nonlinear systems [START_REF] Leith | Survey of gain-scheduling analysis and design[END_REF], [START_REF] Banerjee | H∞ control of nonlinear processes using multiple linear models[END_REF], [START_REF] Tayebi | Iterative learning control for nonlinear systems described by a blended multiple model representation[END_REF] or FDI methods as in the chapter nine of [START_REF] Chen | Robust model-based fault diagnosis for dynamic systems[END_REF] where nonlinear mickael.rodrigues@cran.uhp-nancy.fr dynamic systems are described by a number of locally linearized models based on the idea of Tagaki-Sugeno fuzzy models or as interpolated multiple linear models [START_REF] Murray-Smith | Multiple Model Approaches to Modelling and Control[END_REF]. A great number of gain scheduling strategies are developed in fault-free case [START_REF] Leith | Survey of gain-scheduling analysis and design[END_REF] and we proposed to develop one of them in faulty-case. Various recent FDI/FTC studies, based on a multiple model method have been developed in order to detect, isolate and estimated an accurate state of a system in presence of faults/failures around an unique operating point [START_REF] Maybeck | Multiple model adaptive algorithms for detecting and compensating sensor and actuator/surface failures in aircraft flight control systems[END_REF], [START_REF] Zhang | Integrated active Fault-Tolerant Control using IMM approach[END_REF] and in chapter 7 of [START_REF] Blanke | Diagnosis and Fault-Tolerant Control[END_REF]. Compared to multi-model based reconfigurable control method presented by [START_REF] Astrom | Control of complex systems[END_REF], this paper not consider some redundant hardware which is very useful when failures are supposed to occur on the system. In this paper, an active fault tolerant strategy is developed so as to avoid actuator fault effect on nonlinear system where faults are assumed to be incipient, abrupt but not generate a total actuator fault i.e. a failure. Under the assumption that the fault is detected, isolated and estimated, the developed method preserves the system performances through an appropriate gain scheduling synthesis in faulty case. Compared to recent work applied to similar nonlinear system [START_REF] Theilliol | A multiple model based approach for Fault Tolerant Control in nonlinear systems[END_REF], where a multi-model representation is considered, the proposed FTC strategy is not based on an additional control law but on the redesign of appropriate gain in faulty case allowing stability and performances of the system. The paper is organized as follows. In section II, we introduce a state space representation with both additive and multiplicative fault defined around operating points. A global state space representation of nonlinear system is given through a multi-model approach. In section III, we introduce a pole placement by LMI region and then a gain synthesis for each actuator generate an active global state feedback synthesis. This scheme provides a new active Fault Tolerant Control synthesis. A simulation example is given in section IV to illustrate the proposed method. Finally, concluding remarks are given in the last section.

II. NONLINEAR REPRESENTATION AND FAULT TOLERANT CONTROL STRATEGY

A. Nonlinear representation

Consider a discrete-time nonlinear dynamical system described by:

x(k + 1) = g x k , u k y(k) = h x k , u k (1)
where x k ∈ X ∈ R n represents the state vector, u k ∈ U ∈ R p is the input vector and y ∈ R m is the output vector. We assume that functions g x k , u k and h x k , u k are continuously differentiable.

Definition: [START_REF] Wan | Efficient scheduled stabilizing model predictive control for constrained nonlinear systems[END_REF] Given a set U, a point x 0 ∈ X ⊆ R n is an equilibrium point of the system (1) if a control u 0 ∈ U exists such that x 0 = g(x 0 , u 0 ). We call a connected set of equilibrium points an equilibrium surface. Suppose (x e , u e , y e ) is a point on an equilibrium surface and define a shifted state x = x-x e and a shifted input ū = u-u e , the nonlinear system with respect to (x e , u e ) can be expressed as:

x(k + 1) = g x(k), u(k) -g xe, ue) f x(k), ū(k) ȳ(k) = h x(k), u(k) -h xe, ue) v x(k), ū(k) (2)
It is assumed that dynamic behaviour of the system operating at different operating points can be approximated by a set of N linear time invariant models in chapter 1 of [START_REF] Murray-Smith | Multiple Model Approaches to Modelling and Control[END_REF] and in the following references [START_REF] Rodrigues | Fault Diagnosis on Industrial Systems based on a Multiple Model approach[END_REF], [START_REF] Tayebi | Iterative learning control for nonlinear systems described by a blended multiple model representation[END_REF], [START_REF] Wan | Efficient scheduled stabilizing model predictive control for constrained nonlinear systems[END_REF]. Consider the following state space representation of a nonlinear system around j-th operating point, j ∈ [1, . . . , N ], with additive actuator faults:

x k+1 -x j e = A j (x k -x j e ) + B j (u k -u j e ) + F j f k y k -y j e = C j (x k -x j e ) + D j (u k -u j e ) (3)
Matrices (A j , B j , C j , D j ) are invariant matrices defined around the j th operating point (OP j ) generally obtained from a nonlinear system using a first-order Taylor expansion about (x j e , u j e ) or identification of a nonlinear system around predefined operating points as for example in chemical processes in [START_REF] Ozkan | Control of a solution copolymerization reactor using multi-model predictive control[END_REF], [START_REF] Theilliol | Adaptive filter design for FDI in nonlinear systems based on multiple model approach[END_REF]. The fault distribution matrix is represented by F j ∈ R n×p . f k ∈ R p represents the actuator fault vector and in fault-free case it is obviously equal to zero. In the following, we consider that D j = 0. This linear system (3) can be specified by the set of system matrices as in fault-free case in [START_REF] Murray-Smith | Multiple Model Approaches to Modelling and Control[END_REF], [START_REF] Tayebi | Iterative learning control for nonlinear systems described by a blended multiple model representation[END_REF] and in faulty-case in [START_REF] Rodrigues | Fault Diagnosis on Industrial Systems based on a Multiple Model approach[END_REF]:

S j = A j B j F j C j , ∀j = [1, . . . , N ] (4)
Let S k be a matrix sequence varying within a convex set, defined as:

S k := N j=1 ρ j k S j : ρ j k ≥ 0, N j=1 ρ j k = 1 (5)
In the Multi-Model framework, S k characterizes at each sample the nonlinear system and consequently, the dynamic behavior of nonlinear system can be defined by a convex set of a multiple LTI models. As considered in [START_REF] Murray-Smith | Multiple Model Approaches to Modelling and Control[END_REF] and [START_REF] Theilliol | Adaptive filter design for FDI in nonlinear systems based on multiple model approach[END_REF], a possible model that would be able to catch the full range of operation is made from N weighting local models OP j by interpolation functions ρ j k . These activation functions

ρ j k ∀j ∈ [1, . . . , N ] lie in a convex set Ω = {ρ j k ∈ R N , ρ k = [ρ 1 k . . . ρ N k ] T , ρ j k ≥ 0 ∀j and N j=1 ρ j k = 1}
and these functions are generated via works of [START_REF] Adam-Medina | Fault diagnosis in nonlinear systems through an adaptive filter under a convex set representation[END_REF] and [START_REF] Theilliol | Adaptive filter design for FDI in nonlinear systems based on multiple model approach[END_REF], which permit to generate insensitive residual to faults and some uncertainties. So, activation functions are robust against faults and errors modeling and the dynamic system is well represented. The representation (3) considers additive fault representation but there exists multiplicative representation for specific actuator fault as:

x k+1 = A j (x k -x j e ) + B j (I -γ a )(u k -u j e ) + x j e y k = C j (x k -x j e ) + y j e (6)
So, let consider a local multiplicative actuator fault representation as a blended representation (as in [START_REF] Tayebi | Iterative learning control for nonlinear systems described by a blended multiple model representation[END_REF] in fault-free case):

x k+1 = N j=1 ρ j k [A j (x k -x j e ) + B j (I -γ a )(u k -u j e ) + x j e ] y k = N j=1 ρ j k [C j (x k -x j e ) + y j e ] (7)
with

γ a diag[γ a 1 , γ a 2 , . . . , γ a p ], γ a i ∈ R, such that γ a i = 1 represents a total lost, a failure of i-th actuator,i ∈ [1, . . . , p]
and γ a i = 0 implies that i-th actuator operates normally. The objective of fault detection and diagnosis is to determine the extent of the loss in the control effectiveness by estimating γ a i on-line in real-time [START_REF] Rodrigues | Fault Diagnosis on Industrial Systems based on a Multiple Model approach[END_REF] so that an on-line automatic reconfigurable controller can be synthesized. The matrices (A j , B j , C j ) ∈ S k and the relation between state space representations (3) and ( 6) is equivalent to

F j f k = -B j γ a (u k -u j e ) (8)

B. Fault Tolerant Control design

In closed-loop, the fault occurrence could be detected as described in [START_REF] Noura | Fault-tolerant control in dynamic systems: Application to a winding machine[END_REF] for linear case and [START_REF] Theilliol | A multiple model based approach for Fault Tolerant Control in nonlinear systems[END_REF] for multi-linear systems and Fault Tolerant Control could be performed via an additional control law as in [START_REF] Theilliol | Fault diagnosis and accommodation of three-tank system bsaed on analytical redundancy[END_REF] which permits to avoid fault on a system based on a state space representation as [START_REF] Zhang | Bibliographical review on reconfigurable Fault-Tolerant Control systems[END_REF]. In these papers, the goal was to synthesize a new control law u F T C with a nominal one u nom and additional one u ad . The term u ad was performed in order to vanish fault on the system. The global control law is obtained by interpolating gains of each local controller [START_REF] Leith | Survey of gain-scheduling analysis and design[END_REF] and is defined as:

u j F T C = u j nom + u j ad (9)
with u j nom = -K nom (x kx j e ) + u j e . The gains were only performed for nominal cases and do not take into account fault occurrence. Under the assumption that fault is detected, isolated and estimated, based on a multiplicative fault representation defined in [START_REF] Banerjee | H∞ control of nonlinear processes using multiple linear models[END_REF], we propose a control law u j F T C , around an Operating Point j, which must vanish faults as the following form:

u j F T C = -(I -γ a ) + K(x k -x j e ) + u j e ( 10
)
The objective of fault accommodation consists in a strategy by which the faulty system is controlled in a specific way, so as to achieve the objectives which were (before the fault) achieved by the healthy system. So, note that

u j F T C (k) = [I -γ a ] + u j nom (k) = -[I -γ a ] + K nom (x k - x j e ) + u j e = -K F T C (x k -x j e ) + u j e (11)
In the following, we will note u j F T C (k) as u j F T C so as to relax notations and have a better understanding. We note (.) + the Moore-Penrose matrix inverse and it is assumed that matrix [Iγ a ], which is diagonal, has full column rank i.e. the dimension n of the system is greater than number of p actuators. Assuming that there is no mismatch between fault and their estimation and without considering total fault, this state feedback is inserted in the state space representation (6) around an OP j and leads to:

x k+1 -x j e = A j (x k -x j e ) + B j (I -γ a )(-K F T C (x k -x j e ) + u j e -u j e ) = (A j -B j Knom)(x k -x j e ) (12)
Even if there is an actuator fault on the system, this one still operates normally with the new state feedback gain

K F T C = [I -γ a ] + K nom .
As previously defined in [START_REF] Zhang | Fault Tolerant Control system design with explicit consideration of performance degradation[END_REF], the model probability is viewed as a scheduled variable in the synthesis of the controller like in [START_REF] Leith | Survey of gain-scheduling analysis and design[END_REF], [START_REF] Tayebi | Iterative learning control for nonlinear systems described by a blended multiple model representation[END_REF] and [START_REF] Theilliol | A multiple model based approach for Fault Tolerant Control in nonlinear systems[END_REF], and the new global control law is defined as:

u k = N j=1 ρ j k u j F T C = - N j=1 ρ j k [K F T C (x k -x j e ) + u j e ] (13)
with u j F T C the output of each local controller defined around each OP j . In order to synthesize state feedback for FTC ensuring both active control in multi-model philosophy and quadratic stability, use of LMI provide a well toolbox for these purposes. Total failures are not consider in this paper but only partial ones. We attract attention that the system has to be observable in all around each operating points.

III. FAULT TOLERANT CONTROL ON MULTIPLE OPERATING POINTS

A. Pole clustering

In the synthesis of control system, some desired performances should be considered in addition to stability. In fact, classical stability conditions do not deal with transient responses of the closed-loop system. In contrast, a satisfactory transient response can be guaranteed by confining its poles in a prescribed region. For many real problems, exact pole assignment may not be necessary: it suffices to locate the closed-loop poles in a prescribed subregion in the complex plane. We will discuss about pole clustering by introducing the following LMI-based representation of stability regions.

Definition 1: LMI stability region [START_REF] Chilali | H∞ design with pole placement constraints: an LMI approach[END_REF]. A subset D of the complex plane is called an LMI region if there exist a symmetric matrix α = [α kl ] ∈ R n×n and a matrix β =

[β kl ] ∈ R n×n such that D = {z ∈ C : f D (z) < 0} (14)
where the characteristic function f D (z) is given by

f D (z) = [α kl + β kl z + β lk z] 1≤k,l≤n (15)
(f D is valued in the space of n × n Hermitian matrices). Moreover, LMI regions are convex and symmetric with respect to the real axis since for any z ∈ D : f D (z) = f D (z) < 0. Then, pole location in a given LMI region can be characterized in terms of a block matrix if and only if there exists a symmetric matrix P > 0 such that [START_REF] Chilali | H∞ design with pole placement constraints: an LMI approach[END_REF]:

M D (A cl , P ) = α ⊗ P + β ⊗ (A cl P ) + β T ⊗ (A cl P ) T = [α kl P + β kl A cl P + β lk P A T cl ] 1≤k,l≤m (16)
with A cl = A -BK and M = [µ kl ] 1≤k,l≤n means that M is an n × n matrix (respectively, bloc matrix) with generic entry (respectively bloc) µ kl . Note that M D (A cl , P ) in ( 16) and f D (z) in ( 15) are related by the substitution (P, A cl P, P A T cl ) ↔ (1, z, z). It is easily seen that LMI regions are convex and symmetric with respect to real axis. Specifically, the circular LMI region D is considered:

D = {x + jy ∈ C : (x + q) 2 + y 2 < r 2 } (17)
centered at (-q, 0) with radius r > 0, where the characteristic function f D (z) is given by:

f D (z) = -r z + q z + q -r (18)
Therefore, this circular region puts a lower bound on both the exponential decay rate and the damping ratio of the closed-loop response, and thus is very common in practical control design. It is obvious that well chosen LMI region is needed for ensuring stability and good results: the parameters q, r have to be defined by the engineer.

B. Control law synthesis in fault-free case

Let consider the state space representation (7) of nonlinear system defined around the Operating Points OP j , ∀j = 1, . . . , N

x k+1 = N j=1 ρ j k A j (x k -x j e )+ i∈I B i j (I -γ a )(u k -u j e )+ x j e y k = N j=1 ρ j k [C j (x k -x j e ) + y j e ] (19)
with I : i = 1, . . . , p the actuators for each OP j and matrices (A j , B j , C j ) ∈ S k defined in [START_REF] Zhang | Fault Tolerant Control system design with explicit consideration of performance degradation[END_REF]. Consider the matrix representing total faults in all actuators but the i-th:

B i j = [0, . . . , 0, b i j , 0, . . . , 0] (20)
and

B j = [b 1 j , b 2 j , . . . , b p j , ] with b i j ∈ R n×1 .
It is assumed that each column of B j is full column rank whatever the OP j . The pairs (A j , b i j ), ∀i = 1, . . . , p are assumed to be controllable for all ∀j = 1, . . . , N . Let D, a LMI region defining a disk with a center (-q, 0), and a radius r with (q + r) < 1 for defining pole assignment in the unit circle. Assume that for each B i j , there exist matrices X i = X T i > 0 and Y i , ∀j = 1, . . . , N, ∀i = 1, . . . , p such as:

-rX i qX i + (A j X i -B i j Y i ) T qX i + A j X i -B i j Y i -rX i < 0 (21)
It can be noticed that if q = 0 and r = 1, the previous equation ( 21) is equivalent to solve a classical quadratic stability problem. Based on the assumptions that for each OP j each pairs (A j , b i j ) are controllable, it is possible to find a Lyapunov matrices X i > 0 and state-feedback K i with Y i = K i X i and finally form a global state-feedback gain K nom .

Theorem 1: Consider the system [START_REF] Adam-Medina | Fault diagnosis in nonlinear systems through an adaptive filter under a convex set representation[END_REF] in fault-free case (γ a = 0) defined for all OP j , j = 1, . . . , N : it is possible to develop a mixing of pre-designed state-feedback gains matrices K i = Y i X -1 i for each actuator i with i = 1, . . . , p such that ( 21) holds for all j = 1, . . . , N . The state feedback control for each operating point is given by:

u j nom = -( p i=1 G i Y i )( p i=1 X i ) -1 (x k -x j e ) + u j e (22) = -Y X -1 (x k -x j e ) + u j e = -K nom (x k -x j e ) + u j e with p i=1 G i Y i = Y , X = p i=1 X i and G i = B i+ j B i j
is matrix that has zeros everywhere except in entry (i, i)

where it has a one. The general control law for all OP j could be defined as:

u k = N j=1 ρ j k u j nom = - N j=1 ρ j k [K nom (x k -x j e ) + u j e ] ( 23
)
Proof: Summation of ( 21) for i = 1, . . . , p gives for one equilibrium point

j p i=1 -rX i qX i + (A j X i -B i j Y i ) T qX i + (A j X i -B i j Y i ) -rX i < 0 (24) related to the quadratic D-stability in a prescribed LMI region. Next, denote X = p i=1 X i (with X = X T > 0) to obtain       -rX qX + (A j X - p i=1 B i j Y i ) T qX + (A j X - p i=1 B i j Y i ) -rX       < 0 (25)
Now, denote the l-th row of the matrix Y i as Y l i , i = 1, ..., p and l = 1, . . . , p, i.e.

Y l i = G l Y i (26) Therefore, p i=1 B i j Y i = p i=1 [0, . . . , 0, b i j , 0, . . . , 0]Y i i = B j p i=1 Y i i (27) leading to p i=1 B i j Y i = B j ( p i=1 G i Y i ) (28) Thus, taking Y = p i=1 G i Y i , equation (28) becomes p i=1 B i j Y i = i∈I B i j Y i = B j Y (29)
which, substituted into LMI (25), finally makes

-rX qX + (A j X -B j Y ) T qX + (A j X -B j Y ) -rX < 0 (30)
for all OP j , j = 1, . . . , N . By multiplying each LMI (30) by ρ j k and summing all of them, we obtain

      -rX qX + N j=1 ρ j k (A j X -B j Y ) T qX + N j=1 ρ j k (A j X -B j Y ) -rX       < 0 (31) it is equivalent to -rX qX + (A(ρ)X -B(ρ)Y ) T qX + (A(ρ)X -B(ρ)Y ) -rX < 0 (32) with A(ρ) = N j=1 ρ j k A j ∈ S k and B(ρ) = N j=1 ρ j k B j ∈ S k .
Hence quadratic D-stability is ensured by solving (31) and Y = K nom X quadratically stabilizes the system [START_REF] Adam-Medina | Fault diagnosis in nonlinear systems through an adaptive filter under a convex set representation[END_REF] under the set S k by solving (32) with a state feedback law

u k = N j=1 ρ j k u j nom = - N j=1 ρ j k [Y X -1 (x k -x j e ) + u j e ].
Remark1: It could be noticed that gain synthesis through multiple operating point with such LMI consideration provide only one single gain for all OP due to Bilinear Matrix inequality (BMI) problem in term (2, 1) of LMI [START_REF] Chilali | H∞ design with pole placement constraints: an LMI approach[END_REF]. However, other system such piecewise linear system could use the same approach with a multiple gain synthesis as in [START_REF] Ozkan | Control of a solution copolymerization reactor using multi-model predictive control[END_REF].

Remark2: If all local models have the same B j matrices, i.e. B j = B for all j = 1, . . . , N , it is also possible to use the following parameter-dependent state feedback law instead of (23):

u k = N j=1 ρ j k u j nom = - N j=1 ρ j k [K j (x k -x j e ) + u j e ] (33)
with

K j = Y j Q -1 .
The resulting matrices remain the same except that K j replaces K, B replaces B j and Y j replaces Y .

C. Active Fault Tolerant Control design

By considering the system [START_REF] Adam-Medina | Fault diagnosis in nonlinear systems through an adaptive filter under a convex set representation[END_REF] and based on the previous synthesis control law, the FTC method can be developed in this section where only partial actuator faults are considered under assumptions that fault occurrence and fault magnitude γ a are exactly known.

Theorem 2: Consider the system [START_REF] Adam-Medina | Fault diagnosis in nonlinear systems through an adaptive filter under a convex set representation[END_REF] in faulty case (γ a = 0) coupled with regulators with gains K i = Y i X -1 i for all equilibrium point j = 1, ..., N and for each actuator i with i = 1, 2, .., p. Let introduce the set of indexes of all actuators that are not completely lost, i.e.

Θ {i

: i ∈ (1, 2, .., p), γ a i = 1} The control action is u j F T C = -(I -γ a ) + i∈Θ G i Y i i∈Θ X i -1 (x k -x j e ) + u j e ( 34
) where G i = B i+ j B i j , applied to the faulty system allows to constrain pole placement in prescribed LMI region. Proof: Applying the new control law (34) to the faulty system [START_REF] Banerjee | H∞ control of nonlinear processes using multiple linear models[END_REF], leads to the following equation

B j (I-γ a )u j F T C = B j Γ a ( i∈Θ G i Y i )( i∈Θ X i ) -1 (x k -x j e )+u j e (35) with Γ a = I p-h 0 0 O h (36)
Γ a is a diagonal matrix which contains only entries zero (representing total faults) and one (no fault). But here h = 0, which is the number of actuators completely lost, due to the fact that only the set Θ is considered. Since B j Γ a = i∈Θ B i j models only the actuators that are not completely lost, then performing the summations in the proof of Theorem (1) over the elements of Θ shows that (

p i=1 G i Y i )( p i=1 X i ) -1
is the state-feedback gain matrix for the faulty system (A j , i∈Θ B i j , C j ). The control law in equation (34) implies that

u j F T C = -K F T C (x k -x j e ) + u j e ( 37
)
with

K F T C = (I -γ a ) + i∈Θ G i Y i ( i∈Θ X i ) -1 (38)
The global control law U F T C of the system is realized as:

u k = N j=1 ρ j k u j F T C = - N j=1 ρ j k [K F T C (x k -x j e ) + u j e ] (39)
IV. APPLICATION

A. Process description

The approach presented in this paper has been applied to the well known three tanks benchmark as in [START_REF] Theilliol | A multiple model based approach for Fault Tolerant Control in nonlinear systems[END_REF]. As all the three liquid levels are measured by level sensors, the output vector Y is [l 1 l 2 l 3 ] T . The control input vector is U = [q 1 q 2 ] T . The goal is to control the system around three equilibrium points. Thus, 3 linear models have been identified around each of these equilibrium points and the operating conditions are given in Table (I). The linearized system is described by a discrete state space representation with a sampling period T s = 1s. For each OP, each control matrix pair (A j , b i j ) is controllable. Controllers have been designed for levels l 1 and l 2 to track reference input vector Y r ∈ R 2 . Nominal controllers have been designed through Theorem (1), leading to two state feedback gain matrices K 1 , K 2 (due to 2 actuators) for all the three OP in order to achieve satisfying tracking performances. The simulation of actuator faults on the system does not affect the controllability and observability of the system.

B. Results and comments

Simulations have been performed such as the 3 operating conditions described in Table (I) are reached and weighting functions for each local model are presented in figure [START_REF] Noura | Fault-tolerant control in dynamic systems: Application to a winding machine[END_REF] always close to the dynamic behaviour of the nonlinear system according to the considered operating regimes. Figure [START_REF] Patton | Fault-tolerant control: the 1997 situation[END_REF] shows the time history of the outputs with respect to set-point changes occur at time instant 500s and after at time instant 2500s. In the simulation, gaussian Fig. 2. System outputs in fault-free case with a nominal control law figure [START_REF] Zhang | Bibliographical review on reconfigurable Fault-Tolerant Control systems[END_REF]. A gain degradation of pump 1 (clogged or rusty pump) equivalent to 80% loss of effectiveness is supposed to occur at time instant 1000s. Consequently, the dynamic behaviour of the other levels is also affected by this fault and control system tries to cancel the static error created by the corrupted input. Consequently, the real output is different from the reference input and the control law is different from its nominal value. Since an actuator fault acts on the system as a perturbation, and in spite of the presence of an integral controller, the system outputs can not reach again their nominal values. In the same way, the actuator Fault Tolerant Control method's ability to compensate faults is illustrated in the presence of the same fault. Once the fault is isolated and simultaneously estimated, a new control law (39) is computed in order to reduce the fault effect on the system. Indeed, since the effect of an actuator fault is quite similar to the effect of a perturbation, the system outputs reach again their nominal values, as illustrated in figure [START_REF] Blanke | Diagnosis and Fault-Tolerant Control[END_REF]. A time delay between fault occurrence and fault compensation equals to 10 samples is considered in our simulation. Computation of the tracking error norm in fault-free case, in faulty case without and with FTC underlines the performances of this approach as seen in Table (II). With FTC, the tracking error norm for output l 1 is a bit larger than with the fault-free case but it still widely smaller than the one without FTC. The actuator Fault Tolerant Control is able to maintain performances as close as possible to nominal ones and to ensure the closed-loop stability despite the presence of instruments malfunction. Remark: This proposed scheme could have an interesting development in aeronautic, for example, where redundant actuators are available and where a wide operating range is considered.

V. CONCLUSION

The method developed in this paper emphasises the importance of the active Fault Tolerant Control of nonlinear systems based on multi-model representation. This method is suitable for partial actuator faults on a wide operating range of the system. A robust controller is designed for each separate actuator through an LMI pole placement in faultfree case and faulty case. It allows the system to continue operating safely, to avoid stopping it immediately and to ensure stability. The synthesis of this active state feedback control takes into account the information provided on-line by FDI scheme. The performances and the effectiveness of this active Fault Tolerant Control based on a multiple model approach have been illustrated through a nonlinear benchmark. Futures works will deal with total failures, restructuration and will improve the limits of the strategy.

3 kFig. 1 .

 31 Fig. 1. Weighting functions

2 Fig. 3 .

 23 Fig. 3. System outputs with pump degradation and a nominal controller

3 Fig. 4 .

 34 Fig. 4. System outputs with pump degradation and a FTC controller


M Adam-Medina

M Rodrigues

D Theilliol

H Jamouli

FAULT DIAGNOSIS IN NONLINEAR SYSTEMS THROUGH AN ADAPTIVE FILTER UNDER A CONVEX SET REPRESENTATION

Keywords: Nonlinear system, multiple linear models, fault detection and isolation, stability, adaptive filter

In this paper, the main goal is to design an approach that performs fault detection, isolation and estimation for a large class of nonlinear systems. Fault diagnosis is established by regarding system as a convex combination of linear time invariant (LTI) stochastic models and not as a single global model. The nonlinear representation is based on a bank of decoupled Kalman filters. This paper consists in generating a robust model selection of the "best" representative linear model. Under fault isolation conditions, the main contribution is to design an adaptive filter which makes possible multiple faults detection which appear simultaneously or in a sequential way, isolation and estimation over the whole operating range of nonlinear system. The stability conditions of the adaptive filter are developed. These conditions result in convex linear matrix inequalities (LMIs) that can be solved efficiently with optimization techniques. Performances of the method are tested on an academic example.

Introduction

Fault detection and isolation (FDI) has been developed traditionally with model-based approaches using linear or linearized models. Several papers have been published: these works lead to robust methods of parametric uncertainties and system modelling errors when noise is or is not present [START_REF] Chen | Robust Model-Based Fault Diagnosis For Dynamic Systems[END_REF]. These diagnosis methods are based on residual generation. One solution is to use nonlinear methods. Fault detection and isolation (FDI) problems for nonlinear systems provide a challenging current research topic [START_REF] Frank | Current developments in the theory of fdi[END_REF]. [START_REF] Chen | Robust Model-Based Fault Diagnosis For Dynamic Systems[END_REF] have proposed FDI scheme using linear observers and Takagi-Sugeno configuration for nonlinear system representation in the deterministic case. This technique is based on the decomposition of the original system into a set of linear models. This structure is considered as multiple model (MM) approach. In nonlinear systems, the MM structure has been proposed for control do-main. The robust control using one controller has been presented in [START_REF] Banerjee | h ∞ control of nonlinear processes using multiple linear models[END_REF], the multi-control technique such as gain scheduling interpolation method in control domain [START_REF] Leith | Survey of gainscheduling analysis and design[END_REF] and adaptive or learning control based on neural network have been developed [START_REF] Narendra | Adaptation and learning using multiple models, switching and tuning[END_REF]. However, these methods have not been frequently used for FDI. Under this consideration, the paper aims and contributes to the design of a multiple fault detection, isolation adaptive filter in nonlinear system represented as a convex set of LTI distribution matrices. In our approach, the term "adaptive" should not be associated by readers to an on-line parameter estimation as developed by [START_REF] Narendra | Adaptation and learning using multiple models, switching and tuning[END_REF]. Our goal is to design a scheme which allows FDI and a robust model selection in the nonlinear system represented as a convex combination of the multiple LTI model. To achieve this purpose, an adaptive filter based on an interpolated multiple model is developed. The proposed filter will perform an efficient FDI according to an available convex set representation in faulty case based on decoupled Kalman filter developed by [START_REF] Keller | Fault isolation filter design for linear stochastic system[END_REF]. Moreover, the magnitude, the time of occurrence and the localization of the faults are considered as unknown. The paper is organised as follows: in the second section, the general problem of the robust selection in nonlinear systems described by MM is developed. A solution based on the design of a bank of decoupled Kalman filters is then developed and justified. This section gives us a formulation of the nonlinear system representation in faulty case based in a convex set of the multiple LTI models. In section III, the design of the adaptive filter is developed. The question of stability was addressed in the terms of Lyapunov quadratic stability by using a LMI technique in Section IV. In next Section, an academic example is considered to illustrate the theoretical results. Finally, the last section is devoted to the conclusions and future search.

Fault Diagnosis in nonlinear systems

Problem statement

Dynamic systems such as nonlinear, linear time-invariant, linear time-variant, linear piecewise and hybrid can be represented by a decomposition of the full operating range into a number of possibly overlapping operating regimes [START_REF] Leith | Survey of gainscheduling analysis and design[END_REF]. For each regime, a simple local linear system is defined. Let us assume the following general faulty stochastic system as considered by [START_REF] Varga | Optimal output feddback control: a multi-model approach[END_REF] and [START_REF] Johansen | Multiple Model Approaches to Modelling and Control[END_REF])

X k+1 = A j X k +B j U k +∆ Xj +F j d k +ω j k Y k = CX k +∆ Yj +ν j k (1)
where X ∈ n represents the state vector, U ∈ p is the input vector, Y ∈ m is the measurement vector and d ∈ q is the fault vector. Each linear model is defined around j th operating point, noted

P j ∀j = [1, 2, • • •, M ],
where M is the total number of operating points. Each operating point is defined by a pair of input/output signals (Y P j , U P j ). The terms ω j and ν j are two independent zero mean white noises with variance-covariance matrices defined respectively by Q j and R j . A j , B j , and C are constant matrices with appropriate dimensions. Around the j th operating point it is assumed that ∀j, rank(C) = m. ∆ Xj and ∆ Yj are contant vectors depending on the j th linear model. F j is the fault distribution matrix representing actuator, sensor or component faults with ∀j, rank(F j ) = q. Without loss of generality according to [START_REF] Park | On the representation of sensor faults in fault detection filters[END_REF], in the presence of sensor, actuator or component faults, the system may be represented by a linear system as (1). This linear system (1) can be specified by the set of system matrices as:

S j = A j B j ∆ Xj F j ω j C ∆ Yj ν j , ∀j = [1, 2, ..., M ]
(2) Let S k be a matrix sequence varying within a convex set, defined as:

S k := M j=1 ϕ j k S j : ϕ j k ≥ 0, M j=1 ϕ j k = 1 (3)
In the MM framework, S k characterizes at each sample the nonlinear system. Consequently, the dynamic behavior of nonlinear system can be defined by a convex set of a multiple LTI models, noted Υ Υ := [S 1 , S 2 , ...S M ] .

The state space representation (3) under a convex set can be considered as:

-a conventional modelling approach for non linear smooth plant where ϕ j k is an appropriate weighting functions. The functions ϕ j k , which can be a vector or a scalar, embodies the nonlinearity of the plant as discussed in fault free case [START_REF] Leith | Survey of gainscheduling analysis and design[END_REF] or [START_REF] Johansen | Multiple Model Approaches to Modelling and Control[END_REF].

In order to achieve multi-control and FDI, a bank of Kalman filter is designed. Under assumption that the nonlinear system evolves around the j th operating point, the Kalman filter is described by:

Xi k+1 = A i Xi k +B i U k + ∆ Xi +K i k (Y k -Ŷ i k ) Ŷ i k = C Xi k + ∆ Yi (4)
where Xi k ∈ n denotes the estimated state vector and Ŷ i k ∈ m is the output estimation obtained from the linear filter based on the i th linear model. K i k ∈ n×m is the Kalman filter gain matrix. In this paper, the fault distribution matrix F j is also is represented as F i with the same rank. In the MM framework, the construction of a bank of Kalman filters defined from Υ, ∀i = [1, 2, ..., M ] is considered in order to establish fault detection, isolation and the robust model selection on all the operating range of the system. Based in the needs to develop a robust model selector in faulty or free case, a bank of Kalman filters leads us to obtain the estimated error (ε i k

(ε i k = X k -Xi k ) and the output residual vectors r i k (r i k = Y k -Ŷ i k ))
when the both fault and operating point change occurs (j = i and d = 0). This difference between the representation system to be considered and the filters is represented by equation ( 5) and ( 6) as:

ε i k+1 = (A i -K i k C)ε i k +F j d k (5) -K i k ν j k +ω j k +(∆∆ i Xj -K i k ∆∆ i Yj )ξ i j,k
and,

r i k = C i ε i k + ν j k + ∆∆ i Yj ξ i j,k (6)
where ξ i j,k ∈ (n+p+1)×1 corresponds to the uncertainties magnitude between nonlinear system represented by the j th linear system and the i th linear model used for the Kalman filters computation (i.e., operating point switches). ∆∆ i Xj ∈ n×(n+p+1) and ∆∆ i Yj ∈ m×(n+p+1) are the distribution matrices of uncertainties associated with the state system equation and the output equation, respectively. Equations ( 5) and ( 6) show that residual is sensitive to modelling errors (i.e, a change in the operating point) and at the same time to faults. The use of Kalman filters leads to the following residual properties (∀i

∈ [1, • • • , M ]): r i k ∼ N , if d = 0, i = j r i k N , if d = 0, i = j r i k N , if d = 0, ∀i (7)
In faulty case (d = 0) and whatever the operating point, direct application of nonlinear system using Kalman filters based on a MM framework can be considered, but at the cost of certain degradations in performance. Indeed, in faulty cases, FDI cannot be done correctly since the residual vector is corrupted by two pieces of information at the same time: operating point exchanges and fault occurrences. The probabilistic Bayes method cannot define the best representative model [START_REF] Jamouli | Simultaneous fault diagnosis and robust model selection in multiple linear models framework[END_REF]. Firstly, in this paper, we are interested in knowing which linear model is more representative at each time both in fault free and faulty cases. These considerations yield a new residual generator definition allowing the decoupling of fault effects and modelling errors. This new residual generator gives a first signal insensitive to faults, but sensitive to modelling errors and a second signal sensitive to faults. The new residual generator is expressed as:

ri k = Σ Ξ r i k (8)
where Σ and Ξ are terms introduced in order to decouple the residuals with appropriate dimensions and ri k is the new residual vector. One solution is proposed in the following paragraph.

Robust model selection in free and faulty case.

Under the hypotheses that a fault occurs at time k d (k > k d ), and that operating point switches at time k e (k > k e ), the residual vector of the i th filter is expressed as:

r i k = ri k + ∆∆ i ξ i j,k + ρ k,k d [d k d d k d +1 • • • d k-1 ] (9) + β k,ke [ξ i j,ke ξ i j,ke+1 • • • ξ i j,k-1 ] where ρ k,k d , β k,ke , Γ (k d ,ke) i,k
, L i k are matrices with appropriate dimensions. Equation (9) allows us to confirm that residual is affected by both fault effects and modelling errors. We wish to generate residuals insensitive to fault effects but sensitive only to modelling errors, that is:

A i -K i k C F i = 0 (10)
Supposing that condition ( 10) is fulfilled and the number of fault is strictly lower than the number of outputs (i.e., ∀i, rank(CF i ) = q < m), a solution was proposed by (KELLER, 1999) which defined a Kalman filter gain as:

K i k = ω i Ξ i + Ki k Σ i (11)
with -q)×m is an arbitrary constant matrix defined so that the matrix Σ i is of full row rank. According to (11) and the previous matrices properties, the residual vector ri k can be obtained as suggested in (8):

Ξ i = (CF i ) + , ω i = A i F i , Σ i = α i (I m -CF i Ξ i ) and α i ∈ (m
Σ i (Y k -Ŷ i k ) Ξ i (Y (k)-Ŷ i k ) = Σ i r i k Ξr i k = γ i k Ω i k = ri k ( 12
)
where γ i k ∈ m-q is the residual vector decoupled to fault and Ω i k ∈ q is the residual vector sensitive to fault. Following these assumptions, when nonlinear system operates around the j th operating point, the new residual γ i k insensitive to fault has the following properties:

∀d γ i k ∼ N if i = j γ i k N if i = j ( 13
)
where Θ i k defines the covariance matrix of the residuals γ i k , equal to ( C P i k CT + Vi ). Considering that residual γ i k around j th operating point follows a Gaussian distribution, in the same way as [START_REF] Banerjee | h ∞ control of nonlinear processes using multiple linear models[END_REF] the residuals vector can then be used to compute the probability distribution as:

℘ i k = exp{-0.5γ i k (Θ i k ) -1 (γ i k ) T } [(2π)det(Θ i k )] 1/2 (14)
The mode probability decoupled from faults is expressed as:

ϕ(γ i k+1 ) = ℘ i k ϕ(γ i k ) M l=1 ℘ l k ϕ(γ l k ) (15)
The probability algorithm allows to obtain the i th linear model describing the system behavior in both fault free and faulty cases. The mode probabilities allows us to determine the operating point where nonlinear system is evolving. These probabilities are used to isolate the operating point and consequently define a robust model selector.

The robust model selector is used to represent the plant behavior as a convex set of multiple linear models. This representation is described in equation ( 16):

S k := M i=1 ϕ(γ i k )S i : ϕ(γ i k ) ≥ 0, M i=1 ϕ(γ i k ) = 1 (16)
where S (k) represents the global model and S i is defined as:

S i = A i B i ∆ Xi F i C ∆ Yi , ∀i = [1, 2, ..., M ] (17)
By taking in to account the nonlinear system convex set notation in ( 16), the nonlinear state space representation is then defined as:

X k+1 = A k X k +B k U k +∆ X,k +F k d k Y k = CX k +∆ Y,k (18)
with

A k = M i=1 ϕ(γ i k )A i , B k = M i=1 ϕ(γ i k )B i ∆ X,k = M i=1 ϕ(γ i k )∆X i , ∆ Y,k = M i=1 ϕ(γ i k )∆Y i F k = M i=1 ϕ(γ i k )F i (19)
The equations ( 18) and ( 19) represent the original nonlinear system. In this representation, the noisy function has already been considered in the convex set representation. This convex set representation is used for the design of an adaptive filter which is developed for the fault detection, isolation and estimation (see Fig. 1) in the following section.

% § Σ%ϕL $ § % § * * * * * ϕi ϕi γ 1 k γ 2 k γ M k Ω1 k ΩM k Ω2 k U Y
23(5$7,1* 32,17 '(7(50,1$7,21 To design the adaptive filter, an unique formulation of the convex faulty representation is proposed. In the state space representation (18), a matrix F k is calculated as:

F k = M i=1 ϕ(γ i k
)F i where matrix F i ∈ R n×q is fault distribution matrix of each model i. Faults contribution on the system is described into state space representation as:

M i=1 ϕ(γ i k )F i d k .
According to some F i matrix conditions, the convex combination can be transformed into an unique form:

Definition 1 Matrix F h i (respectively h ) defines the h th column of matrix F i (respectively ). Theorem 1 ∀h ∈ [1..q], ∀i ∈ [1..M ], with Rank[F i ] = q if Rank[F h 1 ..F h i ..F h M ] = 1, then M i=1 ϕ(γ i k )F i d k = f k where d ∈ R q represents "
true" fault vector and f ∈ R q is an image of fault vector, where ∈ R n×q is a constant fault distribution matrix which columns vectors get direction of columns vectors of matrices F i .

Proof (THEILLIOL et al., 2003)

Adaptive filter design

In order to detect and isolate faults, a classical discret filter with a gain K k could be designed according to matrices A k and C defined in ( 18) and ( 19):

Xk+1 = A k Xk + B k U k + ∆ X,k + K k (Y k -Ŷk ) Ŷk = C Xk + ∆ Y,k (20)
where X and Ŷ represent the estimated state and the estimated output.

According to (20) estimation error e k (e k = X k -Xk ) and output residual r k (r k = Y k -Ŷk ) are expressed as:

e k+1 = (A k -K k C)e k + f k r k = C e k ( 21
)
Under the assumption that a fault occurs at time k d (k > k d ), residual vector is defined as:

r k = rk + ρ k,k d [f k d f k d +1 • • • f k-1 ] ( 22
)
where rk represents the residual in fault free case and

ρ k,k d = C     Γ k d +1 k Γ k d +2 k • • •     (23)
with

Γ k d k = k-1 τ =k d L τ (24) L k = (A k -K k C) (25)
As developed by [START_REF] Keller | Fault isolation filter design for linear stochastic system[END_REF], we proposed to design the gain K k such that (A k -K k C) is equal to zero. Under the general classical condition, that the number of faults must not be greater than the number of measurements (i.e. rank(C ) < m), a fault diagnosis adaptive filter is designed with the following gain definition:

K k = ω k Π + Kk Σ (26) with Π = (C ) + , ω k = A k , and Σ = α(I m -C Π)
where α is an arbitrary matrix determined so that matrix Σ is of full row rank.

According to (26), the adaptive filter is defined as:

     X k+1 = A k X k + B k U k + ∆ X,k + (ω k Π + Kk Σ)(Y k -Y k ) Y k = C X k + ∆ Y,k (27)
where X k and Y k are respectively the estimated state and the estimated output.

The gain decomposition, defined in ( 26), involves the following matrices properties

ΠC = I and ΣC = 0 (28)
and makes possible the generation of modified residual vector:

γ k q k = Σ Π r k = Σr k Πr k + f k-1 (29)
It must be noticed that γ k ∈ R m-q is a residual vector insensitive to faults and q k ∈ R q is a residual vector sensitive to faults and defines also an fault estimation vector. The gain Kk , in (26), is the unique degree of freedom in the adaptive filter synthesis. In the multiple model framework, it is designed such as an interpolation of gain Ki (see [START_REF] Stiwell | Interpolation of observer state feedback controllers for gain scheduling[END_REF] or [START_REF] Leith | Survey of gainscheduling analysis and design[END_REF]). Then, in the following, Kk is noted K k . For the determination of K k , each Ki is computed off-line by a classical eigenstructure assignment of ( Āi -Ki C) with C = C and Āi = A i (I -ΠC). We are now expressed necessary and sufficient conditions for adaptive filter stability.

Stability on a convex hull

Using Lyapunov stability definition, the gain Ki , establishing off-line by eigenstructure assignement in order that matrix ( Āi -Ki C) is Hurtwitz, must verified and solved the following inequality :

( Āi -Ki C) T P ( Āi -Ki C) -P < 0 P > 0 (30)
A Schur complement operation transforms the inequality (30) into an LMI and as proposed for LPV system in fault free case by [START_REF] Daafouz | Parameter Dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties[END_REF] or [START_REF] Oliveira | A New Discrete-time Robust Stability Condition[END_REF], this inequality can be extended on convex sets Γ (idem for K k ) and a new LMI is generated:

      P M i=1 ϕ(γ i k )( Āi -Ki C) T P P M i=1 ϕ(γ i k )( Āi -Ki C) P       > 0 (31) Γ = Ā k : Ā k = M i=1 ϕ(γ i k ) Āi , M i=1 ϕ(γ i k ) = 1 (32)
Eq.( 31) is equivalent to:

P ( Ā k -K k C) T P P ( Ā k -K k C) P > 0 (33)
Then, ( 33) is equivalent to Lyapunov quadratic stability in discrete case of matrice (A k -K k C):

(A k -K k C) T P (A k -K k C) -P < 0 P > 0 (34)
Thus, find an unique solution P for all [i = 1...M ] assures the stability of the filter ( 27) and the convergence of error estimation defined as follows:

ēk+1 = (A k -K k C)ē k (35)
The stability of the adaptive filter has been analyzed and correspond to find an unique matrix to solve a set of inequality. To show the performances of this method an academic example is illustrated in the next section.

Example

System representation

The proposed method is applied to an open loop discrete nonlinear system. This stochastic discret system, SIMO (single-input/multiple-output) with X ∈ 4 , U ∈ 1 , and Y ∈ 3 . The j th linear system represents the system behavior around the j th operating point depending directly on the magnitudes of the inputs defined in table 1. Other matrices are the same for each model: ∀j,

B j = [1 0 0 0] T , C =   0 1 0 0 0 0 1 0 0 0 0 1   , Q j = diag([0.9 0 0.5 0.3]) T and R j = I 3×3 .
The faulty distribution matrices are different:

F 1 =     0 0 0.9 0 0 2.8 0 0     , F 2 =     0 0 2.4 0 0 1.4 0 0     and F 3 =     0 0 3.2 0 0 3.7 0 0    
↑ ϕ(γ 3 k ) ϕ(γ 2 k ) ↑ ↑ ϕ(γ 1 k ) (b) Probabilities ϕ(γ i k ).
Figure 2: System input and model selector evolution in faulty case.

Simulation results

The simulation example considered is developed in a multiple faults case (d ∈ 2 ) with a system input U chosen as successive steps with smooth transitions (see Fig. 2a).

In order to evaluate the method, we consider two faults : a first at time k d = 25 seconds with a constant magnitude equal to 2, and a second appears at time k d = 150 seconds with a constant magnitude equal to 5 as shown on Fig. 2a.

A bank of three decoupled Kalman filters in multiple models framework is proposed. It was developed to show the selection of the true model and nonlinear representation, when residual vector is corrupted by two pieces of information: operating point evolution and fault occurrence. In Fig. 2b, we can observe that in the faulty case, how the probabilities evolution allow the robust model selection.

The results obtained show a robust model selector which is insensitive to faults. The same figure shows the transition from model "1" to model "2" and after to model "3" without the probabilities being affected by fault occurrences. The model selection allows to represent the nonlinear system via a convex set of the multiple LTI models. This representation is used to realize the fault detection, isolation and estimation via an adaptive filter. In FDI context and according to adaptive filter design, different conditions are verified: rank(C ) < 3, T heorem 1 is available, ∀i, ( Āi , C) is observable and three gains Ki are computed.

As illustrated in Fig. 3a and after a pseudo-inverse matrix computation [START_REF] Theilliol | Adaptive filter design for fdi in nonlinear systems based on multiple model approach[END_REF] that permits the obtention of accurate fault magnitude estimation where d is a fault estimation of q k . The good performances of the adaptive filter in terms of fault detection, fault isolation is achieved. Based on this residual vector, it is easy to imagine a residual evaluation test with an elementary logic decision in order to develop a multiple faults detection, isolation and estimation module integrated in supervision platform.

Fig. 3b illustrates the residual vector γ k ∈ 1 . In this academic example, where uncertainties or error models are not considered, these residuals vector, designed as a sequence insensitive to faults, are obviously close to zero during the experiment. Meanwhile, in practical case, the residual vector γ k will not be close to zero due to uncertainties or model errors but it can be exploited to indicate the confidence degree of the adaptive filter.

Conclusion

The FDI problem for nonlinear systems based on a multiple models framework has been addressed in this paper.

It is based on a robust model selector that permits to identify the most representative linear model of the nonlinear system in free or faulty case. A robust isolation filter is designed to detect, isolate and estimate faults for a large class of systems through an LTI convex set representation.

In order to guarantee stability of the adaptive filter, a stability analysis has been realized using linear matrix inequality formulation. The developed adaptive filter was demonstrated using a simulation example on an academic model. Moreover, the residual insensitive to fault can be used to supervise the adaptive filter itself. In the presence of model errors or uncertainties, this residual is corrupted by these perturbations and is different from zero. The adaptive filter quality can be evaluated through this residual. Futur work are directly linked to the previous remark, concerns the robustness against uncertainties and will be dedicated to industrial processes.

Figure 1 :

 1 Figure 1: FDI and nonlinear representation scheme

Figure 3 :

 3 Figure 3: Residuals generated by adaptive filter

Table 1 :

 1 Definition of the operating points.

	Operating	j = 1	j = 2	j = 3
	point (P j )	U ∈ [0; 3[ U ∈ [3; 6[ U ∈ [6; 9[
	Y Pj	3.2-5.0 2.58	5.82-8.54 9.88-15.62 4.88 8.34
	U Pj Eigenvalues of A j	2 0.22     0.15 0.37 0.60	   	4 0.01     0.01 0.02 0.04	   	7 0.30     0.20 0.50 0.80	   


D Theilliol
email: didier.theilliol@cran.uhp-nancy.fr

M Rodrigues

M Adam-Medina

D Sauter

ADAPTIVE FILTER DESIGN FOR FAULT DETECTION AND ISOLATION OF NON LINEAR SYSTEMS BASED ON MULTIPLE MODEL APPROACH

Keywords: multi-model, FDI, stability, LMI, convex set

In this paper, the fault diagnosis of non linear systems based on a adaptive filter is developed. The proposed method makes possible the fault detection, isolation and estimation for the whole operating range of the nonlinear system. Based on the assumption that the dynamic behavior of the nonlinear system is defined as a multiple model approach, a set of residual vector insensitive to faults is established and thus generating a global model exploited in an adaptive filter. The convergence and stability conditions of the adaptive filter are developed and its performances are tested in simulation.

INTRODUCTION

The role of a human operator is to preserve normal operating conditions according to several plant parameters, measurements and observations. But complex automated industrial systems are vulnerable to faults in instrumentation as sensors, actuators or controllers. With the growing complexity of modern engineering systems and ever increasing demand for safety and reliability, there has been great interest in the development of Fault Detection and Isolation (FDI) methods. Usually, faults and failures in systems are detected using analytical redundancy when comparing measured and estimated outputs of the system whatever the design used, the residual relations are elaborated so that it is possible to isolate any kind of fault. A large diversity of advanced methods for automated FDI already exists integrating the robustness against parameters uncertainties and model plant mismatches (Chen and Patton, 1999), (Frank and Ding, 1997), (Gertler, 1998). A short historical view on this development can be seen in (Isermann and Ballé, 1996) and currents developments can be observed in (Frank et al., 2000). Rare are the methods dealing with the non linear systems, a great number is established around an operating point or requires the exact knowledge of the nonlinear system (Alcorta Garcia and Frank, 1997), (Hamouri et al., 1999) or as in the chapter nine of Chen and Patton (Chen and Patton, 1999), the nonlinear dynamic systems are described by a number of locally linearized models based on the idea of Tagaki-Sugeno fuzzy models. In the same idea, various studies, based on a multiple model method established through a bank of linear Kalman filters, have been developed in order to detect, isolate and estimated an accurate state of a system in presence of faults/failures when a model is defined around an operating point (Maybeck, 1999), (Rong, Li and Bar-Shalom, 1993) or the whole operating range of a nonlinear systems (Pan et al., 1997). This paper aims at investigating the FDI of nonlinear systems under the assumption that the nonlinear system could be approximated by a finite number of interpolated Linear Time Invariant (L.T.I.) models (Johansen et al., 1998). Under stochastic assumption, the paper aims and contributes to the design of a fault detection and isolation adaptive filter where the distribution matrices, representatives of the nonlinear system, are obtained by probabilistically weighted matrices computation. This computation is based on the properties of residuals insensitive to faults and sensitive to some model errors. These residuals are generated according to a bank of filters (Keller, 1999) decoupled from faults. In this paper, the magnitude, the time of occurrence and the localization of the faults are considered as unknown. The second section is devoted to the L.T.I. model selection in faulty case based on the use of a bank of decoupled Kalman filters. In this section, the design of the adaptive filter is developed. Section 3 is devoted to the simulation of a nonlinear system for various operating points and faults. The simulation results are then shown and commented. Finally, concluding remarks are given.

ADAPTIVE FILTER DESIGN

Positionnement du problème

Under actuator or sensor faults (following some transformations), a general discrete state space representation of MIMO system is described by:

X k+1 = N i=1 µ i (σ k ) A i X k + B i U k + F i d k + ∆X i Y k = N i=1 µ i (σ k ) C i X k + ∆Y i (1)
where X k ∈ R n is the state space vector, U k ∈ R p is the control input vector, Y k ∈ R m is the output vector, and d k ∈ R q is the fault vector to be detected. A i , B i and C i are constant real matrices of appropriate dimensions and F i ∈ R n×q is the fault matrix distribution.

The representation (1) can be considered as : -a conventional modelling approach for non linear smooth plant where µ i (σ) is an appropriate weighting functions. The schedule variable µ i (σ), which can be a vector or a scalar, embodies the nonlinearity of the plant (Leith and Leithead, 1999;Murray-Smith and Johansen, 1997). The N models should described the dynamic behaviour of the plant where N are the number of equilibrium points defined by (X e i , U e i , Y e i ) allowing the determination of the matrices and as follows:

∆X i = -A i X e i -B i U e i ∆Y i = -C i X e i + Y e i (2)
-a faulty linear parameter varying (LPV) system (Szabo et al., 2002) when the terms N i=1 µ i (σ) are time varying parameter dependent and ∆X i = ∆Y i = 0. -an hybrid system when the terms are supposed defining as a system mode sequence with a transition probabilities (Rong, Li X. and Y. Bar-Shalom, 1993). The representation is generally used in linear context ( ) where partial actuator or sensor failures are modelled by multiplying the appropriate column by a factor of effectiveness (Zhang Y. and X.R. Li., 1998).

In this context of N models should represent the evolution of nonlinear systems, the purpose is to isolate, estimate faults under knowledge of the equilibrium point where the system is representing at time t. An answer developped in [START_REF] Adam-Medina | Diagnostic de défauts des systémes non linéaires décrits par des modéles linéaires interpolés[END_REF] use a decoupling Kalman filter that generates two informations: one can show the equilibrium point, the other estimate faults and their magnitude.

In the state space representation (1), the interpolation of the matrix F (p) is made as :

F (p) = N i=1 µ i (σ k )F i where matrix F i ∈ R n * q
are faulty matrix distribution of each model i. Faults events on the system are described into state space representation as :

N i=1 µ i (σ k )F i * d(k) .
According to some previous conditions on direction of columns vectors if matrices F i , this notation can be changed by following notations: Definition 1. F j i and j are defined respectively the jth column of the matrix F i and .

Theorem 1. If the rank of the ith matrix column

F i ∈ R n * q satisfies: ∀j ∈ [1..q],∀i ∈ [1..N ], Rank[F j 1 ..F j i ..F j N ] = 1 with Rank[F i ] = q then: ( N i=1 µ i (σ k )F i ) * d(k) = * f (k) where d(k) ∈ R q
is the true faults vector in which f (k) ∈ R q represents false faults, where ∈ R n * q is a fixed matrix which columns vectors get direction of columns vectors of matrices F i .

Proof As Rank[F j 1 ..F j i ..F j N ] = 1, the jth matrices columns F i get same direction, this condition could be rewritten as F j i = α j i * j where α j i is a scalar corresponding to the jth matrix column F i with a matrix composed by fixed elements. Thus, coefficients α j i can't be equal to zero otherwise rank conditions will not be true. By underlying colinearity of each column, faults in state space representation can be noted as:

N i=1 µ i (σ k )F j i * d j (k) = N i=1 µ i (σ k )(α j i j ) * d j (k) = j * N i=1 µ i (σ k )α j i * d j (k) = j * f j (k)
where d j (k) and f j (k) are jth elements of the column vector of the faults. This equality is repeated q times to get columns of the matrix as well as q elements of the vector f (k). Thus, the given matrix is fixed, full column rank and false faults vector is noted

f (k) = [f 1 (k)f 2 (k)...f q (k)]
T in which each elements f j (k) is an interpolation of the jth element of the true faults vector. Consequently, matrix and false faults f (k) can be represented as:

N i=1 µ i (σ k )F i * d(k) = * f (k) (3) with ∀i = 1 α 1 i F 1 i 1 α 2 i F 2 i ... 1 α q i F q i (4)
and

f (k) =         N i=1 µ i (σ k )α 1 i * d 1 (k) ... N i=1 µ i (σ k )α q i * d q (k)         (5)
Remark: By its construction, matrix is not unic. The scalar α j i is not equal to zero because rank conditions would not be true. Secondly, true faults can be found by simple matrix pseudo inversion. If we get previously:

N i=1 µ i (σ k )F i * d(k) = * f (k) (6)
By using a pseudo inversion, we obtain:

d(k) = N i=1 µ i (σ k )F i + * * f (k) (7) with ( N i=1 µ i (σ k )F i + pseudo-inverse matrix ( N i=1 µ i (σ k )F i . The equation N i=1 µ i (σ k )F i * d(k) = * f (k) (8)
considering the following notation faults can be expressed as

f (k) =         N i=1 µ i (σ k )[α 1 i * d 1 (k)] ... N i=1 µ i (σ k )[α q i * d q (k)]         =         N i=1 µ i (σ k )g 1 i (k) ... N i=1 µ i (σ k )g q i (k)         = N i=1 µ i (σ k )   g 1 i (k) ... g q i (k)   = N i=1 µ i (σ k )g i (k) (9)
where d j (k) is the ith true fault element d(k) and g j i (k) is the jth element of the ith fault column vector g i (k) leading to the ith model. Conditions ( 8) and ( 9) lead to:

N i=1 µ i (σ k )F i * d(k) = * N i=1 µ i (σ k )g i (k) (10)

Adaptive filter design

The state space representation of the system with faults under a convex set Λ corresponds to:

X k+1 = A k X k + B k U k + f k + ∆X k Y k = C k X k + ∆Y k (11)
with

A k = N i=1 µ i (σ k )A i , B k = N i=1 µ i (σ k )B i ∆X k = N i=1 µ i (σ k )∆X i , ∆Y k = N i=1 µ i (σ k )∆Y i C k = C Λ = {µ ∈ R N : µ i (σ k ) ≥ 0 for i = 1, .., N and N i=1 µ i (σ k ) = 1}
By using a classic Kalman filter with a gain K k that could be designed polytopically as matrix A k , the estimation error e(k) = X(k) -X k and a residual

r k = (Y k -Y k ) can be expressed like e(k + 1) = (A k -K k C)e(k) + f (k) r(k) = C e(k) (12)
If we suppose that faults occur at time k d (k > k d ), residual could be defined as [START_REF] Liu | Fault isolation filter design for time invariant systems[END_REF]:

r(k) = r(k) + ρ k,k d [f (k d ) f (k d + 1) • • • f (k -1)] (13) with ρ k,k d = C     Γ k d +1 (k) Γ k d +2 (k) • • •     (14)
where

Γ k d ,ke (k) = k-1 τ =k,ke L ( τ ) (15) L(k) = (A k -K k C) (16)
If the term (A k -K k C) equals to zero, we can obtain (Keller.J.Y, 1999) a sensitive residual to faults

r k = (Y k -Y k ) equals to rk + C f (k -1)
where rk is a residual without any fault and leads to a centered gaussian distribution.

Filter gain Kg k is obtained under matrices assumptions like: Π D = I and Σ D = 0 (17) Adaptive filter generates two residuals:

γ k q k = Σ Π * r k = Σr k Πr k + f (k -1) (18)
Residual γ k ∈ R m-q is insensitive to faults and residual q k ∈ R q is sensitive to faults.

Adaptive filter is described by the following representation:

     X k+1 = A k X k + B k U k + Kk γ gk + ω k q k + ∆X k Y k = C X k + ∆Y k (19)
with X k and Y k are respectively the estimated state and the estimated output of the adaptive filter. The others matrices for the adaptive filter design are then defined as:

A k = N i=1 µ i (σ k )A i , B k = N i=1 µ i (σ k )B i , C k = C, ∆X k = N i=1 µ i (σ k )∆X i , ∆Y k = N i=1 µ i (σ k )∆Y i , K k = ω k Π m + Kk Σ, ω k = A k , Π = D+ , Σ = α(I m -DΠ), C = ΣC, F k = , D = C
where F k is the faulty matrix distribution at time k and α is an arbitrary matrix ∈ R m-q×m designed in order to get Σ full row rank. Gain K is the only free parameter of the filter and it is designed by eigenstructure assignment of ( Ā -K C). We should remark in the adaptive filter design that faulty matrix distribution is fixed as the observation matrix C. These conditions will be very important for following study.

STABILITY ON A CONVEX HULL

In the study which leads to use a decomposition of the system evolution in some operating points, it is worth mentioning the stability and convergence properties of used filters to get a good estimation of the system online. Properties established by [START_REF] Liu | Fault isolation filter design for time invariant systems[END_REF] for each filter is required to apply them on all range of the system.

Local stability

If we consider that a local filter designed for each model ith, its stability is assured by gain K i so as to assure not only faults decoupling but also eigenvalues of A i -K i C are lower than 1. Thus, the error estimation without fault is noted like:

ēi k+1 = (A i -K i C)ē i k (20)
As showed in the previous paragraph, decomposition of gain of each local filter generate that matrix (A i -K i C):

(A i -K i C) = A i -(ωΠ + Ki Σ)C = (A i -ωΠC) -Ki ΣC = (A i -A i ΠC) -Ki C = A i (I -ΠC) Āi -Ki C (21) with ω i = A i , Π = D+ = C Ψ = C (22)
Gains Ki are established by eigenstructure assignment in order to matrix Āi -Ki C be Hurwitz, so error estimation without faults is described as follow:

ēi k+1 = ( Āi -Ki C)ē i k (23)
By using Lyapunov stability on a system like (23), estimation error (Oliveira.M.C. de et al., 1999) leads to:

( Āi -Ki C) T P ( Āi -Ki C) -P < 0 P > 0 ( 24
)
where P is a positive definite matrix with appropriated dimensions.

For adaptive filter design, polytopics matrices are defined as follow:

A(µ) = N i=1 µ i (σ k )A i et K: K(µ) = N i=1 µ i (σ k ) Ki generate directly: ( N i=1 µ i (σ k )A i )(I -ΠC) -( N i=1 µ i (σ k ) Ki ) C = = N i=1 µ i (σ k ) Āi - N i=1 µ i (σ k ) Ki C = Ā(σ) -K(σ) C (25)
with faulty matrix distribution of the interpolated model as represented in (11).

Robust stability of estimation error

In the adaptive filter, through state space representation of system (11) and adaptive filter ( 19) the estimation error is described by the following equations:

ēk+1 = ( Ā(σ) -K(σ) C)ē k ( 26
)
where the dynamics matrix Ā(σ) et K(σ) belong to convex polytopic set defined as:

Λ = { Ā(σ) : N i=1 µ i (σ k ) Āi = 1, N i=1 µ i (σ k ) = 1, µ i (σ k ) ≥ 0} Γ = { K(σ) : N i=1 µ i (σ k ) Ki = 1, N i=1 µ i (σ k ) = 1, µ i (σ k ) ≥ 0} (27) Lemma 1. Uncertain system (3.2) is robustly stable in uncertainty domain (3.2) if ( Āi -Ki Ci ) T P ( Āi -Ki Ci ) -P < 0 P > 0 (28)
for all i = 1..N .

By using Schur complement, inequalities (1) can be described by an LMI as follow:

P ( Āi -Ki C) T P P ( Āi -Ki C) P > 0 (29)
We can extend this result on a convex set defined in (27) and generate:

      P N i=1 µ i (σ k )( Āi -Ki C) T P P N i=1 µ i (σ k )( Āi -Ki C) P       > 0 (30)
As we can see the estimation error checks the quadratic stability conditions depending on parameters. Each Lyapunov matrix is calculated for each vertex of the polytope so as to matrix (ā i -ki c) is stable.

Thus the adaptive filter synthesis extend local stability of each decoupling Kalman filter to a convex set λ representing whole parameters evolution µ. Then adaptive filter is stable on this convex set λ and generates robust residues allowing diagnosis of the system on all operating range.

EXEMPLE ILLUSTRATIF

La méthode proposée est appliquée à une simulation non linéaire discrète en boucle ouverte. Le processus non linéaire considéré, un système SIMO, se compose par un ensemble de trois systèmes stochastiques linéaires avec une dimension identique pour X ∈ 4 , U ∈ 1 et Y ∈ 3 . Le ième modèle linéaire représente le comportement dynamique du système non linéaire autour du ième point de fonctionnement et dépend de la grandeur d'entrée comme définie dans la table suivante : Table 1. Définition des trois points de fonctionnement.

Point de Fonctionnement

i = 1 i = 2 i = 3 U ∈ [0; 3[ U ∈ [3; 6[ U ∈ [6; 9[ Y i 3.2-
   ∀i B i = [1 0 0 0] T , C i =   0 1 0 0 0 0 1 0 0 0 0 1   , R i = 0.1 * I 3×3 .
Il est possible d'avoir des matrices de direction de défauts F i différentes pour chaque modèle i sous les conditions requises et de prendre une seule matrice de direction de défauts pour le filtre adaptatif. Les matrices de direction de défauts sont: 6), on remarque son évolution en trois paliers correspondant chacun à l'un des trois modèles. Le gain Ki k du ième filtre est bien très vite stable car il faut un à deux échantillons pour converger vers la valeur finale du gain Ki du ième filtre de Kalman découplant. Les valeurs de chaque gain peuvent être calculées hors-ligne pour pouvoir lancer l'algorithme du filtre adaptatif avec la valeur finale du gain de chaque filtre. On montre ainsi que reconstruire le gain du filtre adaptatif de manière algorithmique ou de manière polytopique (interpolée) est identique.

F 1 =     0 0 0.9 0 0 2.8 0 0     , F 2 =     0 0 2.4 0 0 1.4 0 0     , F 3 =     0

CONCLUSION

Nous avons présenté une méthode de diagnostic de défauts sur des systèmes non linéaires représentés par un ensemble fini de modèles linéaires invariants dans le temps. Au travers l'utilisation de probabilités bayésiennes, les N filtres permettent alors de sélectionner le meilleur modèle estimant au mieux la dynamique du système non linéaire. Après avoir déterminé le meilleur modèle, la partie diagnostic proprement dite s'effectue au moyen de l'utilisation d'un filtre adaptatif construit avec les matrices polytopiques de la représentation d'état génèrée. La stabilité étant définie localement, elle est étendue sur toute la plage de fonctionnement par le recours aux LMI sous hypothèses de paramètres variants dans un ensemble convexe. Ce filtre génère alors un résidu insensible aux défauts tandis que l'autre sensible aux défauts permet la détection, l'isolation et l'estimation de défauts multiples. Il faut noter que les modèles ont été supposés représentant parfaitement le système quand celui-ci varie dans un point de fonctionnement et par conséquent que les erreurs de modélisation n'ont pas été prises en compte. Dans de futurs travaux, il sera intéressant de prendre en compte les erreurs de modélisation ainsi que d'appliquer cette approche en boucle fermée.

  Fig. 1. Entry evolution Les défauts d(k) ∈ R 2 , rang(C i * ) = 2 < 3 (nombre de défauts inférieur au nombre de sortie)

Fig. 3 .

 3 Fig. 3. Generated faults of adaptive filter L'inversion matricielle qui s'en suit permet alors d'estimer les défauts vrais donnés sur la figure (4). On peut remarquer un léger pic lors du passage d'un modèle à l'autre mais il s'agit d'un seul échantillon dont la présence s'explique par le fait que l'on commute d'un modèle à l'autre sur l'entrée U et que l'évaluation s'effectue sur Y . Remarque: On pourra vérifier que le résidu insensible fig(5) provenant du filtre adaptatif possède une distribution Gaussienne de moyenne nulle sur toute la plage de fonctionnement du système, assurant de ce fait la validité du modèle généré par celui-ci.

Fig. 5 .

 5 Fig. 4. Faults estimation of system

Fig. 6 .

 6 Fig. 6. Gain evolution of adaptive filter


Marie-Josèphe Saurel-Cubizolles

Jean-François Chastang

Gwenn Menvielle

Anette Leclerc

Edisc Daniele Luce

Saurel-Cubizolles Marie-Jos

Menvielle Gwenn

Luce Daniele

Marie-Jos Phe Saurel

Social inequalities in

come

People in lower socio-economic status groups experience poorer health and have shorter lives than those in higher status groups 1-3 and these differences have increased in both sexes in recent years . Published data suggest that social differences are greater for men 4-7 than for women . Comparative studies have shown these social differences in mortality to be wider in France than in other European 8-11 countries . 12-14

The indicators of socio-economic position used in these studies included occupational group, educational level, income or indices based on the characteristics of the residential area. These different indicators show strong mutual associations and are proxy measures of " socio-economic status .

"

The aim of this paper was to compare the magnitude of social inequalities in mortality between men and women aged 30 to 74 years, in the 1990s, based on two complementary indicators of socio-economic status: educational level and occupational group as reported in 1990. We considered all-cause mortality and mortality due to cancer, mortality due to cardiovascular disease and mortality due to external causes (mainly violent causes) these three causes being the three leading causes of death in this population.

-

Methods

The data were obtained from a permanent demographic sample currently including about one million people, corresponding to about 1 of the population, randomly selected on the basis of date of birth (four days in the year). Subjects are included at the time of birth, % marriage (for foreigners with the appropriate birthday if they marry a French national and live in France with them), or at a census, particularly for immigrants. Data are updated at each successive census (1968,1975,1982,1990). A person remains in the sample until death. The French National Institute of Statistics (INSEE) follows the subjects of this sample and collects demographic, social and occupational information from different census schedules and from vital status forms. For the national census, data are collected by trained investigators, who distribute the forms to each household: the respondents fill in the census form and return it to the investigator. In special cases, if necessary, the investigators may fill in the forms themselves, using answers provided orally by the respondents. Causes of death were obtained from the national file of the French Institute of Health and Medical Research (INSERM).

Two indicators of social position were used. Educational level was defined as the highest of five categories achieved: 1) no diploma,

2) primary school level, 3) vocational or technical level, 4) secondary school level and 5) university level. Occupational group was coded into 10 classes, according to the standard classification of occupations in France: 1) professionals and managers, 2) intermediate white-collar workers, 3) farmers, 4) shop keepers and craftsmen, 5) office and sales employees, 6) skilled manual workers, 7) unskilled manual workers, 8) retired or early retired, 9) unemployed job-seekers and 10) others not in the labour force, including housewives.

For educational level, the order of the classes, in terms of frequency, was similar for men and women. For occupational group, the order of the classes depended on all-cause mortality adjusted for age. The first group was intermediate white-collar workers and the " "

second group was professionals and managers for women, whereas the order of these two groups was reversed for men.

" "

Calculations were carried out for overall mortality rates and for mortality rates for the three leading causes of death: cancer socioeconomic categories on a scale from the highest (0) to the lowest [START_REF] Kunst | Occupational class and ischemic heart disease mortality in the United States and 11 European countries[END_REF]. Each category covers a range on the scale proportional to its population size and is given on the scale corresponding to the midpoint of its range . These data are fitted with a Cox regression model 16

(
giving mortality estimates for the entire social distribution. The calculation of RIIs made it possible to take into account the number of subjects in each category and the differences in the distribution of socio-economic status between men and women. The analysis has been performed also for three age subgroups: 30 54, 55 64 and older than 64 years.

--This analysis was limited to French men and women born in France, aged 30 to 64 years at the beginning of the 1990 1999 period. It

Results

Educational level was similarly distributed in men and women, although there tended to be more less educated (without diploma or primary level) women than men ( ). By contrast, the distribution of occupational groups differed considerably between men and table 1 women. There were far more professionally inactive women than men, with almost a third of all women not working. Office and sales personnel accounted for the largest number of employed women, with a quarter of all women working in these sectors. Skilled and unskilled workers were more numerous among men than among women, as were professionals and managers.

For this population aged between 30 and 64 years, the probability of dying during the 1990 1999 period was more than twice as high for men than for women. Crude mortality rates varied from 3.3 for men educated to university level to 13.1 for men without a diploma, % % and from 1.6 for women educated to university level to 5.5 for women without a diploma. In the analysis of overall mortality rates by % % occupation, mortality rates of 2.9 for men working as professionals or managers men and 6.5 for male unskilled manual workers were % % recorded. For women, mortality rates were lowest for professionals or managers (1.5 ) and for intermediate white-collar workers (1.4 )

% %
and highest among farmers (2.9). For both men and women, the probability of dying was much higher among professionally inactive individuals, particularly those who were retired, reaching more than 20 for men and 8.4 for retired and 4.8 for otherwise inactive % % % women.

After adjustment for age, the relative risk of all-cause mortality clearly depended on educational level, with the risk of dying increasing with decreasing educational level ( ). For men, the relative risks for all educational level classes were significantly higher Table 2 than 1, the reference value corresponding to men educated to university level. For women, no differences were found between women educated to university and secondary levels, whereas relative risks significantly greater than one were found for women with a technical or primary education only and for women without a diploma. RII was slightly higher for men (2.96) than for women (2.62) and was significantly higher than 1 for both sexes.

The social inequalities tend to be slightly smaller for the oldest persons. For men the RII related to the educational level was 3. [START_REF] Leclerc | Socioeconomic inequalities in premature mortality in France: have they widened in recent decades?[END_REF] -Similarly strong social differences in mortality were observed if occupational group was used as the indicator. Relative risks were particularly high for the three classes of inactive people: retired, unemployed jobseekers and other inactive people. The RII was much higher for men, at 6.08 (5.54 6.68), than for women, for whom a value of 3.42 (2.96 3.96) was obtained.

--

The proportions of the male and female populations that were inactive differed considerably (4.9 of men inactive and 29.6 of % % women inactive). The heterogeneity of this group was therefore particularly great. Among inactive individuals, the risk of dying varied considerably with educational level, with more marked differences observed among women than among men. For example, the relative risk of dying was 1.83 (95 CI: 1.15 2.94) for men and 2.40 (95 CI: 1.68 3.43) for women without a diploma, taking the risk for

% - % -
individuals educated to university level as one; the RIIs were 1.62 (1.25 2.11) for men and 2.86 (2.28 3.59) for women.

--

We also carried out analyses by cause of death ( ). For cancer mortality, social inequalities were greater for men than for Table 3 women, regardless of the social indicator used: RII of 2.47 (2.17 2.81) for men and 1.63 (1. [START_REF] Steingart | Sex differences in the management of coronary artery disease[END_REF] was significantly higher for women than for men. A similar pattern was observed for occupational group: the relative risks of dying from cardiovascular causes were higher for female skilled and unskilled manual workers than for men in the same occupational category. In both men and women, the relative risk was significantly greater than one.

For the three age subgroups and for the four groups of causes of mortality the RIIs were all significant. Concerning cardiovascular mortality the RIIs were higher for women than for men in the three age subgroups.

In the light of these results for cardiovascular mortality, a more detailed analysis was carried out, separating out different cardiovascular causes of death ( ). For both indicators (educational level and occupational group) and for the three specific causes --

Discussion

This study provides evidence of strong social inequalities in mortality among French men and women at the end of the 1990s, based on an analysis of two indicators: educational level and occupational group. Overall, social inequalities were larger for men than for women, as observed for all-cause mortality, cancer mortality and mortality due to external causes (accidents, suicide, violence). However, the pattern was different for cardiovascular mortality, for which social inequalities seemed to be larger for women than for men, particularly for ischaemic cardiac diseases.

One of the strengths of this analysis is that it includes a very large amount of data, representative of the national situation in France.

These data are of high quality, social status is assessed from census data and the cause of death was reported in 97 of cases.

%

We used the RII relative inequality index to obtain an overview of the inequalities in a global measure. This method overcomes --

the difficulties due to differences in the distributions of socio-economic categories between women and men. This indicator has been used in published papers on the topic of social inequalities , . This index corresponds essentially to a ratio of mortality rate for the lowest 5 17-21 social class (or educational level) to that of the highest social class (or educational level). Higher values for this index indicate greater social inequality in the population. This indicator has already shown a strong increase in the magnitude of social inequalities between 1968 and 1996 for these data . [START_REF] Leclerc | Socioeconomic inequalities in premature mortality in France: have they widened in recent decades?[END_REF] Our results show that social inequalities in mortality were larger among men than among women for the study period. The magnitude of the differences between the sexes may be sensitive to the social measure: larger inequalities were observed with occupational group than with educational level, for both sexes. This finding is consistent with previous reports . The analysis based on occupational group 21 highlighted inequalities related to employment status and within the working population. In both men and women, not working was associated with a higher risk of death. Inactivity is one of the major correlates of poor health, mainly due to the health selection process.

Many women do not work, but this group is highly heterogeneous in terms of educational level.

Suggested explanations for the observed social differences in mortality include an uneven distribution of health risk behaviour (including smoking and alcohol consumption), material factors (such as financial difficulties), occupational exposure and psychosocial resources, and stress-related factors (insecure job, life events). These factors together probably account for much of the inequality in mortality rates . [START_REF] Van Oort | psychosocial, and behavioural factors in the explanation of educational inequalities in mortality in the Netherlands[END_REF] Huisman . , based on data for eight European countries, reported higher educational inequalities for cardiovascular mortality et al 23 among women (for 45-59-year-old women, the rate ratio was 1.74 (1.60 1.90)) than among men (1.51 (1.45 1.57)) whereas the reverse -was observed for cancer mortality (for individuals aged 45 to 59 years, the rate ratio was 1.08 (1.03 1.13) for women and 1.46 (1.40 1.52)

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for men) and total mortality. Our results are consistent with those of this previous study. Similarly, in Norway, Naess . reported et al 24 stronger educational inequalities for death due to coronary heart disease in women than in men. From a nationally representative sample in the United States, Thurston showed also that coronary heart disease risk associated with low education was stronger among et al 25 women than amon men. Women with low levels of education had more concurrent social and psychological risks than did men.

It is difficult to explain why the pattern of cardiovascular mortality differs from that of cancer mortality and all-cause mortality. In mortality studies, it is difficult to elucidate the social differences operating at each step: preventive behaviour, exposure to risk factors, incidence, medical care and survival. Davey Smith observed that cardiovascular disease was the cause of death most strongly et al 21 associated with education, possibly reflecting the particular importance of the role played by socio-economic circumstances in childhood in determining the risk of coronary heart disease and stroke. Obesity is a risk factor for cardiovascular disease. In France, as in other countries, the prevalence of obesity and overweight has increased in recent years . The differences in the frequency of obesity between 26 social groups increased between 1981 and 2003 and were greater among women than among men: obesity rates for men ranged from 10% in the poorest households (the first quartile of living standards) to 9 in the richest households (the last quartile), whereas it ranged from % 13 in the poorest households to 6 in the richest households among women . In the United States also, the body mass index shows a % % 26 stronger educational gradient among women than among men . These trends may partly account for social differences in cardiovascular 25 mortality being greater among women than among men.

Other risk factors, such as diabetes and hypertension, seem be more strongly related to coronary heart disease (CHD) in women than in men. For example, women with diabetes have a much higher relative risk of coronary heart disease with respect to the non-diabetic population than do men with diabetes . Data in Finland have demonstrated a strong impact of diabetes on CHD mortality, with relative 27 death rates of 6.04 for women and 3.42 for men. CHD mortality followed systematic socio-economic trends, with rates higher among the poorest diabetic and non-diabetic individuals . 28

Low social status is associated with arterial hypertension , with greater differences observed among women than among men . In a 29 30 large Dutch sample, Hoeymans observed that the association between hypertension and educational level was stronger among et al 31 women than among men. Moreover, hypertension and type 2 diabetes increase CHD risk independently, but together, they increase the risk considerably, particularly in women . 32

The prevalence of hypercholesteraemia differs considerably with educational level in both men and women . The prevalence of 33 hypercholesterolaemia and low HDL-cholesterol levels seem to be more strongly related to educational level among women than among men . 31

Inequalities in medical care may also play a role. Dong reported that women were less likely than men to undergo cardiac et al 34 surgery in England, regardless of age group. Neither disease severity nor comorbidity could account for these differences. Other studies in the UK or United States have indicated that women are about half as likely to undergo surgery or to be otherwise treated for ischaemic heart disease than men, even if disease severity and comorbidity are taken into account , . Thus, women may be undertreated and 35 36 deprived of effective intervention.

This study provides evidence for persistent social inequalities in mortality in France, in both men and women. These social differences are generally greater among men than among women. However, specific, large social inequalities were observed for mortality by cardiovascular disease in the female population. The reduction of cardiovascular disease mortality in low educational level groups should be treated as a major public health priority. Effective policies are required to decrease cumulative exposure to cardiovascular risk factors, such as smoking, being overweight, hypertension and diabetes. It would be necessary to further understand the different mechanisms involving in the link between educational level and health status.

What this paper adds ?

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  therefore concerned 104 109 men and 109 765 women, and included all deaths occurring between January 1 1990 and December 31 st st 1999 (8 148 men and 3 576 women).

  Cox models were used to calculate relative risks, with age as the time variable to control for age, occupational group or educational level being treated as qualitative variables. The model with the occupational group was not adjusted for the level of education, and this with educational level was not adjusted for the occupational group.

	international classification of diseases (ICD-9) codes 140 209), cardiovascular diseases (CVD; codes 390 459), and external causes --
	(codes E800 E999).
	We calculated relative index of inequality (RIIs) to obtain quantitative global estimates of the magnitude of inequalities in mortality.
	This index is a regression-based summary measure used in research into social inequalities , , . It is calculated by ranking 4 15 16

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  (2.73 3.66) for subjects aged from 30 to 54 years, 3.32(2.87 3.83) for those between 55 and 64 years and 2.44 (2.10 2.83) for men older than The difference between these three RIIs was significant. For women, the RIIs were respectively 2.69 (2.15 3.35), 2.65(2.11 3.33) and

	-	-	-
	64. -	-
	2.51 (1.97 3.19) and this difference was not significant.

  1.96) for women if educational level was was used. A similar pattern was observed for mortality due to external causes (accidents, suicides, violence). Conversely, for cardiovascular mortality, there tended to be larger inequalities among women than among men. The relative risk of dying from cardiovascular causes for people without a diploma was 4.31 (2.52 7.42) for women and 2.72 (2.17 3.41) for men, for example. The RII

	-	-
	used these two RIIs were significantly different-, and of 4.53 (3.94 5.21) for men and 2.09 (1.71 2.56) for women if occupational group ---

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table 4

 4

	considered	ischaemic heart diseases, cerebrovascular diseases and other cardiovascular diseases	higher RIIs were obtained for
	groups are not shown in	table 4	. The RII for ischaemic cardiac disease was 3.21(2.35 4.38) for men and 5.35 (2.43 11.8) for women; that --
	for cerebrovascular disease was 5.28 (3.30 8.45) for men and 5.43 (2.69 11.0) for women and that for other cardiovascular mortality was --
	6.15 (4.32 8.76) for men and 6.43 (3.56 11.6) for women.

--

women than for men. The RII values obtained were particularly high for ischaemic cardiac disease. For each detailed cardiovascular cause of death social inequalities were very large. Due to the small numbers of individuals in some classes, differences between occupational

Table 1

 1 Crude mortality rate (1990 1999) by level of education and occupational group among men and women

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	Men

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[START_REF] Mustard | Gender differences in socioeconomic inequality in mortality[END_REF]

[START_REF] Mustard | Gender differences in socioeconomic inequality in mortality[END_REF]

[START_REF] Mustard | Gender differences in socioeconomic inequality in mortality[END_REF]

This project received financial support from the Programme Sciences biom dicales, Sant et Soci t under the auspices of INSERM, CNRS

and MIRE-DRESS.

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