Didier Theilliol Mickaël Rodrigues

Dominique Sauter

Design of an Active Fault Tolerant Control and Polytopic Unknown Input Observer for Systems described by a Multi-Model Representation

In this paper, an active Fault Tolerant Control (FTC) strategy is developed to systems described by multiple linear models to prevent the system deterioration by the synthesis of adapted controllers. First, a Polytopic Unknown Input Observer is synthesized for providing actuator fault estimation. The actuator fault estimation is used in a FTC scheme which schedules some predefined state feedback gains. These gains are performed 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, a pole placement is designed by pole clustering. The effectiveness and performances of the method have been illustrated in simulation considering a hydraulic system: a three-tank system.

I. INTRODUCTION

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. 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. Various approaches for FTC have been suggested in the literature [START_REF] Patton | Fault-tolerant control: the 1997 situation[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] or Fault Detection and Isolation (FDI) methods as in the chapter nine of [START_REF] Chen | Robust model-based fault diagnosis for dynamic systems[END_REF] where nonlinear 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 estimate 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] and in chapter 7 of [START_REF] Blanke | Diagnosis and Fault-Tolerant Control[END_REF]. Multiple Model Adaptive Estimation (MMAE) [START_REF] Maybeck | Multiple model adaptive algorithms for detecting and compensating sensor and actuator/surface failures in aircraft flight control systems[END_REF] or Interacting Multiple Model (IMM) introduce a multi-model approach for FDI, but these techniques are developed around an unique operating point (OP). Indeed, these methods consider a particular multiple or multi-model approach where each model is dedicated to a specified fault. A polytopic representation is sometimes used in multi-model or piecewise linear models [START_REF] Ozkan | Control of a solution copolymerization reactor using multi-model predictive control[END_REF]. This paper addresses a more general method that could allow to detect actuator fault in a nonlinear system. Compared to multi-model based reconfigurable control method presented by [START_REF] Astrom | Control of complex systems[END_REF], this paper does 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 control 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. The developed method preserves the system performances through an appropriate gain scheduling synthesis in faulty case with a FDI module dedicated to actuator fault estimation in multi-model framework. 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 for nonlinear systems with additive and multiplicative fault around predefined operating points. A Polytopic Unknown Input Observer is synthesized for estimating actuator fault in systems represented by multimodels. In section III, we introduce a pole placement by LMI region and then a gain synthesis for each actuator generates an active global state feedback 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 FDI FOR POLYTOPIC SYSTEMS

A. Nonlinear representation

Consider a discrete-time nonlinear dynamical system in fault-free case 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. Functions g x k , u k and h x k , u k are continuously differentiable.

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 as presented in [START_REF] Rodrigues | Fault Diagnosis on Industrial Systems based on a Multiple Model approach[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 ) (2
) 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 around (x j e , u j e ) with y j e = C j x j e + D j u j e or by an identification procedure 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]. Fault distribution matrix is represented by F j ∈ R n×p . f k ∈ R p represents the actuator fault magnitude vector which in fault-free case it is obviously equal to zero. In the following, we consider that D j = 0. This linear system (2) can be specified by the set of system matrices:

M j = A j B j F j C j , ∀j = [1, 2, ..., N ] (3)
Let M k be a matrix sequence varying within a convex set, defined as:

S k := N j=1 ρ j k M j : ρ j k ≥ 0, N j=1 ρ j k = 1 (4)
In the multi-model framework, M 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 through interpolation or activation functions ρ j k . These activation functions

ρ j k ∀j ∈ [1, 2, .., 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 directly generated via works of [START_REF] Adam-Medina | Fault diagnosis in nonlinear systems through an adaptive filter under a convex set representation[END_REF] which permit to generate insensitive residual to faults and some uncertainties. We will use these works for providing the activation functions as scheduling variable. So, activation functions are robust against faults and the dynamic system behaviour is well represented. based on (2) and around an operating point, the nonlinear system can be represented by the following way as:

x k+1 = A j x k + B j u k + F j f k + ∆x j y k = C j x k (5)
with the term ∆x i , around an jth operating point, equals to:

∆x j = -A j x j e -B j u j e + x j e (6)
As referred in (3), a new set of matrices is designed with the additional term ∆x j such that:

S j = A j B j F j ∆x j C j , ∀j ∈ [1 . . . N ] (7)
In this paper, a diagnosis procedure is developed to detect and isolate a particular fault among several others. In the following a unique matrix B is considered: some plants can be modelled in such way as presented in the last section. While a single residual is sufficient to detect a fault, a set of structured residuals is required for fault isolation. Several approaches have been proposed by [START_REF] Chen | Robust model-based fault diagnosis for dynamic systems[END_REF] to generate structured residuals. Here, a residual generation using unknown input observer scheme is considered in order to be sensitive to fault vector f * k and insensitive to f d k . We consider that only a single actuator fault may occur at a given time, simultaneous faults are not considered. Hence, vector f d k is a scalar and it is considered as an unknown input. In case of an i-th actuator fault, the system can be represented by:

x k+1 = N j=1 ρ j k A j x k + Bu k + F d f d k + F * x f * k + ∆x j ] y k = Cx k (8)
with matrix F d equals to B i which is the ith column of B, and matrix F *

x equals to Bi which is the matrix B without the ith column. In the following, we will consider only one output matrix C for all OP j .

B. Polytopic UIO synthesis for actuator fault estimation

In a general way, the UIO allows to vanish undesirable information by decoupling unknown input. In this FDI scheme, the UIO is used to isolate an actuator fault.

Definition 1: (Polytopic Unknown Input Observer) [START_REF] Rodrigues | Design of a Robust Polytopic Unknown Input Observer for FDI: Application for Systems described by a Multi-Model Representation[END_REF] A polytopic observer is defined as a polytopic unknown input observer (PUIO) for the system described by ( 8) without fault (f d k , f * k = 0), if the estimation error tends asymptotically to zero despite unknown disturbances on the system. This polytopic unknown input observer is defined such that:

z k+1 = N j=1 ρ j k S j z k + T Bu k + K j y k + ∆z j x k+1 = z k+1 + Hy k+1 (9)
where x is the state space estimation of x. The estimation error is equivalent to

e k+1 = x k+1 -x k+1 (10)
where, with notation ( )(ρ) stands for N j=1 ρ j k ( ) j , K(ρ) = K 1 (ρ) + Π(ρ). S(ρ), T, K(ρ) and H are designed so as to ensure the stability and the convergence of the estimation error e k = x kx k without fault on the system (f k = 0). To obtain an exact decoupling, the following conditions should be satisfied:

S(ρ) = T A(ρ) -K 1 (ρ)C Π(ρ) = S(ρ)H (I -HC)F d = 0 T = I -HC T ∆x(ρ) = ∆z(ρ) (11)
The robust polytopic UIO design is realized when relations [START_REF] Theilliol | A multiple model based approach for Fault Tolerant Control in nonlinear systems[END_REF] hold true and S(ρ) is stable. So, estimation error without fault occurrence, denoted ēk , tends asymptotically to zero if all these conditions (11) are satisfied. The necessary and sufficient conditions for the existence of a PUIO are:

(i) Rank(CF d ) = Rank(F d ) (ii) (T A j , C) are detectable pairs, ∀j ∈ [1, 2, .., N ].
These conditions are related in linear case [START_REF] Chen | Robust model-based fault diagnosis for dynamic systems[END_REF]. If condition (i) is true, the synthesis of matrix H, which can permit to avoid unknown input effects, is performed by:

H = F d (CF d ) + (12)
It should be noted that the matrix F d is a constant matrix ∀j ∈ [1, . . . , N ]. If conditions [START_REF] Theilliol | A multiple model based approach for Fault Tolerant Control in nonlinear systems[END_REF] hold true, the estimation error e k and the residual r k are described as:

e k+1 = S(ρ)e k + T F * x f * k r k = Ce k (13)
If conditions [START_REF] Theilliol | A multiple model based approach for Fault Tolerant Control in nonlinear systems[END_REF][START_REF] Rodrigues | Fault Diagnosis on Industrial Systems based on a Multiple Model approach[END_REF] are fulfilled, an unknown input observer provides an estimation of the state vector , used to generate a residual vector [START_REF] Wan | Efficient scheduled stabilizing model predictive control for constrained nonlinear systems[END_REF]). The fault isolation is realized by a common bank of p polytopic unknown input observers. Each residual vector r l k = y k -C x k , produced by the l-th polytopic unknown input observer, may be used to detect a fault according to a statistical test: Page-Hinkley test, limit checking test, generalized likelihood ratio test. The reader could see [START_REF] Noura | Fault-tolerant control in dynamic systems: Application to a winding machine[END_REF] for more details about residual generation and isolation purposes. Some relations have to be verified for detection purposes, ensuring that such synthesis do not affect fault detection:

r l k = y k -C x k (l ∈ [1, . . . , p]), independent from f d k . This means that r k = 0 if f * k = 0 and r k = 0 if f * k = 0 whatever u k and f d k (see
Rank[T F * x ] = rank[F * x ] (14)
This condition allows to verify that de-coupling do not affect fault detection by the estimation error [START_REF] Rodrigues | Design of a Robust Polytopic Unknown Input Observer for FDI: Application for Systems described by a Multi-Model Representation[END_REF]. This condition can be translated in a geometric way such as:

Im(F * x ) ⊆ Im(T T ) (15)
If the union image F * x is included in the image of decoupling matrix T T , then fault detection is possible over all the operating range. It is equivalent to:

Im(F * x ) Ker(T ) = {0} (16)
If condition ( 16)(which can be checked off-line) is fulfilled, it ensures that de-coupling do not affect fault detection. It could be noticed that F * x is supposed to be full column rank in order to avoid fault compensation of non-independent columns.

According to the fault isolation, the fault magnitude estimation of the corrupted element is extracted directly from the l-th polytopic unknown input observer which is built to be insensitive to the l-th fault (f * k = 0). Proposition 1: If ( 9) is a Polytopic Unknown Input Observer, then fault estimation can be done by:

f d k = F + d x k+1 -A(ρ) x k -Bu k -∆x(ρ) (17)
Proof : Then, if ( 9) is a polytopic unknown input observer, x k will coincide either asymptotically or in finite time with x k when f * k = 0. Thus, substituting x k by x k in (8), multiplying left by F + d leads to [START_REF] Chilali | H∞ design with pole placement constraints: an LMI approach[END_REF]. Note that in the presence of an actuator fault, F d is a matrix of full column rank. The stability condition of the polytopic unknown input observer can be performed through the use of LMI.

C. Pole placement of the polytopic UIO

According to [START_REF] Wan | Efficient scheduled stabilizing model predictive control for constrained nonlinear systems[END_REF], the estimation error ēk , without fault, is expressed as

ēk+1 = S(ρ)ē k = T A(ρ) -K 1 (ρ)C ēk = Ā(ρ) -K 1 (ρ)C ēk (18)
Proposition 2: The estimation error (18) is called quadratically stabilizable if there exists matrices R j and if there exists a positive symmetric matrix X > 0 such that ∀j ∈ [1, . . . , N ] :

X ĀT j -C T R T j X Āj -R j C X > 0 (19)
with gain matrices K 1 j equal to X -1 R j . Proof: By weighting each LMI defined in (19) by interpolation functions ρ j defined in the previous section, using notation R j = XK 1 j and summing all of them with

N j=1 ρ j = 1, (19) becomes ∀j ∈ [1, . . . , N ] equivalent to        X N j=1 ρ j ( Āj -K 1 j C) T X X N j=1 ρ j ( Āj -K 1 j C) X        > 0 (20)
With a Schür Complement, LMI in (20) expresses quadratic stability and we find after some computations:

( Āj -K 1 j C) T P ( Āj -K 1 j C) -P < 0 and X -1 = P > 0 that is the quadratic stability in discrete case of the estimation error.

III. FAULT TOLERANT CONTROL SYNTHESIS

The equation ( 8) considers additive fault representation but there exists multiplicative representation for specific actuator fault. So, a local multiplicative actuator fault representation is defined as:

x k+1 = A j x k + B j (I -γ a )u k + ∆x j y k = C j x k (21)
with γ a diag[γ a 1 , . . . , γ a p ], γ a 1 ∈ R, such that γ a i = 1 represents a failure of i-th actuator and γ a i = 0 implies that ith actuator operates normally. The relationship between state space representations (2) and ( 21) is equivalent to

F j f k = -B j γ a u k (22)
where the faulty matrix distribution F j is equal to matrix B j in an actuator fault case. The estimation of γ a is obtained from fault estimation f k provided by the PUIO and availability of nominal control inputs u nom k (in faultfree case). We attract attention that the system has to be observable in all around each operating points and we will now only consider linear models and not affine models, i.e. ∆x j = 0.

A. Nominal case

Let consider the state space representation (21) of nonlinear system defined around the Equilibrium Points (OP j ), ∀j = 1, . . . , N

x k+1 = N j=1 ρ j k A j x k + i∈I B i j (I -γ a )u k y k = N j=1 ρ j k [C j x k ] (23)
with I : i = 1, . . . , p the actuators for each OP j and matrices (A j , B j , C j ) ∈ S k defined in (4). Consider the matrix representing total faults in all actuators but the i-th:

B i j = [0, . . . , 0, b i j , 0, . . . , 0] (24)
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 [START_REF] Chilali | H∞ design with pole placement constraints: an LMI approach[END_REF]. 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
(25) It can be noticed that if q = 0 and r = 1, the previous equation ( 25) is equivalent to solve a classical quadratic stability problem. Due to space limitation basic principes in LMI are omitted. Thus, for more details in LMI pole placement, the reader could refer to [START_REF] Chilali | H∞ design with pole placement constraints: an LMI approach[END_REF]. 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 .
Theorem 1: Consider the system (23) 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 (25) 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 (26) = -Y X -1 x k = -K nom x k 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 = -K nom x k (27)
Proof: Summation of (25) 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 (28) 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 (29)
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 (30) 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 (31) leading to p i=1 B i j Y i = B j ( p i=1 G i Y i ) (32) Thus, taking Y = p i=1 G i Y i , equation (32) becomes p i=1 B i j Y i = i∈I B i j Y i = B j Y (33)
which, substituted into LMI (29), finally makes

-rX qX + (A j X -B j Y ) T qX + (A j X -B j Y ) -rX < 0 (34)
for all OP j , j = 1, . . . , N . By multiplying each LMI (34) 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 (35)
it is equivalent to

-rX qX + (A(ρ)X -B(ρ)Y ) T qX + (A(ρ)X -B(ρ)Y ) -rX < 0 (36) 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 (35) and Y = K nom X quadratically stabilizes the system (23) under the set S k by solving (36) with a state feedback law

u k = N j=1 ρ j k u j nom = -Y X -1 x k . Remark 1:
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 (4):

u k = N j=1 ρ j k u j nom = - N j=1 ρ j k K j x k (37)
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 .

B. Active Fault Tolerant Control design

By considering the system (23) and based on the previous synthesis control law, the FTC method can be developed in this section under assumptions that fault occurrence and fault magnitude γ a are known, provided by the PUIO.

Theorem 2: Consider the system (23) 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, . . . , p. Let introduce the set of indexes of all actuators that are not completely lost, i.e.

Θ {i : i ∈ (1, . . . , 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 (38)
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 (38) to the faulty system (21), 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 with Γ a = I p-h 0 0 O h
Γ a is a diagonal matrix which contains only entries zero (representing total faults) and one (no fault). Due to the fact that only the set Θ is considered and 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

( i∈Θ G i Y i )( i∈Θ 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 (38) implies that

u j F T C = -K F T C x k with K F T C = (I -γ a ) + i∈Θ G i Y i ( i∈Θ X i ) -1
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 = -K F T C x k (39)
IV. APPLICATION

A. Process description

The approach presented in this paper has been applied to the well known three tanks system 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 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 such as assumption l 1 > l 3 > l 2 is assumed to be always true.

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 (1(b)) always close to the dynamic behaviour of the nonlinear system according to the considered operating regimes [START_REF] Adam-Medina | Fault diagnosis in nonlinear systems through an adaptive filter under a convex set representation[END_REF].

m 3 s - 1 l r 1 l r 2 l 1 l 3 l 2 ρ 1 k ρ 2 k ρ 3 k q 1 q 2
Fig. 1. System dynamic behaviour in fault-free case :(a) system outputs, (b) activation functions ρ j k , (c) system inputs Figure [START_REF] Patton | Fault-tolerant control: the 1997 situation[END_REF] shows the outputs with respect to set-point changes occur at time instant 300s and after at time instant 1200s. In the simulation, gaussian noises (N (0, 1e -4 2 )) are added to each output signal. The reference inputs correspond to step changes for l 1 and l 2 . The consequence of an actuator fault is illustrated in figure [START_REF] Blanke | Diagnosis and Fault-Tolerant Control[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 600s. 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. The controller tries to cancel the fault effect with the signal q s1 on figure (2(b) but the real available control signal is described by variable q 1 . 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. The presence of an integrator in the control law allows the system to reach the nominal values as illustrated in the third operating point (after time instant 1200s) in spite of fault persistence as illustrated in figure (4).

The PUIO is synthesized as in first section. The fault is detected, estimated and depicted on figure (4) where we can see its effectiveness. 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] Zhang | Fault Tolerant Control system design with explicit consideration of performance degradation[END_REF]. There exists a little time delay between fault occurrence and fault compensation due to FDI module. 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 for systems based on multi-model representation. This method is suitable for partial actuator faults on a wide operating range of the system. Actuator faults are estimated with a Polytopic Unknown Input Observer. 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 the PUIO. The performances and the effectiveness of this active Fault Tolerant Control based on a multiple model approach have been illustrated through an hydraulic system.

  This work was supported by EC STREP Project NeCST (Networked Control Systems Tolerant to Faults) IST 004303 M. Rodrigues, D. Theilliol and D. Sauter are with Centre de Recherche en Automatique de Nancy -CNRS UMR 7039, Université Henri Poincaré, BP 239 -54506 Vandoeuvre Cedex -France. Phone: +33 383 684 465 -Fax: +33 383 684 462mickael.rodrigues@cran.uhp-nancy.fr

Fig. 2 .Fig. 3 .

 23 Fig. 2. System dynamic behaviour in faulty-case without FTC :(a) system outputs, (b) system inputs

FaultFig. 4 .

 4 Fig. 4. Actuator fault magnitude and its estimation