H Hamdi

M Rodrigues
email: rodrigues@lagep.univ-lyon1.fr

C Mechmeche
email: chokri.mechmeche@esstt.rnu.tn

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

N Benhadj Braiek

Fault Detection and Isolation for Linear Parameter Varying Descriptor Systems via Proportional Integral Observer

Keywords: LPV descriptor systems, polytopic Proportional Integral Observer, fault detection and isolation, Linear Matrix Inequalities (LMIs)

The main contribution of this paper is the design of a polytopic Unknown Input Proportional Integral Observer (UIPIO) for Linear Varying Parameter (LPV) descriptor systems. This observer is used for actuator Fault Detection and Isolation (FDI). The proposed method is based on the representation of LPV descriptor systems in polytopic form where its parameters evolve in a hypercube domain. The designed polytopic UIPIO is also able to estimate the states and the unknown inputs of the LPV descriptor system. Stability conditions of such observer are expressed in terms of Linear Matrix Inequalities (LMI). An example illustrates the performances of such polytopic UIPIO for multiple actuator faults estimation in spite of disturbances.

INTRODUCTION

The topic of Fault Detection and Isolation (FDI) in dynamic systems has been an active research area in the last two decades, as can be seen, in survey papers by [START_REF] Theilliol | Fault diagnosis and accommodation of three-tank system based on analytical redundancy[END_REF] for linear systems, [START_REF] Rodrigues | A Fault Detection and Isolation Scheme for Industrial Systems based on Multiple Operating Models[END_REF] for multi-models representation and [START_REF] Bokor | Fault detection and isolation in nonlinear systems[END_REF] for nonlinear systems. For many processes it is feasible to use differential-algebraic equations to describe the plant dynamic behavior. The use of the physical or chemical laws is often a natural choice. In the most general form, a set of so-called descriptor system equations can be used as discussed by [START_REF] Boutayeb | Observers design for non linear descriptor systems[END_REF]. Model-based FDI techniques use mathematical models of the monitored process and extract features from measured signals, to generate fault indicating signals, that is, the residuals. Hence, generation of residual signals is a central issue in model-based FDI and a rich variety of methods are available for FDI. The basic idea behind the observer-based approaches is to estimate the outputs of the system from the measurements by using observers. Then, the output estimation error is used as a residual. The problem of designing observers has been investigated by many authors. The authors in [START_REF] Darouach | Design of Observers for Descriptor Systems[END_REF] have proposed full and reduced order observers for linear descriptor systems. In [START_REF] Hou | Observer Design for Descriptor Systems[END_REF], the authors have developed an observer design method based on singular value decompositions and the concept of generalized inverse matrix. Observer design for nonlinear descriptor systems was studied in [START_REF] Boutayeb | Observers design for non linear descriptor systems[END_REF]. The authors in [START_REF] Marx | Design of observers for Takagi-Sugeno descriptor systems with unknown inputs and application to fault diagnosis[END_REF] have proposed an unknown inputs multi-observer for nonlinear descriptor systems 3 described by multi-models. As in the conventional systems, in view of the advantages of integral actions, some researchers have introduced the integral term in observer design for descriptor systems [START_REF] Wu | Generalized PI observer design for linear systems[END_REF][START_REF] Koenig | Unknown Input Proportional Multiple-Integral Observer Design for Linear Descriptor Systems: Application to State and Fault Estimation[END_REF]. Full and reduced order Proportional Integral (PI) observers for fuzzy descriptor system subject to parameter variations have been designed in [START_REF] Koenig | Design of Proportional-Integral Observer for Unknown Input Descriptor Systems[END_REF]. These observers allow estimation of both the state and the unknown inputs. The authors in [START_REF] Wang | Robust Fuzzy Control of a Class of Nonlinear Descriptor Systems with Time-Varying Delay[END_REF], [START_REF] Taniguchi | Fuzzy Descriptor Systems and Nonlinear Model Following Control[END_REF] have synthesized robust control laws for fuzzy descriptor systems. More recently, Linear Parameter Varying (LPV) modeling of the monitored system has been considered. Such models can be used efficiently to represent some ordinary nonlinear systems. Some researchers from the fault detection and isolation community have developed model-based FDI methods [START_REF] Grenaille | A method for designing FDI filters for polytopic LPV models[END_REF][START_REF] Zolghadri | Grenaille Fault diagnosis for LPV systems[END_REF] and Fault Tolerant Control [START_REF] Rodrigues | Fault Tolerant Control Design for Polytopic LPV Systems[END_REF] using LPV systems. For polytopic LPV systems, a polytopic unknown inputs observer has been also used for the state estimation of uncertain systems and for FDI of nonlinear systems described by a multi-models representation [START_REF] Rodrigues | Design of a Robust Polytopic Unknown Input Observer for FDI: Application to Nonlinear Systems described by a Multi-Models Representation[END_REF]. The authors in [START_REF] Masubuchi | Gain-Scheduled Controller Design Based on Descriptor Representation of LPV Systems: Application to Flight Vehicle Control[END_REF]17] have designed gain-scheduled controllers for LPV descriptor systems. In spite of that, there are few studies on state reconstruction and fault diagnosis for LPV descriptor systems despite of a method for actuator fault detection and estimation proposed in [START_REF] Hamdi | State Estimation for Polytopic LPV Descriptor Systems: Application to Fault Diagnosis[END_REF]. The main contribution of this paper is associated with the design of a polytopic Unknown Input Proportional Integral Observer (UIPIO) for LPV descriptor systems and for fault detection and isolation. In this modeling framework, the dynamic behavior of the LPV descriptor system is given in a polytopic form, that allows to describe the system as a convex combination of subsystems defined by the vertices of a convex polytope. These submodels are then combined by a convex weighing functions to yield the global model. The first step of this work consists of modeling the LPV descriptor system by a polytopic form. The second step is devoted to the design of a polytopic UIPIO for LPV descriptor systems subjected to external disturbances. The obtained observer is used for state, unknown inputs estimation and for actuator faults detection and isolation. Note that, in one hand, to estimate the state and unknown inputs, all matrices of the LPV descriptor systems are considered with time-varying parameters. On the other hand, to ensure the fault detection and estimation, the control matrix and disturbance matrix of the dynamic systems are considered with static parameters. This paper is organized as follows: The polytopic representation of LPV descriptor systems is presented in Section 2. The polytopic Unknown Input Proportional Integral Observer (UIPIO) design corresponding to the polytopic models and its stability conditions are studied in Section 3. Section 4 is devoted to detection, isolation and estimation of the actuator faults by using a set of polytopic UIPIO for LPV descriptor systems. Finally, an illustrative example is provided to show the effectiveness of the proposed approach. In this paper, the notation is standard. R denotes the set of real numbers; C is the complex plane; A + denotes the generalized inverse of A; Q > 0 or (Q < 0) indicates that the matrix Q is positive (or negative) definite. Also, ∀ means "for all".

POLYTOPIC LPV DESCRIPTOR SYSTEM

Consider the class of LPV descriptor systems:

E ẋ(t) = A(θ(t))x(t) + B(θ(t))u(t) + R(θ(t))d(t) y(t) = Cx(t) (1)
x(t) ∈ R n is the state vector, u(t) ∈ R p is the control input, d(t) ∈ R q is the unknown input vector and y(t) ∈ R m represents the measured output vector. A(•), B(•), R(•) are continuous functions which depend affinely on the time-varying parameter vector θ(t) ∈ R l . It is also assumed that this time-varying parameter vector θ(t) is bounded and lies into a hypercube such that [START_REF] Anstett | Polytopic observer design for LPV systems based on minimal convex polytope finding[END_REF][START_REF] Bruzelius | Linear Parameter-Varying Systems an approach to gain scheduling[END_REF][START_REF] Wu | Control of Parameter Varying Systems PhD Thesis[END_REF]: The LPV descriptor system (1) with the affine parameter dependence (2) allows us to determine a set of matrices Π i such that:

θ(t) ∈ Γ = {θ | θ i (t) ≤ θ(t) ≤ θ i (t)}, ∀t ≥ 0 (2)
Π(θ(t)) = Π 0 + h i=1 θ i (t)Π i , ∀θ(t) ∈ Γ (3)
where

Π = A B R C 0 0 (4)
The system (1) can be transformed into a convex interpolation of the vertices of Γ where the vertices of the polytope are defined such that [START_REF] Rodrigues | Fault Tolerant Control Design for Polytopic LPV Systems[END_REF]

: S i = A i , B i , C, R i , ∀i ∈ [1, . . . , h]
where h = 2 l . The polytopic coordinates are denoted ρ(θ(t)) and vary into the convex set Ω:

Ω = ρ(θ(t)) ∈ R h , ρ(θ(t)) = [ρ 1 (θ(t)), . . . , ρ h (θ(t)] T , ρ i (θ(t)) ≥ 0, h i=1 ρ i (θ(t)) = 1 (5)
Consequently, the polytopic LPV descriptor system with the time-varying parameter vector ρ(θ(t)) ∈ Ω is given by:

   E ẋ(t) = h i=1 ρ i (θ(t))(A i x(t) + B i u(t) + R i d(t)) y(t) = Cx(t) (6)
where

A i ∈ R n×n , B i ∈ R n×p , R i ∈ R n×q
and C ∈ R m×n are time invariant matrices defined for the i th model. E ∈ R nxn is a constant matrix and rank(E) = r < n.

The following section presents a design method of a polytopic Unknown Input Proportional Integral Observer (UIPIO) for LPV descriptor systems subjects to disturbances. This observer is constructed by the interpolation of a set of sub-UIPIO, according to the parameter ρ i (θ(t)). As in ordinary multi-models representation [START_REF] Rodrigues | A Fault Detection and Isolation Scheme for Industrial Systems based on Multiple Operating Models[END_REF] and for multi-observer design, the observability requires that all the subsystems are observable. Then, the following assumptions express the observability properties of the polytopic LPV descriptor system.

Assumption A1:

The matrix C is of full-row rank, i.e., rank(C) = m. Assumption A2: The triple matrix (E, A i , C) is R-observable [START_REF] Koenig | Design of Proportional-Integral Observer for Unknown Input Descriptor Systems[END_REF], ∀i = 1, . . . , h, i.e.,

rank sE -A i C = n, ∀s ∈ C. ( 7
)
Assumption A3: The triple matrix (E, A i , C) is impulse-observable [START_REF] Koenig | Design of Proportional-Integral Observer for Unknown Input Descriptor Systems[END_REF], ∀i = 1, . . . , h, i.e.,

rank   E A i 0 E 0 C   = n + rank(E) (8)
Descriptor system is represented by a combined set of Differential and Algebraic Equations (DAE).

The observability reflects the capacity to estimate the whole state. Then, two types of observability are required:

The R-observability characterizes the capacity to reconstruct only the state of the dynamic part (governed by differential equations).

The impulse observability guarantees the capacity to estimate the state of static part (governed by algebraic equations).

FAULT DETECTION AND ISOLATION FOR LPV DESCRIPTOR SYSTEMS VIA PI OBSERVER 5

POLYTOPIC UNKNOWN INPUT PROPORTIONAL INTEGRAL OBSERVER DESIGN

Polytopic UIPIO of LPV descriptor system ( 6) is described by the following equations:

           ż(t) = h i=1 ρ i (θ(t))(N i z(t) + G i u(t) + L i y(t) + T i d(t)) x(t) = z(t) + H 2 y(t) ḋ(t) = h i=1 ρ i (θ(t))Φ i (y(t) -ŷ(t)) (9)
where x(t) ∈ R n , z(t) ∈ R n and d(t) ∈ R q are the estimated state vector, transformed state vector and estimated unknown inputs vector respectively. N i , G i , L i , T i , H 2 and Φ i are unknown matrices of appropriate dimensions. The unknown inputs can be regarded as signals to be estimated. This polytopic UIPIO ( 9) aggregates all local observers defined at each vertex.

Definition 1

System ( 9) is called a polytopic UIPIO for system (6) if for arbitrary initial conditions x(0) and z(0) and arbitrary input u(t), the following relations hold:

lim t→∞ (x(t) -x(t)) = 0 ( 10
) lim t→∞ (d(t) -d(t)) = 0, ∀ d(0) (11)
The state estimation error is equal to: e(t) = x(t) -x(t); then it follows from ( 6) and ( 9) that

e(t) = x(t) -z(t) -H 2 Cx(t) e(t) = (I n -H 2 C)x(t) -z(t)
Assuming there exists a matrix H 1 ∈ R n×n such that:

H 1 E = I n -H 2 C (12)
which is equivalent to

H 1 H 2 E C = I n (13)
Equation ( 13) has a solution if rank E C = n. By using [START_REF] Koenig | Design of Proportional-Integral Observer for Unknown Input Descriptor Systems[END_REF], the state estimation error becomes:

e(t) = H 1 Ex(t) -z(t) (14)
Now, we assume that the bounded unknown inputs are defined with slow variation, i.e ḋ(t) ≃ 0.

Then, for ζ(t) = d(t) -d(t) the derivative of the unknown input is

ζ(t) = -ḋ(t) (15)
By taking into account ( 6), ( 9) and ( 14), the dynamics of the estimation error is given by the following equation: If the following relations hold true ∀i = 1, . . . , h:

ė(t) = H 1 E ẋ(t) -ż(t) (16)
ė(t) = h i=1 ρ i (θ(t))(N i e(t) + (H 1 A i -L i C -N i H 1 E)x(t) + (H 1 B i -G i )u(t) + T i ζ(t) + (H 1 R i -T i )d(t)) (17)
H 1 A i -N i H 1 E -L i C = 0 ( 18
) G i = H 1 B i (19) H 1 E = I n -H 2 C ( 20
) T i = H 1 R i (21)
and from ( 9), ( 15) and ( 17), the dynamics of the state estimation and the unknown input errors become:

ė(t) = h i=1 ρ i (θ(t))(N i e(t) + T i ζ(t)) (22)
ζ(t) = - h i=1 ρ i (θ(t))(Φ i C(x(t) -x(t))) (23)
ζ(t) = - h i=1 ρ i (θ(t))(Φ i C)e(t) (24)
The above errors dynamics can be represented by connecting [START_REF] Wang | Robust Fuzzy Control of a Class of Nonlinear Descriptor Systems with Time-Varying Delay[END_REF] to [START_REF] Taniguchi | Fuzzy Descriptor Systems and Nonlinear Model Following Control[END_REF] as follows:

ė(t) ζ(t) = h i=1 ρ i (θ(t)) N i T i -Φ i C 0 e(t) ζ(t) (25)
If the above augmented error dynamic equation is stable, the state estimation will converge asymptotically to the real state.

UIPIO Matrices Computing

In order to find matrix H 1 H 2 , it follows that equation ( 13) has a solution since rank E C = n, and the solution is given by

H 1 H 2 = E C + (26)
By substituting [START_REF] Rodrigues | Fault Tolerant Control Design for Polytopic LPV Systems[END_REF] into [START_REF] Orjuela | State estimation for nonlinear systems using decoupled multiple model[END_REF], we obtain ∀i = 1, . . . , h:

H 1 A i = N i (I n -H 2 C) + L i C (27)
Equation ( 27) can be equivalently written as:

N i = H 1 A i -K i C (28)
with

K i = L i -N i H 2 (29)
and matrices L i can be deduced from equation (29) as follows:

L i = K i + N i H 2 (30)
To compute the gains K i and Φ i , the estimated errors equation ( 25) can be written as:

ė(t) ζ(t) = h i=1 ρ i (θ(t)) Āi -Ki C e(t) ζ(t) (31)
FAULT DETECTION AND ISOLATION FOR LPV DESCRIPTOR SYSTEMS VIA PI OBSERVER 7 or equivalently such that:

ėa (t) = h i=1 ρ i (θ(t)) Āi -Ki C e a (t) = A(t)e a (t) (32)
where:

Āi = H 1 A i T i 0 0 , Ki = K i Φ i , C = C 0 , e a (t) = e(t) ζ(t)
and

T i = H 1 R i .
Note that this estimation error (32) is a continuous function which depends on the time-varying parameter vector ρ(t) ∈ Ω.

Obviously, the stability of (32) is ensured if and only if the pair ( Āi , C) is detectable ∀i = 1, . . . , h which is equivalent to:

rank   sE -H 1 A i -H 1 R i 0 sI q C 0   = n + q, ∀ s ∈ C + . ( 33
)
Equation ( 33) is the generalization of R-detectability [START_REF] Koenig | Design of Proportional-Integral Observer for Unknown Input Descriptor Systems[END_REF] for the polytopic descriptor system.

Stability and convergence conditions

The exponential convergence of the estimation error (α-stability of the observer) is a way to ensure a convergence velocity of the estimation error via a decay rate and to improve dynamic performances of the observer [START_REF] Orjuela | State estimation for nonlinear systems using decoupled multiple model[END_REF]. The following theorem provides sufficient conditions for ensuring the exponential convergence of the estimation error (32).

Theorem 1 Let consider the system (6) and the polytopic UIPIO [START_REF] Grenaille | A method for designing FDI filters for polytopic LPV models[END_REF]. The exponential convergence of the estimation error (32) is guaranteed if there exists a symmetric and positive definite matrix Q, matrices W i = Q Ki and a positive scalar α for all ρ ∈ Ω such that:

ĀT i Q + Q Āi -CT W T i -W i C + 2αQ < 0, ∀i ∈ [1, . . . , h] (34)
Proof 1

The proof of this theorem is performed by using the following quadratic Lyapunov function:

V (t) = e T a (t)Qe a (t), Q = Q T > 0 (35)
The exponential convergence of the estimation error is guaranteed if:

∃Q = Q T > 0 : V (t) + 2αV (t) < 0 ( 36
)
where α is called the decay rate. Indeed, the solution of the Equation ( 36) is given by:

V (t) ≤ V (0)exp(-2αt), ∀t ≥ 0 (37) Due to λ min (Q) e a (t) 2 ≤ V (t) ≤ λ max (Q) e a (t) 2
, the norm of the estimation error can be bounded by:

e a (t) ≤ λ max (Q) λ min (Q) exp(-αt) e a (0) , ∀t ≥ 0 ( 38
)
The derivative of (35) with respect to time yields:

V (t) = ėT a Qe a + e T a (t)Q ėa (t) (39) 8
H. HAMDI, M. RODRIGUES, C. MECHMECHE, D. THEILLIOL AND N. BENHADJ BRAIEK that becomes by using equation (32):

V (t) = e a (t) T {A(t) T Q + QA(t)}e a (t) (40)
By using ( 35) and (40), the inequality (36) becomes:

e a (t) T {A(t) T Q + QA(t) + 2αQ}e a (t) (41)
that is a quadratic form in e a (t). Therefore, the above inequality is satisfied if the following inequality holds:

A(t) T Q + QA(t) + 2αQ < 0 (42)
that is a sufficient condition for ensuring the exponential convergence of the estimation error e a (t).

By considering the fact that Q is positive definite matrix and that A(t) is a continuous function of parameter ρ ∈ Ω, and Ω is compact: it is clear that if there exists a positive definite matrix Q for all ρ ∈ Ω it implies that the left hand side of equation ( 42) is uniformly negative definite [START_REF] Wu | Control of Parameter Varying Systems PhD Thesis[END_REF].

Then, from equation (32), let replace A(t) by

h i=1 ρ i (θ(t)) Āi -Ki C = h i=1 ρ i (θ(t))Ψ i .
Hence, the inequality (42) becomes for all ρ i (θ(t)) ∈ Ω:

h i=1 ρ i (θ(t))Ψ T i Q + Q h i=1 ρ i (θ(t))Ψ i + 2αQ < 0 (43)
For any ρ i (θ(t)) ∈ Ω and due to the properties of this compact set [START_REF] Mason | On stability analysis of lilnear discrete-time switched systems using quadratic Lyapunov Functions[END_REF], the previous inequality (43) becomes

h i=1 ρ i (θ(t))Ψ T i Q + Q h i=1 ρ i (θ(t))Ψ i + h i=1 ρ i (θ(t))2αQ < 0 (44)
The inequality (44) holds if the following inequalities are satisfied over all ρ i (θ(t)) ∈ Ω:

Ψ T i Q + QΨ i + 2αQ < 0, ∀i ∈ [1, . . . , h] (45)
or equivalently by substituting Ψ i :

ĀT i Q + Q Āi -Q Ki C -( Ki C) T Q + 2αQ < 0, ∀i ∈ [1, . . . , h] (46)
Let us notice that the above inequalities are bilinear in Ki and Q. Therefore, it is not possible to solve them directly using standard convex optimization algorithms. However, the following change of variable: W i = Q Ki can be useful in order to linearize (46) which becomes:

ĀT i Q + Q Āi -CT W T i -W i C + 2αQ < 0, ∀i ∈ [1, . . . , h] (47)
that are linear inequalities in Q and W i . Now, a solution can be found using classical LMI tools and the gains are computed with

Ki = Q -1 W i , ∀i ∈ [1, . . . , h].
Remark : Exponential convergence of estimation error (32) is a strong form of convergence; it implies asymptotic convergence [START_REF] Wu | Control of Parameter Varying Systems PhD Thesis[END_REF]. Indeed, the asymptotic convergence of the estimation error is obtained by considering a decay rate equal to zero (α = 0).

In the following, the proposed observer is used for fault detection, isolation and estimation of actuator fault with disturbance rejection.

Fault diagnosis is usually performed to accomplish one or more of the following tasks: fault detection (or monitoring), to indicate the fault occurrence; fault isolation, to determine the location of the fault and fault identification, to estimate the fault magnitude. However, the relative importance of these three tasks are obviously subjective even if fault detection and isolation are an absolute must for any practical system. Fault identification (estimation), which is certainly helpful, may not be essential if no reconfiguration action is involved. Hence, fault diagnosis is very often considered as fault detection and isolation. In this section, the polytopic UIPIO are specifically designed for fault detection, isolation and estimation. In the framework of robust fault detection, we are looking forward to generate residual signals that indicate if a fault occurs or not.

Residual generation via polytopic UIPIO

A residual generation algorithm using a polytopic UIPIO is considered in order to be sensitive to an actuator fault vector f ∈ R f and insensitive to the disturbance vector d(t) [START_REF] Theilliol | Fault diagnosis and accommodation of three-tank system based on analytical redundancy[END_REF]. Consider the following polytopic descriptor system with a constant control matrix B i , i.e. B i = B and a constant unknown input matrix

R i = R:    E ẋ(t) = h i=1 ρ i (θ(t))(A i x(t) + Bu(t) + Rd(t) + F f (t)) y(t) = Cx(t) (48)
For residual generation purposes of (48), the polytopic UIPIO has the following form:

               ż(t) = h i=1 ρ i (θ(t))(N i z(t) + G i u(t) + L i y(t) + T f (t)) x(t) = z(t) + H 2 y(t) r(t) = y(t) -ŷ(t) ḟ (t) = h i=1 ρ i (θ(t))Φ i (y(t) -ŷ(t)) (49)
where r(t) is the so-called residual signal generated by the comparison between measured and estimated outputs. The diverse matrices in (49) will be determined to ensure the convergence of the estimated errors. In order to describe the dynamic of polytopic UIPIO (49), the estimation error and the residual equations are defined by: e(t) = x(t) -x(t) and r(t) = Ce(t). It can be shown that:

ė(t) = h i=1 ρ i (θ(t))N i e(t) + (H 1 A i -N i H 1 E -L i C)x(t) + (H 1 B -G)u(t) + H 1 Rd(t) + (H 1 F -T )f (t) + T ζ f (t) (50)
where

ζ f (t) = f (t) -f (t) (51)
The following conditions have to be verified in order to provide robustness to unknown inputs and sensitivity to faults. Then, equations [START_REF] Orjuela | State estimation for nonlinear systems using decoupled multiple model[END_REF] to [START_REF] Rodrigues | Fault Tolerant Control Design for Polytopic LPV Systems[END_REF] and the following equations must be satisfied:

T = H 1 F (52) H 1 R = 0 (53)
Hence, equation (50) becomes: By assuming the fault as a step, we find:

ė(t) = h i=1 ρ i (θ(t))(N i e(t) + T ζ f (t)) (54)
ζf (t) = -ḟ (t) (55)
Thus, from (49) we obtain:

ḟ (t) = - h i=1 ρ i (θ(t))Φ i Ce(t) (56)
Equations ( 54) and ( 56) can be combined in an augmented form as follows:

ė(t) ζf (t) = h i=1 ρ i (θ(t)) N i T -Φ i C 0 e(t) ζ f (t) ( 57
)
The design of the polytopic UIPIO ( 49) is equivalent to solve equations ( 18) to [START_REF] Rodrigues | Fault Tolerant Control Design for Polytopic LPV Systems[END_REF] under constraint (53) and to ensure that the augmented error equation ( 57) is stable. Then, matrices H 1 and H 2 are determined such that constraints [START_REF] Rodrigues | Fault Tolerant Control Design for Polytopic LPV Systems[END_REF] and ( 53) are satisfied. Equations ( 20) and ( 53) are rewritten in an augmented matrix equation as

H 1 H 2 E R C 0 = I n 0 (58)
A solution of (58) exists if [START_REF] Koenig | Unknown Input Proportional Multiple-Integral Observer Design for Linear Descriptor Systems: Application to State and Fault Estimation[END_REF]:

rank E R C 0 = n + rank(R) (59)
Under (59), the general solution of (58) is:

H 1 H 2 = I n 0 E R C 0 + ( 60
)
The remaining matrices G i , N i and L i of this polytopic UIPIO based residual generator are given by ( 19), ( 28) and (30) respectively. The residuals are generated by integrating the estimated output errors and the occurence of faults are detected by its magnitudes. Thus, it is verified that the polytopic UIPIO based residual generator can estimate the state variables and detect faults at the same time.

Actuator Fault Isolation

The fault isolation problem is to determine in which actuator the fault has occurred. The idea is to make each residual sensitive to only one actuator fault and insensitive to all other faults [START_REF] Rodrigues | A Fault Detection and Isolation Scheme for Industrial Systems based on Multiple Operating Models[END_REF].

To design robust actuator fault isolation scheme, only one component of u(t) is considered faulty. The others are assumed to be fault free and considered as unknown inputs. Hence, for each actuator fault, the polytopic LPV descriptor system corrupted by faults and unknown inputs is described by:

   E ẋ(t) = h i=1 ρ i (θ(t)) A i x(t) + Bu(t) + R F r d(t) f d (t) + F j f j (t) y(t) = Cx(t) (61)
where F j is the j th column of B with j ∈ [1, . . . , p], F r is a matrix of the remaining columns of B. f j represents the faults to be isolated and estimated and f d (t) is considered in this FDI scheme as an unknown input. R F r is the new disturbance matrix composed by the original unknown inputs matrix R and the matrix F r . Based on the above system description, p polytopic UIPIO based residual generators can be synthesized for j ∈ [1, . . . , p] such that:

           żj (t) = h i=1 ρ i (θ(t))(N j i z(t) + G j i u(t) + L j i y(t) + T fj (t)) r j (t) = y(t) -ŷ(t) ḟj (t) = h i=1 ρ i (θ(t))Φ j i (y(t) -ŷ(t)) (62)
The observer matrices must satisfy the equations ( 19), ( 28), ( 29) and ( 60) such that the initial matrix R is replaced by R F r . Each residual vector r j (t) produced by the j th polytopic UIPIO, may be used to isolate a fault according to a statistical test [START_REF] Rodrigues | Design of a Robust Polytopic Unknown Input Observer for FDI: Application to Nonlinear Systems described by a Multi-Models Representation[END_REF]. The isolation scheme can only isolate a single fault at the same time. This is based on the fact that the probability for two or more faults to occur at the same time is very small in a real situation.

ILLUSTRATIVE EXAMPLE

Let us consider the following LPV descriptor system defined by

E ẋ(t) = A(θ(t))x(t) + B(θ(t))u(t) + R(θ(t))d(t) y(t) = Cx(t)
where

A(θ) =    -1.75 + θ 2 1 0 0 -1 -1 + θ 1 0 0 -1.8 -1 -0.75 + θ 1 0 -1 0 0 -1 + θ 2    , E =    1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0    , B(θ) =     1 + θ 1 1 1 0.5 + θ 2 1 0 θ 2 0     , R(θ) =     0 0.6 + θ 1 0 1     and C =   1 0 1 0 0 1 0 1 0 0 1 1  
In this example, the polytopic LPV descriptor system is considered where variables θ i vary according to θ 1 ∈ [-0.05, 0.05] and θ 2 ∈ [-0.1, 0.1], l = 2. In this case, the polytopic representation can be written as follows:

   E ẋ(t) = 4 i=1 ρ i (θ(t))(A i x(t) + B i u(t) + R i d(t)) y(t) = Cx(t)
The obtained system evolves in a polytope of 4 vertices corresponding to the extreme values of the parameters θ 1 and θ 2 . The system matrices are given:

A 1 =     -1.85 1 0 0 -1 -1.05 0 0 -1.8 -1 -0.8 0 -1 0 0 -1.1     , A 2 =     -1.65 1 0 0 -1 -1.05 0 0 -1.8 -1 -0.8 0 -1 0 0 -0.9     A 3 =     -1.85 1 0 0 -1 -0.95 0 0 -1.8 -1 -0.7 0 -1 0 0 -1.1     , A 4 =     -1.65 1 0 0 -1 -0.95 0 0 -1.8 -1 -0.7 0 -1 0 0 -0.9     B 1 =     0.95 1 1 0.4 1 0 -0.1 0     , B 2 =     0.95 1 1 0.6 1 0 0.1 0     , B 3 =     1.05 1 1 0.4 1 0 -0.1 0     , B 4 =     1.05 1 1 0.6 0 0 0.1 0    
R 1 = R 2 =     0 0.55 0 1     , R 3 = R 4 =     0 0.65 0 1     and C =   1 0 1 0 0 1 0 1 0 0 1 1  
The weighing functions ρ i (θ(t)) are defined such that:

ρ 1 (θ(t)) = θ 1 -θ 1 θ 1 -θ 1 θ 2 -θ 2 θ 2 -θ 2 = (θ 1 + 0.05)(θ 2 + 0.1) 0.02 , ρ 2 (θ(t)) = θ 1 -θ 1 θ 1 -θ 1 θ 2 -θ 2 θ 2 -θ 2 = (θ 1 + 0.05)(0.1 -θ 2 ) 0.02 ρ 3 (θ(t)) = θ 1 -θ 1 θ 1 -θ 1 θ 2 -θ 2 θ 2 -θ 2 = (0.05 -θ 1 )(θ 2 + 0.1) 0.02 , ρ 4 (θ(t)) = θ 1 -θ 1 θ 1 -θ 1 θ 2 -θ 2 θ 2 -θ 2 = (0.05 -θ 1 )(0.1 -θ 2 ) 0.02

UIPIO Design

The polytopic Unknown Input Proportional Integral Observer is represented by

           ż(t) = 4 i=1 ρ i (θ(t))(N i z(t) + G i u(t) + L i y(t) + T i d(t)) x(t) = z(t) + H 2 y(t) ḋ(t) = 4 i=1 ρ i (θ(t))Φ i (y(t) -ŷ(t))
The design procedure of the polytopic UIPIO is thus given as follows:

(i) Based on the system matrices, assumptions A 2 and A 3 are satisfied.

(ii) Matrices H 1 and H 2 are obtained from equation ( 26).

(iii) By using [START_REF] Rodrigues | Design of a Robust Polytopic Unknown Input Observer for FDI: Application to Nonlinear Systems described by a Multi-Models Representation[END_REF], LMI (47) can be efficiently solved via numerical approach using the LMI toolbox to determine the gains matrices. Therefore, these inequalities are fulfilled with: where K i are the proportional and Φ i are the integral gains matrices. (iv) The remaining polytopic UIPIO matrices G i , N i and L i can be established by ( 19), ( 28) and (29) respectively. The input signals are constant: u 1 (t) = u 2 (t) = 2, the disturbance d(t) is a rectangular signal applied for 7 ≤ t ≤ 10. The initial values of the original states x(t) and the unknown input d(t) are equal to 0 and the initial values of the estimated states are: x(t) = 0.5 1 0.5 -0.5

K 1 =     -1.
T . The state estimation and unknown inputs estimation given by the polytopic UIPIO are shown on the following figures. The behavior of the polytopic UIPIO is shown on figures (1) to (4). It can be observed that the estimated states can closely track the original states of the polytopic descriptor system affected by an unknown input in spite of measurements which are perturbed by a white Gaussian noise with variance 0.01. In addition, as seen on the figure [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory SIAM[END_REF], the polytopic UIPIO provides a correct unknown input estimation in spite of noise and disturbances. The polytopic UIPIO design is easy but the restriction for this polytopic UIPIO is to consider that disturbances have slow variation i.e. ḋ(t) ≃ 0. However, in practice, we can relax this constraint.

Fault diagnosis with polytopic UIPIO

In this paragraph, we will only consider a constant matrix B and two actuator faults with constant magnitudes. Let us consider the model (61) and the polytopic UIPIO (49) with the same previous matrices except for a constant matrix B(θ) :

B(θ) = B = 1 1 1 0 1 0.5 0 0 T and R(θ) = R = 0 0.6 0 1 T .
In this part, we recall that F j is the j th column of B and F r is the matrix B without the j th column.

In order to detect and isolate the actuators faults, two polytopic UIPIO are designed here, each one designed to isolate an actuator fault. Let consider f 1 (t) and f 2 (t), the fault signals which represent control input dysfunctions of u 1 (t) and u 2 (t) respectively and defined by: The first polytopic UIPIO is designed to estimate and isolate the first actuator fault f 1 (t) and a second one is also designed to estimate and isolate the second actuator fault f 2 (t). The performance of the fault diagnosis system is illustrated for two actuator faults as shown below. 6) and ( 7) display the isolation task. One can see that the residual is almost zero throughout the time simulation run for fault-free residuals despite noisy environment and increase in magnitude considerably when actuator faults occur at time t = 10s (Fig.( 6)) and at time t = 15s (Fig. [START_REF] Chen | Robust model-based fault diagnosis for dynamic systems[END_REF]). Then, faults are easily isolated using the information provided by residuals signals. Figures [START_REF] Darouach | Design of Observers for Descriptor Systems[END_REF] and ( 9) display the estimation of f 1 (t) and f 2 (t). An error appears in the estimation of both f 1 (t) and f 2 (t) respectively. This is due to the fact that when those faults stop abruptly, there is a remaining time to give an estimation of f 1 (t) and f 2 (t) owing to the response of this polytopic UIPIO. It is important to note that without noise, the estimation is better but we want to present a more realistic point of view in a noisy environment.

f 1 (t) = 0.

CONCLUSION

This paper has addressed the problem of state estimation, fault detection and isolation for dynamic systems that can be described by polytopic LPV descriptor models subject to disturbances. The state and unknown inputs estimation are obtained by a polytopic Unknown Inputs Proportional Integral Observer which has been used here for the detection, isolation and estimation of actuators faults in spite of disturbances. Stability conditions of this polytopic UIPIO have been formulated and solved within a linear matrix inequality framework. The developed FDI scheme for LPV descriptor systems has been applied into an example in order to estimate both state vector and disturbances. Based on fault estimation, a fault tolerant control scheme for LPV descriptor systems could be synthesized so as to enhance robustness of the system.

4 H

 4 . HAMDI, M. RODRIGUES, C. MECHMECHE, D. THEILLIOL AND N. BENHADJ BRAIEK

6 H

 6 . HAMDI, M. RODRIGUES, C. MECHMECHE, D. THEILLIOL AND N. BENHADJ BRAIEK

10 H

 10 . HAMDI, M. RODRIGUES, C. MECHMECHE, D. THEILLIOL AND N. BENHADJ BRAIEK

12 H

 12 . HAMDI, M. RODRIGUES, C. MECHMECHE, D. THEILLIOL AND N. BENHADJ BRAIEK

Figure 1 .Figure 2 .Figure 3 .

 123 Figure 1. Original state x 1 (t) and its estimated x1 (t)

Figure 4 .

 4 Figure 4. Original state x 4 (t) and its estimated x4 (t)

Figure 5 .

 5 Figure 5. Unknown input d(t) and its estimated d(t)

Figure 6 .

 6 Figure 6. the residual signal r 1 (t)

Figure 7 .Figure 8 .Figure 9 .

 789 Figure 7. the residual signal r 2 (t)

Figures (

  Figures[START_REF] Bruzelius | Linear Parameter-Varying Systems an approach to gain scheduling[END_REF] and[START_REF] Chen | Robust model-based fault diagnosis for dynamic systems[END_REF] display the isolation task. One can see that the residual is almost zero throughout the time simulation run for fault-free residuals despite noisy environment and increase in magnitude considerably when actuator faults occur at time t = 10s (Fig.(6)) and at time t = 15s (Fig.[START_REF] Chen | Robust model-based fault diagnosis for dynamic systems[END_REF]). Then, faults are easily isolated using the information provided by residuals signals.Figures[START_REF] Darouach | Design of Observers for Descriptor Systems[END_REF] and (9) display the estimation of f 1 (t) and f 2 (t). An error appears in the estimation of both f 1 (t) and f 2 (t) respectively. This is due to the fact that when those faults stop abruptly, there is a remaining time to give an estimation of f 1 (t) and f 2 (t) owing to the response of this polytopic UIPIO. It is important to note that without noise, the estimation is better but we want to present a more realistic point of view in a noisy environment.

Copyright c 2011 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. (2011) Prepared using acsauth.cls DOI: 10.1002/acs

Copyright c 2011 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. (2011) Prepared using acsauth.cls DOI: 10.1002/acs FAULT DETECTION AND ISOLATION FOR LPV DESCRIPTOR SYSTEMS VIA PI OBSERVER


Iuniana Oprescu

Mickaël Meulle
email: michael.meulle@orange-ftgroup.com

Philippe Owezarski

dVirt: a Virtualized Infrastructure for Experimenting BGP Routing

Keywords: simulation, virtualization, routing, BGP

In this paper we propose dVirt, a distributed virtualized infrastructure enabling network operators to better grasp the routing mechanisms and anticipate network behavior. dVirt offers an emulated layer 2 connectivity that allows accurate simulation of IP routing and protocol dynamics. Using virtualization and Quagga routing software, we achieve scalable simulations while keeping a realistic model of the routing layer.

To illustrate the benefits of simulating networks with dVirt, we study a well-known simple Border Gateway Protocol (BGP) topology subject to routing oscillations. dVirt allows for quick deployment of this topology and easy monitoring of BGP message exchanges.

I. INTRODUCTION

The Internet is a collection of more than 35000 networks called Autonomous Systems (ASes), each AS being under the authority of a specific administrative entity. The Border Gateway Protocol (BGP) is the de facto standard used for communications between the ASes, whereas the routing within a given AS is handled by an Interior Gateway Protocol (IGP).

Initially designed for exchanging reachability information among several research networks, BGP plays a crucial role in the current Internet: it handles inter-domain routing for protocols that support a wide range of aplications with different traffic requirements. Facing an increasing number of Internet users, however, BGP needs to adapt and keep up with demands regarding complex architectures and new features [START_REF] Schrieck | BGP Add-Paths: The scaling/performance tradeoffs[END_REF].

Transitions to new routing conditions can be complex and unexpected issues may arise. Despite the plethora of software tools available to model and experiment BGP configurations, there is no dedicated tool offering an automated testbed that allows for accurate simulations and interactions with the underlying protocol layers. dVirt federates multiple functionalities into a flexible tool enabling the user to automatically deploy and evaluate a network.

The ultimate goal for dVirt is to be able to "clone" a full ISP network on top of a virtualized infrastructure running on a smaller number of servers. dVirt aims to reproduce the actual events in a network, experiment with the real configurations and addressing schemes. The proposed framework can be used to check correctness (e.g., avoid oscillations), compare convergence time for different setups or even implement and test additional features on top of the existing protocol stack.

dVirt relies on virtualization techniques and routing tools: we use the Xen [START_REF]The Xen Project[END_REF] hypervisor and virtual machines to represent a large number of network equipments and we put to work the Quagga [START_REF] Ishiguro | Quagga Software Routing Suite[END_REF] software routing suite for simulating the multiple routing instances. We take advantage of sbgp [START_REF] Labovitz | MRT: Merit's Multi-Threaded Routing Toolkit[END_REF], a simple BGP4 speaker and listener, to mimic the behavior of neighboring ASes by injecting external route advertisements into the edge routers of the simulated network.

II. RELATED WORK

Previous inquiry methods fall back into three main categories: real testbeds, simulation and emulation. A network testbed consists of the actual equipment being deployed, configured and dedicated to conducting experiments. Despite the accurate results provided, a real testbed is ill-suited for large scale tests and has other inconvenients such as the cost of the equipment, the time necessary for setting up the testbed and tearing it down after the experiment.

Initiatives such as Cloonix [START_REF] Cloonix | [END_REF], Virconel [START_REF] Benchaïb | VIRCONEL: A new emulation environment for experiments with networked IT systems[END_REF], NetKit [START_REF]Netkit[END_REF], AutoNetKit [START_REF] Knight | How to build complex, large-scale emulated networks[END_REF], Virtual Network User Mode Linux (VNUML) [START_REF] Galán | Scenario-based virtual network infrastructure management in research and educational testbeds with VNUML[END_REF], Virtual Square with its Virtual Distributed Ethernet [START_REF] Bigliardi | Design e implementazione del nuovo framework Virtual Distributed Ethernet: analisi delle prestazioni e validazione sulla precedente architettura[END_REF] framework or the Xen-based virtual integrated network emulator viNEX [START_REF] Mukwevho | A Virtual Integrated Network Emulator on Xen (viNEX)[END_REF] provide tools and general frameworks for simulation or emulation purposes. The objectives of these networking environments are diverse and often target specific aspects of networking, whereas dVirt realistically emulates the entire protocol stack starting with layer 2.

In the category of training platforms, Dynamips [START_REF] Fillot | Dynamips[END_REF] and its front-end Dynagen [START_REF] Anuzelli | Dynagen[END_REF] offer an emulation of specific Cisco equipment, running IOS [START_REF]Cisco IOS[END_REF] software on commodity hardware. Several instances of virtual routers make it possible to build a small network and run educational experiments.

A different environment consists of emulated networks located on geographically distributed nodes. The shared testbeds offer physical network equipment on which researchers can conduct experiments in a granted slot. A drawback of using such a testbed is the lack of control in the environment which leads to unpredictable and unrepeatable experiments.

Among the many BGP tools available (see for example [START_REF]The Border Gateway Protocol Advanced Internet Routing Resources[END_REF]), the C-BGP solver [START_REF] Quoitin | Modeling the routing of an autonomous system with c-bgp[END_REF] shows some limitations due to the complexity of the model. C-BGP takes into account the logic of the BGP decision process and can compute the outcome of the path selection mechanism for networks of several routers. It is possible to build a model of a large network and investigate changes in the routing or in the topology of such networks. The tool's purpose does not include handling the TCP connections, packet exchanges between the peers or BGP timers such as the MRAI.

Developed by J. Qiu as a lighweight event-driven simulator, simBGP [START_REF] Qiu | SimBGP : Python Event-driven BGP simulator[END_REF] is able to perform large scale simulations while ignoring the underlying protocols such as TCP or IP. Although protocol dynamics can be investigated, simBGP does not offer the full environment for an accurate study of the whole communication mechanism.

III. DVIRT: A BGP SIMULATION FRAMEWORK

dVirt is a software library written in Python for automatically deploying a given BGP network. It can be controlled from a single machine through simple and flexible inputs that avoid the individual provisioning of resources and configuration of routing protocols. The typical use of dVirt is to simulate the entire topology of one or more large ISP networks and incorporate realistic configurations.

A. dVirt Overview

Compared to previous tools, dVirt is a heavier simulator but removes many barriers thanks to its full customization. dVirt relies on open-source software and runs real operating systems, it supports many real network conditions such as addressing, multitasking of the routing processes and interprotocol interactions.

dVirt creates an Ethernet topology of virtual machines (VMs) running on top of hypervisors that are mutually reachable at the IP layer. dVirt emulates virtual point-to-point Ethernet connectivity between pairs of router interfaces using virtual switches provided by the Open vSwitch software. Open vSwitch enables virtual Ethernet connectivity between two routers located either on a single or two distinct hypervisors by encapsulating Ethernet traffic inside GRE tunnels.

The OSPF and BGP topologies are automatically configured to enable full reachability inside each AS and setup (monohop) eBGP sessions. Routers are running the Quagga routing software with OSPF and BGP daemons to simulate the demanded network. External neighbors of the deployed topology are emulated using sbgp software instances running in one or more additional VMs.

dVirt emulates the full protocol in each router and allows the study of the BGP protocol dynamics by directly running Quagga with all the implemented features. The tool can also be used to deploy modified versions of the Quagga software and therefore handles many routing protocol testing scenarios. dVirt simplifies the instrumentation of experiments conducted according to a simulation scenario. The user can directly use python bindings to execute existing or user-defined functions. Network monitoring functions run on routers and provide information about the state of the BGP routers, allowing thus to obtain exact measurement data. To allow permanent communication between the user and the routers, dVirt separates the infrastructure in two distinct networks: a management network for remote access and a test network for the actual simulation. Each VM has a local IP address configured on the interface attached to the management bridge defined on each hypervisor as seen in Fig. 1.

B. dVirt Management Network

To interact with the remote routers, dVirt provides two libraries to exchange files and do remote calls: SSH and RPC. The SSH library allows to exchange files and send commands to the VMs over an ssh connection with text output.

The RPC (remote procedure call) library enables the creation of a TCP tunnel between the user and any hypervisor in order to execute requests directly on the hypervisor with a simple function call. The output is a python object that is serialized and sent back to the user over the TCP session. The dVirt RPC library has the particular ability to allow transparent execution of RPC requests from the user to a VM or router. The RPC resorts to the hypervisor as an intermediate point that forwards the request in an embedded call directed to the virtual router, as shown in Fig. 2.

dVirt comes with a set of pre-defined functions for the RPC server-side for interaction with hypervisors, VMs, routers and their installed software. The user can easily improve the existing library by adding new functions to the python files in the library. During the next deployment, dVirt will automatically update the RPC library of each hypervisor and each VM making available the new user-defined functions.

C. Virtual Routers

dVirt relies on the Xen open-source software to achieve the virtualization of x86 CPU architectures. The Xen hypervisor allows one physical machine to run multiple router instances Fig. 2. The user launches a request that can be executed either by the hypervisor or the virtual machines. Note that for the calls on VMs, an embedded request is forwarded by the hypervisor to the corresponding VM. by acting as an abstraction layer to the bare hardware and isolating the virtual machines from the external networks.

Xen handles the concurrent access of the VMs to the resources and manages the execution of the guest OSes. In Xen terminology, Dom0 is the first operating system that boots automatically and receives privileged rights regarding hardware access and management. From the Dom0, the administrator can launch new virtual machines, called DomU s and manage all the existing guest machines. dVirt uses virsh management interfaces of the libvirt [START_REF]Libvirt -The virtualization API[END_REF] API to create machines from a customizable xml file where memory and CPU allocation can be changed for any router.

By default, each VM in dVirt is a router or it hosts sbgp software instances to simulate external BGP neighbors. Each VM has a dedicated SWAP filesystem, a CPU, a dedicated memory of 512 MB, and a generic pre-installation of the Linux Debian Lenny operating system (distribution 2.6.26-2-xen-686) customized with the required software such as Quagga or Python.

D. Virtual Ethernet

In the Dom0 of each hypervisor, dVirt configures the management network through virsh and uses Open vSwitch to emulate point-to-point links in the experimental network. Quagga runs as a regular application on each virtual machine and takes over the kernel routing of the virtual machine. As seen in Fig. 3, for a pair of source-destination routers, two scenarios are possible: if the routers run on the same hypervisor, they interconnect through a dedicated virtual switch (e.g., R2 and R3 linked with grebr3 on hypervisor B); otherwise the two ends of the link are on distinct hypervisors and dVirt needs to define two Open vSwitches, one for the test interface of each router. The traffic between the routers is then transparently forwarded inside a GRE tunnel (e.g., R1 and R2 connect respectively to grebr1 on hypervisor A and B).

A GRE tunnel is setup between two hypervisors only if two distant routers share a point-to-point link. Multiple links between the same two hypervisors can take the same tunnel since isolation is guaranteed by Open vSwitch.

E. Simulated BGP Network

OSPF handles routing within the AS, whereas BGP interconnects different ASes through external BGP (eBGP) sessions. OSPF achieves full intra-AS reachability and external neighbors are directly connected on the specified interfaces, later redistributed inside the OSPF network.

For inter-domain routing, Quagga implements many BGP features, going from different types of BGP sessions (iBGP or eBGP, route reflector or route-server) to ACLs, filtering, prefix aggregation, etc. During the test phase, the routers are fully capable of forwarding traffic, performing the BGP best path decision process as well as receiving or sending OSPF and BGP protocol messages.

F. dVirt Typical Usage

dVirt requires privileged access to a set of hypervisors running Xen with pre-installed software (an SSH server, Python interpreter, Open vSwitch, Quagga and libvirt).

To automatically deploy the testbed, dVirt needs as an input a topology file describing the actual network. Table I illustrates the elements for defining the simulation: routers, links, BGP sessions. dVirt uses the specified attributes to instantiate a VM for the router RR3, with all the required configuration parameters (distinct management and test addresses, BGP loopback and AS number, etc.). A link between the routers RR3 and Rc is another entry in the topology file, just as the different types of BGP sessions with the desired options.

Once a network has been deployed with dVirt, the user can perform specific tasks by running customized code: simulate network events such as incoming routes, link failures, etc. It is possible to run any software application or traffic generator on any of the existing virtual routers or in additional virtual machines. Opposed to most of the existing BGP simulators or emulators, dVirt does not restrict the set of potential experiments on top of the deployed BGP network.

By default, dVirt simulates external BGP neighbors of routers with the sbgp software. One or more dedicated virtual machines can host sbgp software instances, where each instance emulates one external BGP neighbor. Sbgp can inject BGP routes from a customizable mrt file but dVirt also includes functions to randomly generate routes.

Another feature of dVirt is that it enables different monitoring strategies (tap the traffic over network interfaces, query the Quagga routing daemon periodically or call functions through the command line) to collect protocol and router behavior data.

IV. EXAMPLE

We setup an example BGP topology to illustrate the dVirt framework. Fig. 4 depicts a classical example of a "no solution" topology that permanently oscillates [START_REF] Griffin | On the correctness of iBGP configuration[END_REF].

Consider a route to a given prefix ρ advertised by Rx, Ry and Rz. The border routers Ra, Rb and Rc will prefer their direct eBGP path. Due to the specific topology and the IGP metrics on the links, the route reflectors RR1, RR2 and RR3 will never reach an agreement about the best path to ρ. To setup the routers in the topology, the user provides the input file containing the description of the BGP network: routers, links and sessions.

Once all the VMs are instantiated and the Quagga routing software is configured, the simulation can start with the injection of prefixes into the border routers Ra, Rb and Rc using three sbgp instances simulating Rx, Ry and Rz.

The routing oscillation from Fig. 4 can be observed with dVirt. Indeed, each of the RRs has one client, but the IGP configuration makes RR1 prefer RR2's client, Rb; RR2 prefers RR3's client, Rc; RR3 prefers the client of RR1, Ra due to lower metrics. Initially, all RRs know the route advertised by their own client and they advertise it to their peers. But as soon as they each receive the routes from their peers, they select as best path the one advertised by the peer and hence they each withdraw their own path. Simultaneously, the neighbor withdraws in its turn the path it had advertised, so the current best path becomes unavailable. Every RR switches back to its own client route and the situation continues indefinitely. We analyze the messages exchanged between the routers and find that RR3 keeps updating and withdrawing its advertised routes as seen in Fig. 5.

V. CONCLUSION

In this work we present dVirt, a distributed virtualized testbed for experimenting network scenarios related to BGP routing. We implement features allowing for automated deployment of a BGP topology over a virtual Ethernet. Each BGP router is emulated by routing software in a dedicated virtual machine and links are emulated through virtual switches.

To the best of our knowledge, dVirt is the first BGP simulator able to keep an accurate model of the protocol stack. It can also reproduce routing dynamics, replay network traces in realistic conditions and take into account protocol interactions between OSPF and BGP as illustrated in the example of an oscillating topology.

Fig. 1 .

 1 Fig. 1. An overview of the dVirt components: the user interacts through the RPC with the remote hypervisors and the corresponding virtual machines.

Fig. 3 .

 3 Fig. 3. An overview of the communication in point-to-point mode: the bridges that have the source-destination pair on distinct hypervisors will be encapsulated in the GRE tunnel between the hypervisors.

Fig. 4 .

 4 Fig. 4. Topology of experimentation.

Fig. 5 .

 5 Fig. 5. Route oscillation as seen on RR3: the messages sent by RR3 to its peers, RR1 and RR2.


C M Astorga-Zaragoza

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

J C Ponsart

M Rodrigues

Fault diagnosis for a class of Descriptor Linear Parameter Varying systems

Keywords: fault estimation, observers, LPV systems, descriptor systems

In this paper, a model-based fault estimation method for a particular class of discrete-time Descriptor Linear Parameter Varying (D-LPV) systems is developed. The main contribution of this work consists in the design of an observer that performs simultaneously both, the states estimation and the fault magnitude vectors, considered as unknown inputs. The conditions for the existence of such observer are given. Such conditions guarantee the observer stability and they are proved through a Lyapunov analysis combined with a Linear Matrix Inequalities (LMI) formulation. The fault estimation scheme is evaluated through numerical simulations.

INTRODUCTION

Descriptor systems (also known as singular systems) are a special class of systems that include ordinary algebraic equations, (differential) state-space equations, combinations of algebraic and differential equations and noncausal systems [START_REF] Luenberger | Dynamic equations in descriptor form[END_REF]. From the point of view of process modeling, descriptor systems can be considered as a generalization of state-space linear systems where algebraic relations resulting from the interaction of the process dynamics exist. Several works dealing with linear and nonlinear descriptor systems exist in literature (see for instance [START_REF] Dai | Singular control systems[END_REF] for linear systems and [START_REF] Müller | Stability and optimal control of nonlinear descriptor systems: a survey[END_REF] for the nonlinear case). Some recent works dealing with descriptor systems are [START_REF] Sun | Optimal and self-tuning fusion Kalman filters for discrete-time stochastic singular systems[END_REF][START_REF] Ishihara | H∞ filtering for regular discrete-time descrptor systems[END_REF][START_REF] Darouach | H∞ unbiased filtering for linear descrptor systems via LMI[END_REF] where different observer approaches are used for this kind of systems. Different fault diagnosis techniques for descriptor systems are recently synthesized in [START_REF] Frisk | Sensor placement for fault isolation in linear differential-algebraic systems[END_REF][START_REF] Hu | Fault detection and diagnosis for singular stochastic systems via B-spline expansions[END_REF].

On the other hand, Linear Parameter Varying (LPV) systems (which can be considered as a particular case of Linear Time Varying systems LTV) can be used to approximate nonlinear systems, hence systematic and generic available theoretical results for LPV systems can be applied to derive nonlinear control laws for nonlinear systems. For instance, in [START_REF] Wei | Gain scheduled H∞ control for air path systems of diesel engines using LPV techinques[END_REF] the authors use the LPV approach to model and control diesel engines. In [START_REF] Grenaille | A method for designing fault diagnosis filters for LPV polytopic systems[END_REF] a method for designing structured fault diagnosis filters (with guaranteed H ∞ nuisance attenuation and fault transmission H -gain) for polytopic LPV systems is developed and simulated for a secondary circuit of a nuclear power plant. An application of fault detection and isolation to a Boeing 747-100/200 aircraft is presented in [START_REF] Ganguli | An application of H∞ fault detection and isolation to a transport aircraft[END_REF].

In recent years, fault detection and isolation for LPV systems has been investigated for several authors. The case of robust fault detection and isolation of multiple simultaneous faults for affine LPV systems affected by disturbances is presented in [START_REF] Armeni | Robust fault detection and isolation for LPV systems under a sensitivity constraint[END_REF], where the authors design a residual generator filter that maximizes the DC-transmission gain from a specific fault to the residual while attenuating the effect caused by the nuisances. The authors in [START_REF] Abdalla | Fault detection and isolation filter design for linear parameter varying systems[END_REF] present an H ∞ approach for fault detection and isolation (and estimation), where a linear fractional transformation (LFT) representation of the LPV system is adopted. In [START_REF] Balas | Failure detection for LPV systems -a geometric approach[END_REF][START_REF] Bokor | Detection filter design for LPV systems -a geometric approach[END_REF], failure detection for LPV systems is investigated by using a geometric approach, where a procedure was derived to obtain the detection filter gain via the construction of a suitable family of invariant subspaces. Parameter-dependent observers with pole assignment are used to synthesize the residual generators for LPV systems in [START_REF] Bara | Parameter-dependent state observer design for affine LPV systems[END_REF][START_REF] Millerioux | Bounded state reconstruction error for LPV systems with estimated parameters[END_REF][START_REF] Casavola | A fault detection, filter-design method for linear parametervarying systems[END_REF]. A robust state-based approach to estimate faults in uncertain LPV systems can be found in [START_REF] Henry | Robust fault diagnosis in uncertain linear parameter-varying sistems[END_REF].

Although the idea of merging descriptor and LPV systems is not new (see for instance [START_REF] Masubuchi | Synthesis of output feedback gain-scheduling controllers based on descriptor LPV system representation[END_REF]), there are very few works dealing with observer and controller synthesis, stabilization, fault diagnosis and other topics on automatic control for this kind of systems. Some recent papers on this topic dealing with the problem of control and stabilization are [START_REF] Chadli | Static output stabilisation of singular LPV systems: LMI formulation[END_REF][START_REF] Masubuchi | Gain-scheduled controller synthesis based on new LMIs for dissipativity of descriptor LPV systems[END_REF], whereas the authors in [START_REF] Bouali | H 2 gain scheduled observer based controllers for rational LPV systems[END_REF] addresses the problem of observer-based controllers.

The aim of this work is to develop a fault estimation method for discrete-time descriptor linear parameter varying systems (in the sequel referred as D-LPV systems), described by a polytopic system representation. In order to achieve this objective, an observer for D-LPV systems is synthesized. The observer is based on the observer proposed in [START_REF] Darouach | Design of observers for descriptor systems[END_REF] where a method to design full-order observers for LTI descriptor systems is considered. The existence conditions of the observer are given. Such conditions guarantee the observer convergence proved through a combined method based on the original approach proposed by [START_REF] Darouach | Design of observers for descriptor systems[END_REF] and a Lyapunov analysis.

By taking into account some simple matrix manipulations, this observer can estimate both, the states and the unknown inputs, simultaneously. In order to achieve a robust fault detection and identification, the unknown input estimation feature is used to build a fault diagnosis scheme based on a bank of observers. The effectiveness and performance of the proposed scheme are illustrated through a numerical example.

PROBLEM FORMULATION

Consider the following discrete-time polytopic D-LPV system where the terms k and k + 1 are the simplified form to write kT s and kT s + T s respectively, and T s is the sampling time:

Ē x(k + 1) = M i=1 ε i (ρ(k)) Āi x(k) + Bi u(k) + F v(k) y(k) = C x(k) (1) where x(k) ∈ R n , u(k) ∈ R q , y(k) ∈ R p , v(k) ∈ R d are
the state, the measured input, the measured output and the unmeasured input vectors which can be classically considered as fault vectors, respectively, Ē ∈ R m×n , Āi ∈ R m×n , Bi ∈ R n×q , C ∈ R p×n , F ∈ R n×d , are constant matrices, ρ(k) is a time-varying bounded parameter vector assumed to be measured on-line and M is the total number of weighting functions ε i (ρ(k)) defined as

M i=1 ε i (ρ(k)) = 1, ε i (ρ(k)) ≥ 0 (2)
Classicaly, the term polytopic comes from the fact that the vector ε(ρ(k)) evolves over the convex set:

Ω = col i : M i=1 ε i (ρ(k)) = 1, ε i (ρ(k)) ≥ 0
The following assumptions are considered ( [START_REF] Darouach | Design of observers for descriptor systems[END_REF]):

(A1) rank Ē = r < n. (A2) rank Ē C = n.
Assumption A1 simply states that Ē is a singular matrix. Assumption A2, guarantees the existence of a nonsingular matrix Γ:

Γ = ᾱ β γ ξ such that ᾱ Ē + β C = I n (3) γ Ē + ξ C = 0 (4)
where ᾱ, β, γ and ξ are constant matrices of appropriate dimensions which can be found by the singular value decomposition of Ē C .

The problem is to synthesize an observer for system (1) such that the estimation error ẽ(k) = x(k) -x(k) converges to zero as k → ∞.

In this paper bias are considered constant faults, called bias, are considered, i.e.

(A3) v(k) = v(k -1).
This assumption may be considered as restrictive, however in many practical situations, once a fault appears, it remains constant during a significant time period that is long enough to estimate the fault. In this way, the system (1) can be rewritten in the following form:

Ex(k + 1) = M i=1 ε i (ρ(k)) (A i x(k) + B i u(k)) y(k) = Cx(k) (5)
A i = Āi 0 m×d 0 d×n I d (6)
B i = Bi 0 d×q (7)
C = C 0 p×d (8)
Following the idea presented by Darouach [START_REF] Darouach | H∞ unbiased filtering for linear descrptor systems via LMI[END_REF] for state and unknown input estimation, the expanded state vector is formed by the usual state variables and the faults at the precedent time instant

x(k) = x(k) v(k -1) ∈ R (m+d)×(n+d) (10)
In this way, the faults can be estimated simultaneously with the state variables. The problem becomes to design an adequate observer for the D-LPV system [START_REF] Sun | Optimal and self-tuning fusion Kalman filters for discrete-time stochastic singular systems[END_REF].

Assumption A2 becomes

(A4) rank E C = n + d.
Considering Assumption A4, it exists a nonsingular matrix Γ:

Γ = α β γ ξ such that αE + βC = I n+d (11) γE + ξC = 0 (12)
where α, β, γ and ξ are constant matrices of appropriate dimensions which can be found by the singular value decomposition of E C .

The following additional assumption is considered:

(A5) The pair αA i , γA i C is detectable.
As proposed in [START_REF] Darouach | Design of observers for descriptor systems[END_REF], if Assumptions A4 and A5 holds, an observer for the system (5) should have the following form:

z(k + 1) = M i=1 ε i (ρ(k)) [N i z(k) + L 1i y(k) + G i u(k) + L 2i y(k)] x(k) = z(k) + βy(k) + Kξy(k) (13)
where z(k) ∈ R n+d is the observer state vector; N i , L 1i , L 2i , G i and K, i = 1, . . . , M are constant matrices of appropriate dimensions.

In the next section, sufficient conditions are given, such that system ( 13) is an observer for system (5).

THE OBSERVER SYNTHESIS

For the sake of simplicity, the following notation is used:

Ω(ρ k ) = M i=1 ε i (ρ(k))Ω i i = 1, . . . , M (14)
Thus, system (13) can be rewritten in the simplified form:

z(k + 1) = N (ρ k )z(k) + L 1 (ρ k )y(k) + G(ρ k )u(k) + L 2 (ρ k )y(k) x(k) = z(k) + βy(k) + Kξy(k) (15)
Let the observer error e(k) ∈ R n+d be defined as:

e(k) = x(k) -x(k) (16)
The difference equation of the error is

e(k + 1) = [(α + Kγ) A(ρ k ) -N (ρ k ) (α + Kγ) E -L 1 (ρ k )C -L 2 (ρ k )C] x(k) + [(α + Kγ) B(ρ k ) -G(ρ k )] u(k) + N (ρ k ) [(α + Kγ) Ex(k) -z(k)] e(k) (17)
If the following conditions are fulfilled

G(ρ k ) = (α + Kγ) B(ρ k ) (18) L 1 (ρ k ) = N (ρ k ) (β + Kξ) (19)
and

N (ρ k ) = KγA(ρ k ) + αA(ρ k ) -L 2 (ρ k )C (20)
then [START_REF] Bara | Parameter-dependent state observer design for affine LPV systems[END_REF] reduces to

e(k + 1) = N (ρ k )e(k) (21)
Equation ( 21) can be written as:

e(k + 1) = M i=1 ε i (ρ(k))( Ãi + Ki Ci )e(k) (22)
where Ãi = αA i , Ki = (K -L 2i ) and Ci = γA i C .

Equation ( 22) confirms the necessity of Assumption A5. The following subsection is devoted to demonstrated the stability and convergence of the error dynamics given in Equation ( 22).

Observer stability

Theorem 3.1. Considering Assumptions A4-A5, the system (15) is a stable observer for the system (5) if there exist symmetric positive definite matrices H i , nonsingular matrices J i , and matrices L 2i , and K such that

  J T i + J i N T i J T i N i H j 0 J i 0 H -1 i   > 0 ( 23
)
∀ i = 1, ..., M , j = 1, ..., M , where N i = KγA i G i + αA i G i -L 2i CG i .
Proof:

Consider the following Parameter Dependent Lyapunov Function (PDLF)

V (e(k), ξ(ρ(k))) = e T (k)P (ρ k )e(k) (24)
where according to notation [START_REF] Abdalla | Fault detection and isolation filter design for linear parameter varying systems[END_REF]:

P (ρ k ) = M i=1 ε i (ρ(k))P i (25)
P i are symmetric positive definite constant matrices. According to the Lyapunov stability theorem, this PDLF must satisfy:

∆V (k) = V (k + 1) -V (k) = e T (k)(N T (ρ k )P (ρ k+1 )N (ρ k ) -P (ρ k ))e(k) ≤ 0 (26)
It can be seen that the function ∆V (k) is negative definite if

N T (ρ k )P (ρ k+1 )N (ρ k ) -P (ρ k ) < 0 (27)
The following notation is used for P (ρ k+1 ):

P (ρ k+1 ) = M i=1 ε i (ρ(k + 1))P i = M j=1 ε j (ρ(k))P j (28)
By substituting ( 25) and ( 28) in [START_REF] Gao | Improved H∞ control of discrete-time fuzzy systems: a cone complementary linearization approach[END_REF], it can be deduced that the negative definiteness of ( 27) is guaranteed if [START_REF] Daafouz | Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties[END_REF]:

N T i P j N i -P i < 0, ∀ i = 1, . . . , M, j = 1, . . . , M. (29)
The authors in [START_REF] Daafouz | Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties[END_REF] have demonstrated that poly-quadratically stability of the system e(k + 1) = N (ρ k )e(k) (which ensures the stability of the observer) is guaranteed if and only if there exist symmetric positive definite matrices H i , and matrices (of appropriate dimensions) Ψ i , i = 1, . . . , M , such that

Ψ i + Ψ T i -H i Ψ T i N T i N i Ψ i H j > 0 (30)
∀ i = 1, . . . , M, j = 1, . . . , M , with

P (ρ k ) = M i=1 ε i (ρ(k))H -1 i (31)
It is worth to note from equations ( 25), ( 28) and ( 31) that H i = P -1 i and H j = P -1 j . In this way, the feasibility problem of inequalities ( 29) can be replaced for the feasibility problem of inequality (30). If so, then ( 26) is a negative definite and decreasing sequence.

Replacing N i from ( 20) into (30):

Ψ i + Ψ T i -H i (⋆) T KγA i Ψ i + αA i Ψ i -L 2i CΨ i H j > 0 (32)
∀ i = 1, . . . , M, j = 1, . . . , M . It can be seen that the main difficulty to solve the inequality (32) is the bilinear term KγA i Ψ i , where K and Ψ i are unknown matrices.

Assuming that Ψ i is nonsingular, the positive definiteness of (30) is equivalent to the positive definiteness of the following congruent matrix

Ψ -1 i T 0 0 I Ψ i + Ψ T i -H i Ψ T i N T i N i Ψ i H j Ψ -1 i 0 0 I > 0 (33) or Ψ -1 i T + Ψ -1 i -Ψ -1 i T H i Ψ -1 i N T i N i H j > 0 (34)
Equation ( 34) is the Schur complement of the following positive definite matrix

  J T i + J i N T i J T i N i H j 0 J i 0 H -1 i   > 0 ( 35
)
with J i = Ψ -1 i . This completes the proof. 2

The inequality [START_REF] Masubuchi | Gain-scheduled controller synthesis based on new LMIs for dissipativity of descriptor LPV systems[END_REF], implies that H j H -1 i = I when i = j, or simply

H i P i = I (36)
(where P i = H -1 i ). Thus, inequality (23) can be written as:

  J T i + J i N T i J T i N i H j 0 J i 0 P i   > 0, H i P i = I. (37)
Among various methods to solve LMI problem under a LME constraint, the cone complementary technique has been considered. The following facts are used to give a solution procedure to solve the inequality (37): Fact 3.1. The feasibility of the LMI

H i I I P i ≥ 0, H i > 0; P i > 0 i = 1, . . . , M
implies that Tr(H i P i ) ≥ n, where the equality Tr(H i P i ) = n holds if and only if H i P i = I (see [START_REF] Gao | Improved H∞ control of discrete-time fuzzy systems: a cone complementary linearization approach[END_REF]).

The following matrices are introduced

H = diag (H 1 , . . . , H M ) (38)
P = diag (P 1 , . . . , P M ) (39)
By taking into account Fact 3.1, it is easy to deduce the following: In this way, the problem of finding a solution of (37) is equivalent to solve the following minimization problem: min Tr(HP) subject to

                         J T i + J i N T i J T i N i H j 0 J i 0 P i   > 0 H I I P > 0 H > 0 P > 0 (40)
∀ i = 1 . . . , M and ∀ j = 1 . . . , M . The bilinear objective problem: min Tr(HP), in (40) is solved by using the cone complementary algorithm proposed in [START_REF] El-Ghaoui | A cone complementarity linearization algorithm for static outputfeedback and related problems[END_REF].

The following notation is introduced:

H [k]
is used to denote the k-th element of the matrix sequence H [0] , H [1] , . . . , H [kopt] .

The cone complementary algorithm propose to transform the bilinear objective problem given in (40) by a linear objective problem of the form

min Tr(H [k-1] P [k] + P [k-1] H [k] ) subject to                                J [k] i T + J [k] i N [k] i T J [k] i T N [k] i H [k] j 0 J [k] i 0 P [k] i     > 0 H [k] I I P [k] > 0
H [k] > 0 P [k]
> 0 (41) for k = 1, . . . , k opt , and to find recursively the values of H = H [kopt] , P = P [kopt] ,

J i = J [kopt] i , N i = N [kopt] i
, where k opt represents the iteration number where the sequence converges to min Tr(H

[k-1] P [k] + P [k-1] H [k] ) = 2nM or equivalently min Tr(HP) = nM , in other words min Tr(H [k-1] i P [k] i + P [k-1] i H [k] i ) = 2n or equivalently min Tr(H i P i ) = n.
A small enough value of a constant stopping criterion ε > 0 should be considered with the purpose of reduce the convergence time of the algorithm, i.e., once the value of Tr(H [k] P [k] ) < nM + ε, the algorithm stops. As indicated [START_REF] El-Ghaoui | A cone complementarity linearization algorithm for static outputfeedback and related problems[END_REF], each step of the algorithm has a simple LMI problem. The recursive sequence is bounded and decreases if and only if H [k] P [k] is at the optimum. For more details concerning the used of such method, the authors recommend to read carefully [START_REF] El-Ghaoui | A cone complementarity linearization algorithm for static outputfeedback and related problems[END_REF] and associated papers.

The initial matrices H [0] and P [0] , are obtained by computing an initial feasible point of the simple LMIs

    J [0] i T + J [0] i N [0] i T J [0] i T N [0] i H [0] j 0 J [0] i 0 P [0] i     > 0 H [0] I I P [0] > 0 H [0] > 0 P [0] > 0 (42)
∀ i = 1 . . . , M and ∀ j = 1 . . . , M .

Observer gain design based on a pole placement

The observer should generate an accurate state and unknown inputs estimations based on an appropriate gain design. Among various kind of LMI regions, such as disk, cone, subplane, a pole placement through a disk as proposed by [START_REF] Kim | Pole placement of uncertain discrete systems in the smallest disk by state feedback[END_REF] has been considered. In fact, any region included inside the disk D(0, 1) can be chosen. However, if pole placement design is not considered, then the algorithm used to solve (41) could compute any H i , J i , L 2i and K satisfying the LMI [START_REF] Masubuchi | Gain-scheduled controller synthesis based on new LMIs for dissipativity of descriptor LPV systems[END_REF], such that the matrices N i , i = 1, ..., M have eigenvalues anywhere inside the unit circle, disregarding if they are near from the origin or near from the border of the unit circle. Then, in order to ensure an adequate pole placement of matrices N i , i = 1, . . . , n, the following inequalities can be taken into account instead of the inequalities shown in ( 29):

(N i -λ i I) T P j (N i -λ i I) -δ 2 i P i < 0, ∀ i = 1, . . . , M, j = 1, . . . , M. (43)
By doing so, the poles of matrices N i are assigned in a disk D(λ i , δ i ) defined by the radius δ i > 0 and the real number

λ i ∈ R such that δ i ≤ 1 -|λ i |, ∀ i = 1, .
. . , M . The parameters associated to a disk region are chosen from the nominal dynamic behavior of the system in order to stabilize and to guarantee an efficient state estimation.

The inequality shown in (43) can be written in the simplified form:

N T i P j Ni -Pi < 0, ∀ i = 1, . . . , M, j = 1, . . . , M. ( 44
)
which have the same form of inequality ( 29) with Ni = (N iλ i I) and Pi = δ 2 i P i . According to Theorem 3.1, the feasibility of inequality ( 44) is equivalent to the feasibility of finding symmetric positive definite matrices Hi , and matrices (of appropriate dimensions) Ψ i , i = 1, . . . , M , such that

Ψ i + Ψ T i -Hi Ψ T i N T i Ni Ψ i H j > 0 ( 45
) ∀ i = 1, . . . , M, j = 1, . . . , M , with P (ρ k ) = M i=1 ε i (ρ(k)) H-1 i ( 46
)
where Hi = P -1

i = δ -2 i P -1 i = δ -2
i H i Replacing Hi and Ni in (45), and by following an analogous procedure to that described in the proof of Theorem 3.1, the following corollary can be easily derived Corollary 3.1. Considering Assumptions A4-A5, the system (15) is a stable observer for the system (5) if there exist symmetric positive definite matrices H i and nonsingular matrices J i such that 18)-( 20). The poles of N i are located in a disk D(λ i , δ i ) defined by the radius δ i > 0 and the real number

  J T i + J i (N i -λ i I) T J T i (N i -λ i I) H j 0 J i 0 δ 2 i H -1 i   > 0 ( 47
) ∀ i = 1, . . . , M, j = 1, . . . , M , where N i = KγA i + αA i -L 2i C and L 2i . The matrices G(ρ k ), L 1 (ρ k ) and N (ρ k ) are deduced from (
λ i ∈ R such that δ i ≤ 1 -|λ i |.
Finally, the cone complementary algorithm transforms the problem of feasibility of the inequality shown in (47) in the following linear objective minimization problem:

min Tr(H [k-1] P [k] +P [k-1] H [k] ) subject to                                J [k] i T + J [k] i (N [k] i -λ i I) T J [k] i T (N [k] i -λ i I) H [k] j 0 J [k] i 0 δ 2 i P [k] i     > 0 H [k] I I P [k] > 0
H [k] > 0 P [k] > 0 ( 48
) which is solved as described above. If (48) is feasible, then the poles of N i are located in a disk D(λ i , δ i ) defined by the radius δ i > 0 and the real number

λ i ∈ R such that δ i ≤ 1 -|λ i |.

Extended case

In this subsection, the design of a full-order observer for descriptor systems with measurement noises and perturbations is presented.

Let us consider the signal w(t) characterized as the output of the following linear discretetime system:

x w (k + 1) = A w x w (k) + B w u w (k) w(k) = C w x w (k) (49) where x w (k) ∈ R w , u w (k) ∈ R, w(k) ∈ R
y m (k) = y(k) + w(k) = [C C w ] x(k) x w (k)
The state-space models of systems ( 5) and (49) can be rewritten in the compact form:

Ẽ ẋ(k + 1) = Ã(ρ k )x(k) + B(ρ k )ū(k) y m (k) = C x(k) (50)
where

Ẽ = E 0 0 I w Ã(ρ k ) = A(ρ k ) 0 0 A w B(ρ k ) = B(ρ k ) 0 0 B w C = C C w x(k) = x(k) x w (k) ū(k) = u(k) u w (k)
The procedure to design an observer for system (50) is equivalent to design an observer for system (5) extended to a classical robust residual generators method [START_REF] Ding | Model-based fault diagnosis techniques -Design schemes, algorithms, and tools[END_REF] .

ILLUSTRATIVE EXAMPLE

Consider a descriptor LPV system of the form (1) in the presence of two structured faults with unknown fault magnitudes given by:

Ā1 =     2.
    Ā2 =     1 1.4 -0.07 -0.2 1.5 2 -0.15 -0.3 0.4 0.1 0.5 0.2 0 0 -0.5 -0.2     (51) B1 =     0.6047 1.644 -1.709 -1.732     B2 =     0.5 1 -1.5 -1     C = 1 0 0 0 0 1 0 0 (52) Ē =     1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1     F =     1 0 0 0 0 1 0 0     (53)
where matrices A 1 , B 1 , E and C were taken from [START_REF] Chadli | Static output stabilisation of singular LPV systems: LMI formulation[END_REF] with structured fault matrix F .

The system (1) can be rewritten in the form of system [START_REF] Sun | Optimal and self-tuning fusion Kalman filters for discrete-time stochastic singular systems[END_REF], where the augmented matrices E and C are:

E = Ē -F 0 2×4 I 2 C = C 0 2×2 (54)
It can be verified that

rank E C = n + d = 6 (55)
The matrices α, β, γ and ξ, satisfying [START_REF] Grenaille | A method for designing fault diagnosis filters for LPV polytopic systems[END_REF] and ( 12) are computed by using the Singular Value Decomposition property of (E C) T :

α =        
0.333 0 0 0 0.333 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 -0.333 0 0 0 0.667 0 0 0 0 0 0 1

        β =         0.667 0 0 1 0 0 0 0 0.333 0 0 0         (56)
γ = -0.471 0.577 0 0 -0.471 0 0.333 0.817 0 0 0.333 0 ξ = 0.471 0 -0.333 0 (57)

At this point, it is worth to note that the pair αA i ,

γA i C is detectable ∀ i = 1, . . . , 2.
An observer of the form ( 13) can be designed. The disks to place the poles of the matrices N i are D(0.2, 0.8) for i = 1, 2.

Following the procedure described in Section 3, the following matrices are obtained: 18)- [START_REF] Henry | Robust fault diagnosis in uncertain linear parameter-varying sistems[END_REF].

P 1 =         1.
The effectiveness of the proposed observer scheme is illustrated by the system studied in open-loop. In this simulation, the time-varying parameter ρ(k) is the measured input, and it is shown in Fig. 1 in Fig. 2. The time evolution of these functions are in accordance with (2). It can be seen that the system is evaluated over the entire operating envelope. Fault-free dynamic behaviour of the outputs is shown in Fig. 3. It can be noted that a gaussian noise is added to the measured outputs. Two severe faults were simulated. The first fault occurs at the sampled time t k = 50 and the second fault occurs at t k = 130. The faults acts as two unknown inputs to the system, affecting the dynamics of the outputs and the states (see Figs. 45). The state observer tracks well both, the system states, despite the variations on the input (and consequently the changing operating points), the noisy measurements (see Fig. 4), and the faults. In Fig. 6, it can be appreciated that the observer detects these faults.

As expected, the observer is noise sensitive. However, these simulation results show that the observer is very effective in detecting and isolating the fault over the entire operating envelope. The estimation of the faults should be used to generate the residuals which can be evaluated through classical statistical methods with the purpose of generate alarms. The fault diagnosis scheme constructed in this way is able to detect, to isolate and to estimate the fault and it represents an efficient tool in the operator's decision.

CONCLUSION

In this paper, a model-based fault diagnosis method for discrete-time D-LPV systems is introduced. For that, an observer that performs fault detection and fault magnitude estimation over the entire operating envelope of the system. The observer synthesis is an extension of the work presented in [START_REF] Darouach | Design of observers for descriptor systems[END_REF] where an observer for LTI descriptor systems is reported. By taking into account some simple matrix manipulations, this observer can estimate simultaneously both, the states and the fault magnitude vectors, considered as unknown inputs. Sufficient conditions are stated to ensure the existence and the stability of the proposed observer by using a combined Lyapunov analysis based on LMI formulation. An observer gain synthesis has been also proposed to guarantee the efficiency of the model-based fault diagnosis technique. The proposed method is evaluated successfully via numerical simulations.

Fact 3 . 2 .

 32 The feasibility of the LMI H I I P ≥ 0, H > 0; P > 0 implies that Tr(HP) ≥ nM , where the equality Tr(HP) = nM holds if and only if HP = I

  are the state, the input and the output of the perturbation model. A w , B w and C w are constant matrices of appropriate dimensions. The signal w(k) can represent either a perturbation signal, or a noise signal. The output y m (k) is given by

Figure 1 .Figure 2 .

 12 Figure 1. Dynamic behaviour of the input

Figure 3 . 2 Figure 4 .

 324 Figure 3. Fault-free dynamic behaviour of the outputs y1 and y2

Figure 5 .

 5 Figure 5. Dynamic behaviour of the states x1, x2, x3 and x4 and their estimated values x1, x2, x3 and x4

1 Figure 6 .

 16 Figure 6. Dynamic behaviour of the faults d1 and d2 and their estimated values d1 and d2

Copyright c 2002 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. 2002; 00:1-6 Prepared using acsauth.cls

ACKNOWLEDGEMENTS

The authors would like to thank Prof. Mohamed DAROUACH for most fruitful discussions and Dr. Samir ABERKANE for their recommendations on LMIs. We also thank to PROMEP, Mexico for financial support to Dr. Astorga-Zaragoza during his postdoctoral stage to Centre de Recherche en Automatique de Nancy, France, in 2009. Moreover, the authors would like to thank the anonymous reviewers for their very helpful comments