Tahar Bouarar

Didier Maquin Benoît Marx

José Ragot

Fault tolerant tracking control for continuous Takagi-Sugeno systems with time varying faults

This paper deals with Fault Tolerant Control design for continuous nonlinear Takagi-Sugeno faulty systems. The goal is to ensure both state and fault estimation and the state reference tracking even if faults occur. In this study, the faults affecting the system behavior are considered as time varying functions modeled by exponential functions or first order polynomials. Based on descriptor redundancy property, solutions are proposed for both cases, exponential and polynomial faults, in terms of Linear Matrix Inequalities (LMI). In order to illustrate the applicability and the effectiveness of the proposed approaches, numerical examples are considered.

I. INTRODUCTION

The modern control of physical systems uses new algorithm approaches in order to increase the performances and the safety requirements. The classical control methodologies for nonlinear complex systems can lead to unsatisfactory performances, or even in some cases, to the instability if any malfunction occurs in the process components (as sensors or actuators). To overcome these drawbacks, new control system techniques have been developed in order to guarantee the overall system stability and acceptable performances, despite the failure situation. This new strategy of system control is called Fault Tolerant Control (FTC). The main idea of this strategy relies on the adaptation of the control law on the basis of the estimation of the faults affecting the system. Many works dealing with FTC design have been developed. For example, in the linear framework, the FTC problem is widely treated [START_REF] Gao | Reconfigurable control system design via perfect model following[END_REF][2][3][4] [START_REF] Marx | A Robust fault tolerant control for descriptor systems[END_REF][6] [START_REF] Zhang | Bibliographical review on reconfigurable faulttolerant control systems[END_REF] [START_REF] Zuo | Fault tolerant control for singular systems with actuator saturation and nonlinear perturbation[END_REF]. However in practice, most of physical systems are nonlinear, hence, it is primary to consider the FTC design for nonlinear systems. Some approaches dealing with this problem are proposed by [START_REF] Shtessel | Tailless aircraft flight control using multiple time scale reconfigurable sliding modes[END_REF][10] [START_REF] Theilliol | Fault diagnosis and accommodation of a three-tank system based on analytical redundancy[END_REF]. Nevertheless, these approaches are applied only when the operating points change or the fault occurs. Thus, the approaches taking into account the changes caused by both operating point variations and fault are not yet available (for more details, the reader can refer to [START_REF] Zhang | Bibliographical review on reconfigurable faulttolerant control systems[END_REF]). Indeed, a way to overcome the existing FTC approach drawbacks is to consider the nonlinear Takagi-Sugeno models [START_REF] Takagi | Fuzzy identification of systems and its application to modeling and control[END_REF]. This class of models allows representing exactly the overall nonlinear systems on its operating region in the state space. Takagi-Sugeno models are described by a set of Linear Time Invariant (LTI) models, each LTI model represents the nonlinear system behavior in a particular region in the state space. The This work was supported by the GIS 3SGS All the authors are with Centre de Recherche en Automatique de Nancy (CRAN) UMR 7039. Nancy-Université, CNRS, 2, avenue de la forêt de Haye, 54516 Vandoeuvre-Lès-Nancy, France.

{tahar.bouarar, benoit.marx, didier.maquin, jose.ragot,}@ensem.inpl-nancy.fr overall nonlinear system dynamics is obtained by blending together all the LTI models with nonlinear functions. Since introducing Takagi-Sugeno models, many works dealing with stability, stabilization and estimation have been developed [13][14][15] [START_REF] Sala | On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems[END_REF]. Nevertheless, the FTC design problem has been few treated. For example, the trajectory tracking FTC design for Takagi-Sugeno model subject to actuator constant fault has been developed by [START_REF] Bouarar | Trajectory tracking fault tolerant controller design for Takagi-Sugeno systems subject to actuator faults[END_REF], when the premise variables are measurable and by [START_REF] Ichalal | Fault tolerant control for Takagi-Sugeno systems with unmeasuable premise variables by trajectory tracking[END_REF][19] when they are unmeasurable. In these studies, the estimation of the constant faults is obtained by using proportional integral observers. Many papers reported that, on a practical point of view, even if the considered faults are not constant but only slow varying (with regards the dynamics of the system), the proposed observers provide good results.

However, in this paper, our goal is to study the problem of FTC design for Takagi-Sugeno models subject to time varying faults. The idea is to take into account an a priori knowledge about the faults acting on the system and investigate how the fault estimation can take advantage of that. Two particular fault models are considered: exponential function and first order polynomial. Thus, two approaches are proposed to ensure the state tracking between the healthy system and the faulty ones.

The following notations are considered to improve the paper readability: X µ = ∑ r i=1 µ (ξ (t))X i . In a block matrix, a star, * , denotes the transposed element in the symmetric position and diag (Θ 1 , ..., Θ r ) is a block diagonal matrix which diagonal entries are defined by Θ 1 , ..., Θ r . The following lemmas are needed to provide LMI conditions. Lemma 1 [START_REF] Zhou | Robust stabilization of linear systems with norm-bounded time-varying uncertainty[END_REF]: Consider two real matrices Π and Z with appropriate dimensions, for any positive scalar τ the following inequality holds:

Π T Z + Z T Π ≤ τ Π T Π + τ -1 Z T Z( 1 )
Lemma 2 [START_REF] Boyd | Linear Matrix Inequalities, System and Control Theory[END_REF]: Consider the matrices T i = T T i , i ∈ {0, ..., k}. The following expressions are equivalent:

∀ζ , ζ T T 0 ζ ≥ 0a n d ζ T T i ζ ≥ 0, ∀i ∈ {1, ..., k} (2) ∃ ρ 1 ≥ 0, ..., ρ k ≥ 0s u c ht h a t∀ζ , T 0 - k ∑ i=1 ρ i T i ≥ 0( 3 )

II. PROBLEM FORMULATION

Let us consider the following reference model:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ẋ(t)= r ∑ i=1 µ i (ξ (t))(A i x(t)+B i u(t)) y(t)= r ∑ i=1 µ i (ξ (t))(C i x(t)+D i u(t)) (4)
where x(t) ∈ R n , y(t) ∈ R p and u(t) ∈ R m are respectively the state, the output and the nominal control vectors. The r submodels are defined by the matrices

A i ∈ R n×n , B i ∈ R n×m , C i ∈ R p×n and D i ∈ R p×m . µ i (ξ (t))
are the nonlinear functions depending on the variable ξ (t) which can be measurable (u(t) or y(t)) or unmeasurable (x(t)). These nonlinear functions satisfy the convex sum property:

∀ t ⎧ ⎪ ⎨ ⎪ ⎩ 0 ≤ µ i (ξ (t)) ≤ 1, i = 1, ..., r r ∑ i=1 µ i (ξ (t)) = 1 (5)
Let us consider the following faulty system:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ẋ f (t)= r ∑ i=1 µ i (ξ (t))(A i x f (t)+B i u f (t)+G i f (t)) y f (t)= r ∑ i=1 µ i (ξ (t))(C i x f (t)+D i u f (t)+W i f (t)) (6) where x f (t) ∈ R n , y f (t) ∈ R p , f (t) ∈ R q and u f (t) ∈ R m
are respectively the faulty state, the faulty output, the fault affecting the system and the fault tolerant control vectors. G i ∈ R n×q and W i ∈ R p×q are the transfer matrices of the faults on the system. The fault tolerant controller design methodology is based on the following scheme. In order to ensure the tracking between the healthy system states and the faulty system ones, we consider the following FTC law given by:

y f (t) System f (t) u f (t) u(t) + + f (t) x(t) + - x f (t) Observer Controller Reference model
u f (t)=u (t)+ r ∑ i=1 µ i (ξ (t)) K i x (t) -x f (t) -K f i f (t) (7)
where K i ∈ R m×n and K f i ∈ R m×q are the state feedback gain matrices to be determined. In this study, an observer is needed to estimate simultaneously the faults and the faulty system states. The observer structure is given by:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ẋ f (t)= r ∑ i=1 µ i (ξ (t)) A i x f (t)+B i u f (t)+G i f (t)+ϕ i (t) ŷ f (t)= r ∑ i=1 µ i (ξ (t)) C i x f (t)+D i u f (t)+W i f (t) ḟ (t)= r ∑ i=1 µ i (ξ (t)) H 2 i y f (t) -ŷ f (t) -H 3 i f (t) (8) with ϕ i (t)=H 1 i y f (t) -ŷ f (t) . The observer's gain matri- ces H 1 i ∈ R n×p , H 2 i ∈ R q×p and H 3 i ∈ R q×q
are to be designed. With this FT controller structure, one can remark that fault detection and isolation are performed since an estimate of the fault affecting the system is available.

III. FAULT TOLERANT CONTROLLER DESIGN

Along of this study, we assume that: 1) The faults and the system states are observable from the output.

2) The nonlinear functions depend only on the measurable variables y(t) or u(t).

3) The considered fault functions are differentiable.

A. FTC for exponential faults

Assume that the faults affecting the system are modeled by exponential function ( f i (t) ≡ e α i t+β i , with α i , β i ∈ R, i = 1, ..., q. In the next, we consider that α i = α 0,i + Δα i , which allows defining a set of exponential functions, α 0,i and Δα i representing respectively the nominal and the uncertain parts of α i . Let us define α = diag(α 1 , ..., α q ), α 0 = diag(α 0,1 , ..., α 0,q ),a n dΔα = diag(Δα 1 , ..., Δα q ).T h e uncertain part can be bounded as:

(Δα) T Δα ≤ ν (9)
where ν ∈ R q×q is a known diagonal positive definite matrix. Let us respectively define the state and fault estimation errors as: e s (t)=x f (t)x f (t) and e d (t)= f (t) -f (t).L e t us also define the state tracking error e p (t)=x (t)x f (t), the output error e y (t)=y f (t)ŷ f (t) and the error between the nominal and FTC control signals e u (t)=u (t)u f (t).

According to the above notations, the dynamics of e p (t) and e s (t) are given by:

ėp (t)=A µ e p (t)+B µ e u (t) -G µ f (t) (10) ės (t)=A µ e s (t)+G µ e d (t) -H 1 µ e y (t) (11)
Since ḟ (t)=α f (t), the dynamics of the fault estimation e r r o ri sg i v e nb y :

ėd (t)=α f (t) -H 2 µ e y (t)+H 3 µ f (t) (12)
By adding and subtracting H 3 µ f (t) in [START_REF] Takagi | Fuzzy identification of systems and its application to modeling and control[END_REF], one can obtain:

ėd (t)=-H 2 µ e y (t) -H 3 µ e d (t)+ α + H 3 µ f (t) (13)
The substitution of the expression of e y (t) in ( 11) and ( 13) and the expression of e u (t) in [START_REF] Hess | Sliding mode control applied to reconfigurable flight control design[END_REF], leads to introducing multiplications between the system matrices and the ones of the observer and the controller. This coupling leads to conservative results. A way to overcome this problem is to introduce a "virtual dynamics" in the e y (t) and e u (t) equations [START_REF] Guelton | Robust dynamic output feedback fuzzy Lyapunov stabilization of Takagi-Sugeno systems -A descriptor redundancy approach[END_REF] [22][23]. This craftiness allows decoupling the system, observer and FTC controller matrices in the expressions of the error dynamics. Hence, e y (t) and e u (t) can be rewritten as: 0ė y (t)=-e y (t)+C µ e s (t)+W µ e d (t) [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF] where 0 ∈ R p×p is a zero matrix.

Adding and subtracting K µ x f (t) and K f µ f (t) in e u (t), one can obtain:

0ė u (t)=-K f µ f (t)+K f µ e d (t)+K µ e p (t)+K µ e s (t)+e u (t) (15
) where 0 ∈ R m×m is a zero matrix. The concatenation of ( 10), ( 11), ( 13), ( 14) and ( 15) leads to the following descriptor representation:

E ė (t)= õ ẽ (t)+ Bµ f (t) (16)
where E = diag (I n I n I q 0 p 0 m ),

ẽT (t)= e T p (t) e T s (t) e T d (t) e T y (t) e T u (t) , õ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ A µ 000 B µ 0 A µ G µ -H 1 µ 0 00 -H 3 µ -H 2 µ 0 0 C µ W µ -I 0 -K µ -K µ -K f µ 0 -I ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ and BT µ = -G T µ 0 α + H 3 µ T 0 K f µ T .
From the structure of the matrices E and õ , we remark that the descriptor ( 16) is impulse-free. The main interest of the descriptor approach is to avoid products of system, observer and controller gains in the LMI, and thus obtain more tractable and less restrictive conditions.

The main provided results are summarized in the following theorem 1. Theorem 1: The system (16) that describes the different estimation errors is stable and the L 2 -gain from the faults to the state tracking error, the state and fault estimation errors is bounded by √ γ, if there exists matrices P 1 = P T 1 0, P 7 = P T 7 0, P 13 = P T 13 0, P 16 , P 17 , P 18 , P 19 , P 25 , F i , R i , S i , Q i and M i and positive scalars γ and τ such that the following LMI are verified, for i = 1, ..., r:

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ϒ 1,1 i * * * * * 0 C T i P 16 ϒ 2,2 i * * * 00 W T i P 16 ϒ 3,2 i ϒ 3,3 i * * * * -P 16 ϒ 4,2 i ϒ 4,3 i ϒ 4,4 00 0 ϒ 5,1 i -F i -Q i 0 ϒ 5,5 * 0 -G T i P 1 0 ϒ 6,3 i 0 Q T i (τ -γ)I 0 00 P 13 00 0 -τνI ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ < 0 (17)
where

ϒ 1,1 i = P 1 A i + A T i P 1 + I, ϒ 2,2 i = P 7 A i + A T i P 7 + P T 17 C i +C T i P 17 + I, ϒ 3,2 i = P T 18 C i + G T i P 7 +W T i P 17 , ϒ 3,3 i = -M T i -M i + P T 18 W i +W T i P 18 + I, ϒ 4,2 i = P T 19 C i -P 17 -S i , ϒ 4,3 i = P T 19 W i -P 18 -R T i , ϒ 4,4 = -P 19 -P T 19 , ϒ 5,1 i = -F i + B T i P 1 , ϒ 5,5 = -P 25 -P T
25 and ϒ 6,3 i = α 0 P 13 + M i .

Proof. : To study the stability of ( 16), we consider the following quadratic Lyapunov function candidate:

V (e p (t) , e s (t) , e d (t)) = ẽT (t) EPẽ (t) (18)
with

EP = P T E 0 ( 19
)
The choice of the Lyapunov function [START_REF] Ichalal | Fault tolerant control for Takagi-Sugeno systems with unmeasuable premise variables by trajectory tracking[END_REF] will ensure the stability of the tracking error and of the state and fault estimations. To attenuate the fault effect on the error dynamics, one considers the L 2 constraint [START_REF] Boyd | Linear Matrix Inequalities, System and Control Theory[END_REF] given by:

t f 0 ẽT (t)E ẽ (t) γ 2 t f 0 f T (t) f (t) (20)
where t f and γ represent respectively the final time and the attenuation level. The tracking error e p (t), state e s (t) and fault e d (t) estimation errors must therefore satisfy the following inequality: [START_REF] Ichalal | Observer based actuator fault tolerant control in nonlinear Takagi-Sugeno fuzzy systems: LMI approach[END_REF] and substituting ( 16) in [START_REF] Guelton | Robust dynamic output feedback fuzzy Lyapunov stabilization of Takagi-Sugeno systems -A descriptor redundancy approach[END_REF], one can obtain:

ėT (t) EPẽ (t)+ ẽT (t) EP ė (t)+ ẽT (t) E ẽ (t)-γ 2 f T (t) f (t) < 0 (21) Considering
ẽT (t) f T (t) T ÃT µ P + P T õ + E * BT µ P -γ 2 ẽ (t) f (t) < 0 (22)
The inequality [START_REF] Bouarar | Static Output Feedback Controller Design for Takagi-Sugeno Systems -A Fuzzy Lyapunov LMI Approach[END_REF] is fulfilled if:

ÃT µ P + P T õ + E * BT µ P -γ 2 < 0 ( 23
)
The structure of the Lyapunov matrix P is chosen as follows:

P = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
P 1 0000 0 P 7 000 00 P 13 00 P 16 P 17 P 18 P 19 0 0000

P 25 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ (24)
Clearly, this structure is not the general one but it limits the coupling between the unknown submatrices to be determined and allows the convergence conditions to be expressed using LMI. According to [START_REF] Ichalal | Observer based actuator fault tolerant control in nonlinear Takagi-Sugeno fuzzy systems: LMI approach[END_REF], this latter imposes that P 1 = P T 1 0, P 7 = P T 7 0, P 13 = P T 13 0a n dP 16 , P 17 , P 18 , P 19 P 25 are free slack matrices with appropriate dimensions. Considering [START_REF] Boyd | Linear Matrix Inequalities, System and Control Theory[END_REF] and the matrices defined in [START_REF] Sala | On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems[END_REF], the mathematical development of ( 23) leads to:

Ξ µ + Γ T Ω + Ω T Γ < 0 (25)
where Γ = 00000Δα , Ω = 00P 13 000 and

Ξ µ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ Ψ 1,1 µ 0 K f µ T .
As well as for the case of exponential faults, the obtained desciptor (30) is impulse-free.

The main provided results are summarized in the following theorem 2. Theorem 2: The system (30) that describes the different estimation errors is stable and the L 2 -gain from the faults to the state tracking error, the state and fault estimation errors is bounded by √ γ, if there exists matrices P 1 = P T 1 0, P 7 = P T 7 0, P 13 = P T 13 0, P 16 , P 17 , P 18 , P 19 , P 25 , F i , R i , S i , Q i and M i and positive scalars γ, ρ and σ such that the following LMI are verified:

for i = 1, ..., r ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ Φ 1,1 i * * * * * 00 Φ 2,1 i Φ 2,2 i * * * 00 0 Φ 3,1 i Φ 3,2 i Φ 3,3 i * * * * 0 -P 16 Φ 4,2 i Φ 4,3 i Φ 4,4 000 0 Φ 5,1 i -F i -Q i 0 Φ 5,5 * 00 Φ 6,1 i 0 M i 0 Q T i -γI 00 00 Φ 7,3 i 000 Φ 7,7 * 0 00000 P 13 -σ I ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ < 0 (31)
where

Φ 1,1 i = ϒ 1,1 i + ρI, Φ 2,1 i = C T i P 16 , Φ 2,2 i = ϒ 2,2 i + ρI, Φ 3,1 i = W T i P 16 , Φ 3,2 i = ϒ 3,2 i , Φ 3,3 i = ϒ 3,3 i +(σν + ρ)I, Φ 4,2 i = ϒ 4,2 i , Φ 4,3 i = ϒ 4,3 i , Φ 4,4 i = ϒ 4,4 i + ρI, Φ 5,1 i = ϒ 5,1 i , Φ 5,5 i = ϒ 5,5 i + ρI, Φ 6,1 i = -G T i P 1 , Φ 7,3 = λ 0 P 13 , Φ 7,7 = -ερI.
Proof. Considering ( 18), ( 19), (30) and following the same path as for the proof of theorem 1 (( 18)-( 22)), one can obtain:

⎛ ⎝ ẽT f T I ⎞ ⎠ T ⎛ ⎝ ÃT µ P + P T õ + E * * BT µ P -γ 2 I 0 N T P 00 ⎞ ⎠ ⎛ ⎝ ẽ f I ⎞ ⎠ < 0
(32) To ensure the asymptotic convergence to a ball of radius ε of the error dynamics, we consider the following constraint:

ẽ 2 2 ≥ εI ( 33
)
with ε is a known small positive scalar. The inequality (33) is equivalent to:

⎛ ⎝ ẽT f T I ⎞ ⎠ T ⎛ ⎝ I 00 00 0 00-εI ⎞ ⎠ ⎛ ⎝ ẽ f I ⎞ ⎠ ≥ 0 (34)
Applying S-procedure lemma 2 [START_REF] Boyd | Linear Matrix Inequalities, System and Control Theory[END_REF] on (33) and (34), one can obtain:

⎛ ⎝ ẽT f T I ⎞ ⎠ T ⎛ ⎝ Ψ µ * * BT µ P -γ 2 I 0 N T P 0 -ερI ⎞ ⎠ ⎛ ⎝ ẽ f I ⎞ ⎠ < 0 (35)
where Ψ µ = ÃT µ P + P T õ + E + ρI.

Considering the matrix N defined in (30) and P, (35) is fulfilled if:

χ µ + Σ T φ + φ T Σ < 0 (36)
where Σ = 000000Δλ , φ = 00P 13 0000 and

χ µ = ⎛ ⎝ Ψ µ * * BT µ P -γ 2 I 0 00λ 0 P 13 00 0 -ερI ⎞ ⎠ .
Applying lemma 1 on the inequality (36), this latter can be written as:

χ µ + σ Σ T Σ + σ -1 φ T φ < 0 (37)
Following the same steps of the proof of theorem 1 from [START_REF] Bouarar | Robust Fuzzy Lyapunov Stabilization for Uncertain and Disturbed Takagi-Sugeno Descriptors[END_REF] to the end, then applying the Schur complement [START_REF] Boyd | Linear Matrix Inequalities, System and Control Theory[END_REF] on the term σ -1 φ T φ , thus, the sufficient LMI conditions proposed in theorem 2 hold IV. SIMULATION EXAMPLE In this section, the effectiveness and the applicability of the proposed approach are illustrated. An academic example is considered since the main purpose of this paper is theoretical. Let us consider the following Takagi-Sugeno model:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ẋ (t)= 2 ∑ i=1 µ i (ξ (u (t))) A i x f (t)+B i u f (t)+G i f (t) y (t)= 2 ∑ i=1 µ i (ξ (u (t))) C i x f (t)+D i u f (t)+W i f (t) with A 1 = ⎛ ⎝ -51-3 1 -32 11 -4 ⎞ ⎠ , A 2 = ⎛ ⎝ -31 1 0.5 -32 0.51-5 ⎞ ⎠ , B 1 = ⎛ ⎝ 5 3 2 ⎞ ⎠ , B 2 = ⎛ ⎝ 5 4 5 ⎞ ⎠ , G 1 = ⎛ ⎝ 5 5 0 ⎞ ⎠ , G 2 = ⎛ ⎝ 5 5 5 ⎞ ⎠ , W 1 = -0.5 0.5 , W 2 = -1 0.5 , C 1 = 0.50 .50 00 .50 , C 2 = 10 .50 0.510 , D 1 = -0.8 0 , D 2 = -0.8 -0.5 , µ 1 (u (t)) = 1-tanh(0.5-u(t))
2 and µ 2 (u (t)) = 1 -µ 1 (u (t)).The nominal input signal is u (t)=sin (cos (0.5t)t).T h eL M I conditions given in theorems 1 and 2 have been solved by using Matlab LMI Toolbox.

A. Case of exponential fault

Let us consider the following exponential fault affecting the system behavior occurs at 8sec ≤ t ≤ 11sec:

f (t)=e 1.15t-14 (38)
The controller and the observer are synthesized for α 0 = 1 and Δα = 0.2. Thus, the simulation results are illustrated by the figures 2 and 3.

From these latter, one can see that the synthesized observer and FTC controller showed their effectiveness, since the fault and the system states are estimated and the tracking between the faulty system states and the reference model ones is ensured.

B. Case of first order polynomial fault

The fault affecting the system occurs at 9sec ≤ t ≤ 12.5sec and is given by the following first order polynomial:

f (t)=0.2t -1 (39)
Note that the FTC controller and observer gains are computed for ε = 10 -3 , λ 0 = 0.11 and Δλ = 0.25. Thus, the simulation results are illustrated by the following figures.

It can be seen that the synthesized FTC controller compensates the fault and ensures the tracking between the reference and the faulty system states, even if the parameter of the occuring fault is not perfectly known for observer and controller designs.

V. C ONCLUSION

In this paper, fault tolerant tracking controller design approaches for continuous Takagi-Sugeno models are proposed. They concern the case when the fault affecting the system behavior are time varying. In this study, the considered faults are modeled by exponential function or first order polynomial. The provided conditions are easily formulated in term of LMI using the well known descriptor redundancy property. This latter allows introducing free slack variables in the Lyapunov function candidate leading to less conservative LMI conditions. The effectiveness of the proposed approaches are illustrated with an example. Only an academic example was considered, since the present paper provides preliminary theoretical results, more efforts will be made to improve applicability to real processes.

Fig. 1 .

 1 Fig. 1. Tracking fault tolerant controller design scheme

Fig. 2 .Fig. 3 .

 23 Fig. 2. Reference model states vs. faulty system ones with FTC

Fig. 4 .Fig. 5 .

 45 Fig. 4. Reference model states vs. faulty system ones with FTC

Applying lemma 1 on (25), then considering ( 9), (25) can be bounded as:

with Λ = 00000ν . To provide LMI conditions, we consider the following bijective changes of variables:

, then applying Schur complement [START_REF] Boyd | Linear Matrix Inequalities, System and Control Theory[END_REF] on the term τ -1 Ω T Ω, thus, the sufficient LMI conditions proposed in theorem 1 hold.

B. FTC for first order polynomial faults

Let us now consider that the fault affecting the system are modeled by the first order plynomial given by:

where λ i ∈ R and δ i ∈ R, i = 1, ..., q. As well as for exponen- tial function, λ = λ 0 + Δλ , with Δλ verifying:

where ν ∈ R q×q is a known diagonal positive definite matrix. According to [START_REF] Tanaka | Fuzzy control systems design and analysis: a linear matrix inequality approach[END_REF], the fault estimation error dynamics is given by:

The combination of ( 10), ( 11), (29), ( 14) and ( 15) leads to the following descriptor representation:

where the matrices E and õ are defined in equation ( 16) and N T = 00λ T 00 , BT µ = -G T µ 0 H 3 µ T


T Bouarar

B Marx

D Maquin

J Ragot

Design of fault tolerant control for nonlinear systems subject to time varying faults

In this paper, a Fault Tolerant Control (FTC) problem for discrete time nonlinear systems represented by Takagi-Sugeno (T-S) models is investigated. The goal is to design a fault tolerant controller taking into account the faults affecting the overall system behavior in order to ensure the system stability. The principal idea is to introduce a Proportional Integral (PI) observer to detect and to estimate an eventual fault occuring in the system. Based on Lyapunov theory, two new approaches are proposed in term of Linear Matrix Inequalities (LMI) leading to synthesize an FTC laws ensuring the tracking between the reference model states and the faulty system ones. These results concern the case of time varying faults modeled by exponential function and first order plynomial. To illustate the effectiveness of the proposed approaches, an academic example is considered.

INTRODUCTION

Generally speaking, there exists two strategies for faulty systems control: the passive strategy and the active one. In the case of the passive strategy, also called robust control, the controller design problem has been widely studied in the literature and many approaches have been proposed for linear and nonlinear systems. The objective is to ensure simultaneously the stability of the system and the insensitivity to certain faults. Nevertheless, robust control methodology concerns a specific class of faults characterized by a bounded norm. The active control or Fault Tolerant Control (FTC) has been introduced to overcome the passive control drawbacks. Indeed, the FTC method allows improving the system performances for a large class of faults. The principal idea of this strategy is to reconfigure the control law according to the fault detection and estimation performed by an observer to allow the faulty system to accomplish its mission. Since the introduction of FTC techniques, several works have been developed. In linear system frameworks, a FTC approach based on pseudo-inverse technique has been proposed by [START_REF] Gao | Reconfigurable control system design via perfect model following[END_REF]. The main idea of this technique is to minimise a Frobenius norm leading to determine the controller gains. Thereafter, an extension of this approach has been proposed by [START_REF] Staroswiecki | Fault tolerant control: the pseudoinverse method revisited[END_REF]. In [START_REF] Liu | Eigenstructure Assignment for Control Systems Design[END_REF], the FTC gains have been determined such that the eigenvalues of the controlled faulty system and those of a reference model are identical.

In the case of linear descriptor systems described by differential and algebraic equations, an approach has been proposed by [START_REF] Marx | Robust fault tolerant control for descriptor systems[END_REF]). In the last decades, Takagi-Sugeno nonlinear systems [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF] have attracted a great deal attention, since they allow extending the linear systems theory to nonlinear ones [START_REF] Tanaka | Fuzzy control systems design and analysis. A linear matrix inequality approach[END_REF][START_REF] Feng | A survey on analysis and design of modelbased fuzzy control systems[END_REF]). Thus, many problems dealing with stability, stabilization, observer design and diagnosis have been widely studied. Nevertheless, the FTC problem based on this kind of model is not largely treated. Some works have been introduced in recent years, for instance, trajectory tracking FTC design approach for Takagi-Sugeno systems subject to actuator faults has been developed by [START_REF] Ichalal | Estimation et diagnostic de systèmes non linéaires décrits par un multimodèle de Takagi-Sugeno[END_REF]. Nevertheless, this approach may be conservative and some results should be improved by obtaining more relaxed conditions. More recently, new less conservative approach has been developed by [START_REF] Bouarar | Trajectory tracking fault tolerant controller design for Takagi-Sugeno systems subject to actuator faults[END_REF]. Note that these approaches concern the Takagi-Sugeno systems with measurable premise variables (i.e. premise variables depending on the the input or the output). In the other hand, when the premise variables are unmeasurable (depend on the states of the system), the FTC design problem has been studied by (Ichalal et al. 2010a[START_REF] Ichalal | Observer based actuator fault tolerant control in nonlinear Takagi-Sugeno fuzzy systems: LMI approach[END_REF]). In the above studies, the considered faults affecting the system behavior are modeled by a constant function. However, in practice, the faults are often time variant.

Based on Lyapunov theory, two approaches dealing with FTC design for nonlinear systems represented by discrete Takagi-Sugeno systems with measurable premise variables are proposed. The objective is to ensure the tracking between a healthy reference nonlinear model and the eventually faulty nonlinear system. The proposed approaches are formulated in terms of Linear Matrix Inequalities (LMI) and they respectively concern the cases when fault dynamics is modelled by exponential function and first order polynomial. Moreover, the developed approaches does not require knowledge of the considered fault varying functions coefficients. To illustrate the applicability and the effectiveness of the proposed approaches, an academic example is considered. To simplify the mathematical expressions and to improve the paper readability, we consider the following notations:

Γ µ = r ∑ i=1 µ i (ξ (k)) Γ i , Γ µ µ = r ∑ i=1 r ∑ j=1 µ i (ξ (k)) µ j (ξ (k)) Γ i j
, in a bloc matrix, an asterisk * denotes the transposed element in the symetric position, in the mathematical expressions χ (k + ) is equivalent to χ (k + 1) and diag ( Λ 1 • • • Λ r ) represent a block diagonal matrix. The following lemmas are needed to provide LMI conditions.

Lemma 1 [START_REF] Zhou | Robust stabilization of linear systems with norm-bounded time-varying uncertainty[END_REF]: Consider two real matrices Φ and Ξ with appropriate dimensions, for any positive scalar τ the following inequality holds:

Φ T Ξ T + Ξ T Φ ≤ τ Φ T Φ + τ -1 Ξ T Ξ T (1)
Lemma 2 [START_REF] Boyd | Linear Matrix Inequalities, System and Control Theory[END_REF]): Consider the matrices T i = T T i , i ∈ {0, ..., k}. The following expressions are equivalent:

∀ζ , ζ T T 0 ζ ≥ 0 and ζ T T i ζ ≥ 0, ∀i ∈ {1, ..., k} (2) ∃ ρ 1 ≥ 0, ..., ρ k ≥ 0 such that ∀ζ , T 0 - k ∑ i=1 ρ i T i ≥ 0 (3)
2 TAKAGI-SUGENO MODEL A Takagi-Sugeno (T-S) model allow the representation of a nonlinear system behavior by the interpolation of a set of linear submodels [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF], [START_REF] Tanaka | Fuzzy control systems design and analysis. A linear matrix inequality approach[END_REF]. Each submodel contributes to the global behavior of the nonlinear system through a weighting function ξ l (k). The T-S structure is given by:

     x (k + ) = r ∑ i=1 µ i (ξ (k)) (A i x (k) + B i u (k)) y (k) = r ∑ i=1 µ i (ξ (k)) (C i x (k) + D i u (k)) (4)
where r represent the number of local linear submodels,

ξ (k) = ( ξ 1 (k) • • • ξ j (k)
) represent the vec- tor of premise variables which can be measurable (input u(t) or/and output of the system y(t) ) or unmeasurable (the system state x(t)). A i ∈ R n×n , B i ∈ R n×m , C i ∈ R p×n and D i ∈ R p×m are the matrices of the i th linear submodel representing the plant behavior in the local region.

The weighting functions satisfy the convex sum property, i.e:

   0 ≤ µ i (ξ (k)) ≤ 1 r ∑ i=1 µ i (ξ (k)) = 1 (5)
The T-S model defined in ( 4) is a nonlinear system since the weighting functions blinding together the r linear submodels are nonlinear.

The Takagi-Sugeno model shown its interest both in theoretical and practice fields. Indeed, the T-S model can represent exactly a nonlinear system in operating region of the state space. In other hand, thanks to convex sum property of the weighting functions ( 5), it is possible to extend the linear control theory to the nonlinear case.

PROBLEM STATEMENT

Let us consider (4) as a reference model corresponding to the healthy system. Let us also consider the faulty T-S model given by:

     x f (k + ) = r ∑ i=1 µ i (ξ (k)) A i x f (k) + B i u f (k) + G i f (k) y f (k) = r ∑ i=1 µ i (ξ (k)) C i x f (k) + D i u f (k) +W i f (k) (6)
where G i ∈ R n×q and W i ∈ R p×q describe the distribution matrices of the faults acting on the system.

x f (k) ∈ R n , y f (k) ∈ R p , u f (k) ∈ R m and f (k) ∈ R q
represent respectively the faulty state, the faulty output, the FTC law and the faults affecting the T-S model.

To ensure the tracking between the faulty T-S model ( 6) and the healthy one (4), consider the following FTC law:

u f (k) = u (k) + u c (k) (7)
where

u c (k) = r ∑ i=1 µ i (ξ (k)) K i x (k) -x f (k) -f (k) ,
with K i ∈ R m×n are the state feedback gain matrices to be determined.

Note that this control law is an extension of a classical linear state feedback law. It is obtained by an interpolation using the same weighting functions as the model, and use an estimate of the fault.

The considered FTC law methodology is based on the scheme described in Figure 1. The conception of the FTC controller needs the knowledge of

y f (k) u(k) f (k) + + u f (k) System f (k) x(k) + - x f (k) Observer Controller u c (k)

Reference model

Figure 1: Tracking fault tolerant controller design scheme both faults and state estimates. Thus, the following Proportional Integral (PI) observer is considered.

             x f (k + ) = r ∑ i=1 µ i (ξ (k)) A i x f (k) + B i u f (k) + ϕ i (k) ŷ f (k) = r ∑ i=1 µ i (ξ (k)) C i x f (k) + D i u f (k) +W i f (k) f (k + ) = r ∑ i=1 µ i (ξ (k)) H 2 i y f (k) -ŷ f (k) + f (k) (8)
where

ϕ i (k) = G i f (k) + H 1 i y f (k) -ŷ f (k) , H 1 i ∈ R n×p and H 2 i ∈ R q×p
are gain matrices of the PI observer to be determined.

Along of this work, we assume that:

• The system states and the faults are observable from the output.

• The nonlinear weighting functions depend only on measurable premise variables (i.e. y(t) and/or u(t)).

FTC DESIGN LMI CONDITIONS

In this work, two kind of time varying faults are considered. The first one concerns the faults modeled by exponential function. The second one concerns the case of faults modeled by a first order polynomial.

Case of faults approximated by an exponential function

Let us consider that the system behavior is affected by several faults in the form:

f i (k) = e a i k+b i (9)
where a i , b i ∈ R, for i = 1, ..., q. The variation of the faults is given by:

f i (k + ) = e a i f i (k) (10)
Let us consider that a i = a 0,i + ∆a i , with a 0,i and ∆a i representing respectively the nominal and the uncertain parts of the parameter a i . This structure leads defining a set of exponential functions describing the faults affecting the T-S model behavior.

Let us also define:

a = diag ( e a 1 • • • e a q ) (11) a 0 = diag ( e a 0,1 • • • e a 0,q ) (12) ∆a = diag e ∆a 1 • • • e ∆a q (13)
The uncertain part can be bounded as:

(∆a) T ∆a ≤ λ ( 14
)
where λ ∈ R q×q is a known diagonal positive definite matrix.

Let us define the state tracking, the state and the fault estimation errors given respectively as:

   e p (k) = x (k) -x f (k) e s (k) = x f (k) -x f (k) e d (k) = f (k) -f (k) (15)
The dynamics of e p (k), e s (k) and e d (k) are respectively given by:

e p (k + ) = M µ µ e p (k) -χ µ µ e s (k) -B µ e d (k) + Ω µ f (k) (16)
where

M µ µ = A µ -B µ K µ , Ω µ = B µ -G µ and χ µ µ = B µ K µ . e s (k + ) = Π µ µ e s (k) + Θ µ µ e d (k) (17)
with

Π µ µ = A µ -H 1 µ C µ and Θ µ µ = G µ -H 1 µ W µ . e d (k + ) = -H 2 µ C µ e s (k) + Σ µ µ e d (k) + α f (k) (18)
where Σ µ µ = I -H 2 µ W µ and α = a -I. The combination of ( 16), ( 17) and ( 18) leads to the following expression:

e (k + ) = Āµµ e (k) + Z µ f (k) (19)
where

Z µ = Ω µ 0 α , e (k) = e p (k) e s (k) e d (k) and Āµµ =   M µ µ -χ µ µ -B µ 0 Π µ µ Θ µ µ 0 -H 2 µ C µ Σ µ µ   .
Remark 1: Along this work, we consider that the matrices B i and G i of the system (6) and the observer (8), have the same dimensions.

The conditions leading to FTC design controller are proposed in the following theorem 1.

Theorem 1: The state tracking error e p (k), the state and fault estimation errors e s (k) and e d (k) converge assymptotically to zero and the L 2 -gain from the faults to the errors e p (k), e s (k) and e d (k) is bounded by √ γ, if there exists matrices

X 1 = X T 1 ≥ 0, X 2 = X T 2 ≥ 0, X 3 = X T 3 ≥ 0, K i , L 1
i and L 2 i and positive scalars γ and τ such that the following LMI, for i = 1, 2, ..., r hold

ϒ i j < 0 (20)
The matrix ϒ i j is defined in the next page.

Proof. To study the asymptotic convergence to zero of the above errors e p (k), e s (k) and e d (k), we consider the following Lyapunov function candidate:

V (k) = e T (k) Xe (k) (21)
with

X = X T ≥ 0 (22)
Let us consider the following L 2 constraint minimizing the fault effect on e p (k), e s (k) and e d (k).

N ∑ k=0 e T (k)Qe (k) ≤ γ 2 N ∑ k=0 f T (k) f (k) (23)
where N denotes the final step time, γ represents the attenuation level and Q is a known symmetric positive definite weighting matrix.

The error dynamics expressed in ( 19) is stable under the L 2 constraint (23) if:

Q -X 0 0 -γ 2 I + ĀT µ µ Z T µ X Āµµ Z µ < 0 (24)
To provide easly LMI stability conditions for (19), we choose the matrix structure X as:

X = diag ( X 1 X 2 X 3 ) (25)
According to ( 22), matrices X 1 , X 2 and X 3 are symmetric positive definite matrices.

By applying the Schur complement on (24), one can obtain:

  Q -X 0 * 0 -γ 2 I * X Āµµ XZ µ -X   < 0 (26)
Considering the matrices defined in ( 19), the mathematical developement of ( 26) leads to:

Ψ µ µ + φ 1 T φ 2 µ µ + φ 2 µ µ T φ 1 + φ 1 T φ 3 µ µ + φ 3 µ µ T φ 1 + φ 4 T φ 5 + φ 5 T φ 4 < 0 (27)
where φ 1 = ( 0 0 0 0 X 1 0 0 ), φ 2 µ µ = ( -B µ K µ 0 0 0 0 0 0 ), φ 3 µ µ = ( 0 -B µ K µ 0 0 0 0 0 ), φ 4 = ( 0 0 0 0 0 0 X 3 a 0 ), φ 5 = ( 0 0 0 ∆a 0 0 0 ) and Ψ µ µ is given in the next page To provide LMI conditions, consider the following bijective change of variables: γ = γ 2 , L 1 µ = X 2 H 1 µ and L 2 µ = X 3 H 2 µ . By applying lemma 1, ( 27) can be rewritten as:

Ψ µ µ +τ 1 φ 1 T φ 1 +τ -1 1 φ 2 µ µ T φ 2 µ µ +τ 2 φ 1 T φ 1 + τ -1 2 φ 3 µ µ T φ 3 µ µ + τ φ 4 T φ 4 + τ -1 φ 5 T φ 5 < 0 (28)
Considering τ 1 =τ 2 =1 and applying the Schur comple- ment on (28), thus the sufficient LMI conditions proposed in theorem 1 hold.

Case of faults approximated by a first order polynomial

Let us consider the faults occuring in the system are modeled by a first order polynomial as:

f i (k) = a i k + b i ( 29
)
where a i , b i ∈ R, for i = 1, ..., q.

In the same way as the first case, we define a, a i and ∆a i as follows:

   a = diag ( a 1 • • • a q ) a 0 = diag ( a 0,1 • • • a 0,q ) ∆a = diag ( ∆a 1 • • • ∆a q ) (30)
Let us consider that the uncertain part is bounded as:

(∆a) T ∆a ≤ δ (31)
where δ ∈ R q×q is a known diagonal positive definite matrix.

In this case, the fault estimation dynamics is given by:

e d (k + ) = -H 2 µ C µ e s (k) + Σ µ µ e d (k) + a (32)
The combination of ( 16), ( 17) and (32) leads to:

e (k + ) = Āµµ e (k) + E µ f (k) + P ( 33
)
where e (k) and Āµµ are defined in equation ( 19),

E = Ω µ 0 0 and P = 0 0 a . ϒ i j =                  Q 1 -X 1 0 0 0 * 0 0 0 0 * 0 0 Q 2 -X 2 0 0 0 * * 0 0 0 * 0 0 Q 3 -X 3 0 * * * 0 0 0 0 0 0 0 τ -1 λ -γI * 0 * 0 * 0 0 X 1 A i 0 -X 1 B i X 1 (B i -G i ) -X 1 0 0 0 0 0 0 0 X 2 A i -L 1 j C i X 2 G i -L 1 j W i 0 0 -X 2 0 0 0 0 0 0 L 2 j C i X 3 -L 2 j W i -X 3 0 0 -X 3 * 0 0 0 0 0 0 0 0 0 a 0 X 3 -τ -1 I 0 0 0 0 0 0 0 X 1 0 0 0 -2I 0 0 B i K j 0 0 0 0 0 0 0 0 -I 0 0 B i K j 0 0 0 0 0 0 0 0 -I                  Ψ µ µ =            Q 1 -X 1 0 0 0 * 0 0 0 Q 2 -X 2 0 0 0 * * 0 0 Q 3 -X 3 0 * * * 0 0 0 -γ 2 I * 0 * X 1 A µ 0 -X 1 B µ X 1 B µ -G µ -X 1 0 0 0 X 2 A µ -H 1 µ C µ X 2 G µ -H 1 µ W µ 0 0 -X 2 0 0 -X 3 H 2 µ C µ X 3 I -H 2 µ W µ -X 3 0 0 -X 3           
The main provided results are given in the following theorem 2.

Theorem 2: The state tracking error e p (k), the state and fault estimation errors e s (k) and e d (k) converge assymptotically to zero and the L 2 -gain from the faults to the errors e p (k), e s (k) and e d (k) is bounded by √ γ, if there exists matrices X 1 = X T 1 ≥ 0, X 2 = X T 2 ≥ 0, X 3 = X T 3 ≥ 0, K i , L 1 i and L 2 i and positive scalars γ, τ and ρ such that the following LMI are verified, for i = 1, 2, ..., r

Φ i j < 0 (34)
where Φ i j is given in the next page, with: 21), ( 22), ( 23) and ( 33), then following the same steps of the theorem 1 from ( 21) to (26), one can obtain:

Φ 1,1 = ρI + Q 1 -X 1 , Φ 2,2 = ρI + Q 2 -X 2 Φ 3,3 = ρI + Q 3 -X 3 , Φ 5,5 = -ρεI + τ -1 δ I Φ 6,4 i = X 1 (B i -G i ), Φ 7,2 i j = X 2 A i -L 1 j C i Φ 7,3 i j = X 2 G i -L 1 j W i and Φ 8,3 i j = X 3 -L 2 j W i Proof. Considering (
  ĀT µ µ X Āµµ + Q -X * * E T X Āµµ E T XE -γ 2 I * P T X Āµµ P T XE P T XP   < 0 (35)
To transform (35) to a feasable problem, we consider the following inequality ensuring the asymptotic convergence of the error dynamics to a ball of radius ε.

e (k) 2 2 ≥ εI ( 36
)
where ε is a knows small positive scalar.

The mathematical developpement of (36) leads to:

  e T (k) f T (k) I   T I 0 0 0 0 0 0 0 -εI e (k) f (k) I ≥ 0 (37)
By applying S-procedure lemma 2 on ( 35) and (37), one can obtain:

  ρI + Q -X 0 0 0 -γ 2 I 0 0 0 -ρεI   +   ĀT µ µ E T P T   X Āµµ E P < 0 (38)
By applying Schur complement on (38), this latter becomes

   ρI + Q -X * * * 0 -γ 2 I * * 0 0 -ρεI * X Āµµ XE XP -X    < 0 (39)
Following the same path as for the proof of theorem 1 from (27) to the end, thus the sufficient LMI conditions proposed in theorem 2 hold.

SIMULATION EXAMPLE

Let us consider the nonlinear T-S model ( 6) described by the following matrices and weighting nonlinear functions:

Φ i j =                     Φ 1,1 0 0 0 0 * 0 0 0 * 0 0 0 Φ 2,2 0 0 0 0 * * 0 0 * 0 0 0 Φ 3,3 0 0 * * * 0 0 0 0 0 0 0 -γI 0 * 0 0 0 0 0 0 0 0 0 0 Φ 5,5 0 0 * 0 0 0 0 X 1 A i 0 -X 1 B i Φ 6,4 i 0 -X 1 0 0 * 0 0 0 0 Φ 7,2 i j Φ 7,3 i j 0 0 0 -X 2 0 0 0 0 0 0 -L 2 j C i Φ 8,3 i j 0 X 3 a 0 0 0 -X 3 0 0 0 * 0 0 0 0 0 X 1 0 0 -2I 0 0 0 B i K j 0 0 0 0 0 0 0 0 -I 0 0 0 B i K j 0 0 0 0 0 0 0 0 -I 0 0 0 0 0 0 0 0 X 3 0 0 0 -τ -1 I                     A 1 = -0.5 0.1 -1 -1 , A 2 = 0 0.2 -0.45 -0.7 , B 1 = 0.4 0.5 , B 2 = 0.6 0.4 , G 1 = 0.2 0.4 , G 2 = 0.5 0.5 , C 1 = ( 0.2 0 ), C 2 = ( 0.4 0.1 ), W 1 = -0.3, W 2 = -0.4, the nominal input signal u(k) = 0.5cos(sin(0.1k)0.1k).
The activation nonlinear functions are depending on the known nominal input u(k), they are given by: µ

1 (u (k)) = 1 -tanh (0.5 -u (k)) and µ 2 (u (k)) = 1 -µ 1 (u (k)).
Let us consider that the fault affecting the system at 9 ≤ k ≤ 17 is given by: f (k) = e 0.11k-10 (40)

The observer and the controller are designed for a 0 = e 0.1 leading to ∆a 0 = e 0.01 . For simulation, the parameter λ defined in ( 14) is chosen equal to 1.3.

Remark 2: To show the robustness of the synthesized FTC controller and observer, the parameter value of the fault acting in the system are augmented. indeed, the following results given by Figures 2 to 6 proaches has been illustrated by considering a numerical example. Indeed, the fault occurring in the system has been taken into account by the synthesized FTC controller allowing to ensure the tracking between the healthy system states and the faulty ones.

Figure 2 :

 2 Figure 2: Reference model states vs. faulty system ones with FTC

Figure

  Figure 4: Fault and its estimation

Figure 5 :Figure 6 :

 56 Figure 5: Nominal and FTC control input signals


Femke ; Hannes

Jeroen Van Houdt

Oliver W Quarrell

Martin Poot

Ron Hochstenbach

Jean-Pierre Fryns

Joris R Vermeesch
email: joris.vermeesch@med.kuleuven.be

Telomere Healing Following DNA polymerase Arrest Induced Breakages is likely the Main Mechanism Generating Chromosome 4p Terminal Deletions

Keywords: terminal deletions, telomere healing, telomerase, DNA polymerase arrest sites, Wolf-Hirschhorn syndrome, Chromosome 4p terminal deletions, telomere healing, telomerase, DNA polymerase arrest sites, Wolf-Hirschhorn syndrome, Chromosome 4p

Constitutional developmental disorders are frequently caused by terminal chromosomal deletions. The mechanisms and/or architectural features that might underlie those chromosome breakages remain largely unexplored. Since telomeres are the vital DNA protein complexes stabilizing linear chromosomes against chromosome degradation, fusion and incomplete replication, those terminal deleted chromosomes acquired new telomeres either by telomere healing or by telomere capture. To unravel the mechanisms leading to chromosomal breakage and healing, we sequenced nine chromosome 4p terminal deletion boundaries. A computational analysis of the breakpoint flanking region, including 12 previously published pure terminal breakage sites, was performed in order to identify architectural features that might be involved in this process. All terminal 4p truncations were likely stabilized by telomerase mediated telomere healing. In the majority of breakpoints multiple genetic elements have a potential to induce secondary structures and an enrichment in replication stalling site motifs were identified. These findings suggest DNA replication stalling induced chromosome breakage during early development is the first mechanistic step leading towards terminal deletion syndromes.

Introduction

Terminal chromosomal deletions are the most common class of subtelomeric abnormalities and are often associated with mental retardation and multiple congenital anomalies [START_REF] Ballif | The clinical utility of enhanced subtelomeric coverage in array CGH[END_REF]. The most frequent terminal deletion syndromes include the 1p36 deletion syndrome (MIM 607872), the 4p terminal deletion leading to Wolf-Hirschhorn syndrome (MIM 194190), the 5p terminal deletions causing Cri-du-chat syndrome (MIM 123450), the 16p terminal deletion leading to alpha thalassaemia (MIM 141750), 9q34 deletion syndrome (MIM 610253) and the 22q terminal deletion syndrome (MIM 606232). Recent studies show that many apparent simple terminal deletion breakpoints have proved difficult to sequence [START_REF] Rooms | Diverse chromosome breakage mechanisms underlie subtelomeric rearrangements, a common cause of mental retardation[END_REF][START_REF] Yatsenko | Molecular mechanisms for subtelomeric rearrangements associated with the 9q34.3 microdeletion syndrome[END_REF][START_REF] Angelo | Further delineation of nonhomologous-based recombination and evidence for subtelomeric segmental duplications in 1p36 rearrangements[END_REF]. In a recent study on chromosome 9q deletions, only 2 out of 14 apparently pure terminal deletions could be sequenced [START_REF] Yatsenko | Molecular mechanisms for subtelomeric rearrangements associated with the 9q34.3 microdeletion syndrome[END_REF] . The inability to analyze the breakpoints may relate to the complexity of the genomic sequence at the breakpoint such as inverted and tandem duplications or intervening sequence of unknown or ectopic origin [Flint et al., 1994;Varley et al., 2000;Ballif et al., 2003;[START_REF] Ballif | The clinical utility of enhanced subtelomeric coverage in array CGH[END_REF][START_REF] Rooms | Diverse chromosome breakage mechanisms underlie subtelomeric rearrangements, a common cause of mental retardation[END_REF][START_REF] Gajecka | Monosomy 1p36 deletion syndrome[END_REF]. Another reason for the inability to identify precisely the genomic location of the telomeric breakpoint is its location within repetitive sequences [D' Angelo et al., 2009]. Overall, relatively few breakpoints have been analyzed at the sequence level and systematic studies mapping larger series of broken chromosomes are nearly non-existent.

Moreover, the finding of mainly pure terminal deletions at chromosome 16pter versus more complex rearrangements at 1p, 22q and 9qter might imply that different chromosomes Apparently simple 4p terminal deletions are the most frequent cause of WHS (55%-75%) [START_REF] Lurie | The Wolf-Hirschhorn syndrome[END_REF][START_REF] Shannon | An epidemiological study of Wolf-Hirschhorn syndrome: life expectancy and cause of mortality[END_REF][START_REF] South | Comprehensive analysis of Wolf-Hirschhorn syndrome using array CGH indicates a high prevalence of translocations[END_REF][START_REF] Zollino | On the nosology and pathogenesis of Wolf-Hirschhorn syndrome: genotype-phenotype correlation analysis of 80 patients and literature review[END_REF]. Thus far, no chromosome 4 terminal deletions breakpoints have been mapped at sequence level. Here, we sequence systematically 9 consecutive terminal deletions of the short arm of chromosome 4 and explore the genomic architecture surrounding the breakpoints. In contrast to studies at other chromosomes, we were able to map and sequence all 9 telomere junction sites. Our data suggest that all telomeres were added de novo to the chromosome specific sequence.

Furthermore, analysis of the breakage sites provide evidence that specific sequence motifs potentially induce chromosomal breaks leading up to terminal deletion syndromes.

Materials and Methods

Human subjects

Patients with developmental anomalies and mental retardation were ascertained over a period of five years of screening. Blood samples were collected after informed consent. Seven patients were seen in Leuven (BE), one in Sheffield (UK) and one in Utrecht (NE). Except for the latter patient were all patients diagnosed with Wolf-Hirschhorn syndrome. The phenotype and genotype of the latter patient have been described elsewhere [START_REF] Engbers | Wolf-Hirschhorn syndrome facial dysmorphic features in a patient with a terminal 4p16.3 deletion telomeric to the WHSCR and WHSCR 2 regions[END_REF]. All patients, but one, had normal high resolution G-banded chromosome analysis and were ascertained through either abnormal Fluorescent in situ hybridization (FISH) results using a commercially available WHS probe (Cytocell) [Van Buggenhout et al., 2004] and/or 1 Mb BAC array following the protocol as described elsewhere [START_REF] Menten | Emerging patterns of cryptic chromosomal imbalance in patients with idiopathic mental retardation and multiple congenital anomalies: a new series of 140 patients and review of published reports[END_REF]. Unbalanced 4p translocations as well as interstitial 4p deletion carriers were excluded.

Ultra-high resolution oligonucleotide mircoarray

A tiling path oligonucleotide microarray containing 385.000 probes (Roche NimbleGen systems) covering the region between 1.0 Mb to 11.8 Mb on chromosome 4p was custom designed. Both olfactory receptor gene clusters, located respectively around 3.9 Mb and 8.8 Mb, were avoided from the design. The average probe resolution was 30 bp. In addition, for one patient with the largest deletion, a HD2 whole genome oligonucleotide microarray with 2.1 million probes was used (Roche NimbleGen systems). For this array the average probe resolution was 1.5 kb.

The labeling, hybridization, washing and scanning were performed according the manufacturer's instructions (Roche NimbleGen systems). Data extraction and calculation was done via NimbleScan (Roche NimbleGen systems). The log2 ratios were calculated following

Quantitative PCR (qPCR)

qPCR primers were designed with Universal ProbeLibrary (Roche Applied Science).

Primers were chosen free from any single nucleotide polymorphisms or repeats using repeatmasker provided by the UCSC browser (http://genome.ucsc.edu/). qPCR was performed using the lightcycler 480 instrument (Roche Applied Science) working in a total volume of 15 µl including 7.5 µl of mastermix, 2.5 µl of primermix (2.5 µM) and 50 ng template DNA. The following amplification conditions were used: 95°C for 5 min, and 40 cycles at 95°C for 10 s and 60°C for 20 s. After the amplification protocol, a melting curve was obtained at 95°C for 30 s, 60°C for 30 s and 95°C for a continuous mode (5 acquisitions per °C) and finally cooling down to 40°C for 30 s. Next, data was analyzed with Excel (Microsoft) according to the comparative ddCt method (Sequence Detection System bulletin 2 [Applied Biosystems]).

Telomere anchored PCR and breakpoint sequencing

A unique primer, located in the normal copy region, was chosen per breakpoint in combination with a telomere specific primer (5'TATGGATCCCTAACCCTGACCCTAACCC3') to amplify across the junction. A total volume of 50 µl contained platinum Taq polymerase (Invitrogen), 1.5 mM MgCl 2 , 200 µM of dNTPs, 0.25 µM forward and reverse primer (Eurogentec) and 100 ng of template DNA. The PCR program is defined as follows: 95°C-5 min, 35 cycles of (95°C-30s, 58°C-30s, 72°C-1.30 min) with a final extension of 72°C-10 min. After amplification, the PCR products were analyzed on a1% agarose gel and sequenced on an ABI 3130 automated capillary DNA sequencer using the BigDye Terminator v. The Blat tool provided by the UCSC browser (http://genome.ucsc.edu/cgibin/hgBlat?command=start) [START_REF] Kent | BLAT--the BLAST-like alignment tool[END_REF] was used to map the obtained sequences to the human genome for determining the exact location of the breakpoint. Furthermore, we collected from the literature, 14 additional pure terminal breakpoints that have been mapped at the sequence level including seven breakpoints at chromosome 16p [Flint et al., 1994;Horsley et al., 2001], one breakpointcluster at chromosome 22q containing three terminal deletion breakpoints [Wong et al., 1997;Bonaglia et al., 2006], two at chromosome 9q34 [START_REF] Yatsenko | Molecular mechanisms for subtelomeric rearrangements associated with the 9q34.3 microdeletion syndrome[END_REF] and two single breakpoints at respectively chromosomes 7q [Varley et al., 2000] and 1p36 [Ballif et al., 2003]. To exclude a bias during our statistical analysis when using all three terminal breakpoints at chromosome 22q, only one breakpoint flanking region [Wong et al., 1997] was taken into account. In total, 12 terminal breakpoints were used to complete our analysis (Supplementary table 1).

All breakpoint flanking regions for further in silico analyses were defined by 150 bp including 75 bp flanking both sides of the breakpoint. In order to assess whether the breakpoint regions (4p and others) were enriched for certain genomic features, we randomly selected 500 sequences of 150 bp from the human genome (build 18), distributed over the different chromosomes and avoiding gaps and centromeric regions. Different online tools were used (online resources) to analyze these 150 bp random human sequences and the breakpoint sequences surrounding each junction (including 75 bp at both sides of the exact breakpoint).

RepeatMasker (http://www.repeatmasker.org/) (Smit, AFA, Hubley, R & Green, P.

RepeatMasker Open-3.0) was used to identify the highly repetitive sequences like longinterspersed nuclear elements (LINEs), Short-interspersed tandem nuclear elements (SINEs) (http://bioinfo.cgrb.oregonstate.edu/zDNA/) [START_REF] Champ | Distributions of Z-DNA and nuclear factor I in human chromosome 22: a model for coupled transcriptional regulation[END_REF] and QGRS (http://bioinformatics.ramapo.edu/QGRS/analyze.php) [START_REF] Kikin | QGRS Mapper: a web-based server for predicting Gquadruplexes in nucleotide sequences[END_REF]. REPuter (http://bibiserv.techfak.uni-bielefeld.de/reputer/submission.html) [START_REF] Kurtz | REPuter: the manifold applications of repeat analysis on a genomic scale[END_REF] allows identifying different repeat types, we recorded direct, mirror, complementary and inverted repeats. The presence of known motifs was analyzed with Fuzznuc (http://inntemp.weizmann.ac.il/cgi-bin/emboss/fuzznuc) (EMBOSS package, [START_REF] Rice | EMBOSS: the European Molecular Biology Open Software Suite[END_REF]). The motifs presented by Abeysinghe and co-workers were used for further analysis [START_REF] Abeysinghe | Translocation and gross deletion breakpoints in human inherited disease and cancer I: Nucleotide composition and recombination-associated motifs[END_REF]. Melina (http://melina2.hgc.jp/public/index.html) [START_REF] Poluliakh | Melina: motif extraction from promoter regions of potentially co-regulated genes[END_REF][START_REF] Okumura | Melina II: a web tool for comparisons among several predictive algorithms to find potential motifs from promoter regions[END_REF] was used for identifying novel common motifs and Weblogo (http://weblogo.berkeley.edu/logo.cgi) [START_REF] Crooks | WebLogo: a sequence logo generator[END_REF] was used to generate sequence logo's which are the graphical representation of a nucleotide multiple sequence alignment.

Weblogo enabled us to screen for the presence of common nucleotides at certain positions from the exact breakpoint.

For all features we recorded the number of observations in the breakpoint sequences (4p and others from literature) and the random human sequences (Table 1). In both sets of sequences we counted the number of sequences with at least one hit for a given feature (Table 1). We performed a Fisher exact test in order to determine whether the occurrence of a feature was significantly different for the breakpoint sequences and the random human sequences.

Results

Sequence analysis of terminal breakpoints

DNA of 8 patients were hybridized on specific tiling path oligonucleotide microarrays of chromosome 4p which enabled further fine mapping of the breakpoints to an approximately 1A). The breakpoint in one larger terminal deletion, mapped at approximately 15 Mb according the 1 Mb BAC array, was further characterized using a whole genome HD2 NimbleGen array. This breakpoint is mapped to an approximately 12 kb interval. Since terminal deletions are often reported to be associated with submicroscopic terminal duplications [START_REF] Zuffardi | Inverted duplications deletions: underdiagnosed rearrangements??[END_REF] the probes flanking the contiguous stretch of probes with reduced intensity ratios due to the terminal deletion were inspected for increased intensity ratios. No increased intensities were identified near the breakpoints thus excluding the presence of inverted deletion duplications. Nevertheless, in breakpoint n° 6, 7 and 8, a group of probes around 4 Mb, more specifically between 3,854,200 bp and 4,322,200 bp, showed an apparently normal copy number within a larger terminal deleted region (Fig. 1A).

The reason for this elevation in log 2 values is most likely the presence of repetitive regions flanking the olfactory receptor gene clusters which have multiple copies scattered within the genome. Hence, a deletion of a few copies will not be detected and will appear as normal copy number value.

To validate the results of the high resolution arrays and further fine map the location of the breakpoint, a series of qPCRs were performed in the purported breakpoint region.

Quantitative PCR using different primer sets showed a contiguous stretch present as a single allele while a flanking set of primer pairs showed the presence of two alleles and thus no additional complex rearrangements flanking the breakpoint could be registered. The region inbetween those primer pairs with altered copy number contains the terminal breakpoint. Once the breakpoint region was refined down to 200-800 bp, a breakpoint spanning PCR was designed. To this end, telomere anchored PCR was performed using a telomere specific primer combined with the reverse primer from the first primer pair with a diploid copy number state. A positive control without a 4p deletion was performed in parallel to distinguish the correct PCR product from the other bands as seen in the positive control. All amplicons F o r P e e r R e v i e w 10 were smaller than 600 bp and subsequently sequenced (Fig. 1B). In all sequenced products, 4 to 27 TTAGGG hexamers were obtained (Fig. 1B). The chromosome specific sequences flanking the telomeric repeats were aligned with the human genome reference sequence and revealed the immediate proximity of the telomeric repeats to the chromosome specific sequence in all fragments (Fig. 1C). De novo telomere addition to a DNA fragment is thought to be mediated by telomerase. Telomerase mediated addition of telomeric repeats onto a DNA fragment is stimulated by the presence of a telomeric repeat sequence to which the RNA template of telomerase can bind [START_REF] Muller | New telomere formation after developmentally regulated chromosomal breakage during the process of chromatin diminution in Ascaris lumbricoides[END_REF][START_REF] Melek | Chromosome healing: spontaneous and programmed de novo telomere formation by telomerase[END_REF][START_REF] Bottius | Plasmodium falciparum telomerase: de novo telomere addition to telomeric and nontelomeric sequences and role in chromosome healing[END_REF]]. Hence, we determined whether such telomeric repeats are located at or near the telomeric breakpoint. In 7/9 junctions we observed, in frame with the TTAGGG repeat, microhomology of 2 to 5 bp with the RNA component of telomerase (Fig. 1C and Table 2). In addition to terminal breakages at chromosome 4p16, microhomology in frame with the telomere sequence was also detected in 9/12 breakpoint flanking sequences previously reported.

The enzymatic part of telomerase also known as TERT contains mainly 3 functional domains: the active site located at the C-terminus, the TERC binding site is located centrally, and the N-terminus interacts with the ssDNA end. This N-terminal anchor site will determine the efficiency by which telomerase can conduct his function. Recently, Sealy and colleagues showed a ssDNA length and sequence dependency of the telomerase enzyme within a short template sequence. Moreover the interaction of the N-terminus preferentially interacts with oligonucleotides of minimal 13 nt in length and a G-rich character at certain positions in the oligonucleotide [Sealey et al., ]. In order to characterize such a sequence specificity flanking the breakpoint at the junctions that could define the interaction with telomerase, a 16 bp fragment flanking each breakpoint was analyzed via Weblogo. No common nucleotides at certain positions flanking the breakpoint could be detected on chromosome 4p16 or others.

Low copy repeats and highly repetitive elements

The mechanism(s) generating terminal chromosomal deletions remain(s) largely unknown. If the genomic architecture influences the generation of chromosomal breaks leading up to terminal deletions, specific sequences or DNA structures might be present near the deletion breakpoint both at chromosome 4p16 and others. To investigate this possibility, in silico analysis was conducted on the 150 bp breakpoint junction interval defined by 75 bp on either side of the breakpoint. The breakpoint flanking sequences were first analyzed for the presence of low copy repeats (LCRs) and highly repetitive elements. LCRs are mediators of non-allelic homologous recombination [START_REF] Stankiewicz | Genome architecture, rearrangements and genomic disorders[END_REF], and could be involved in chromosome instability leading up to terminal deletions. No LCRs were detected in the nine 4p deletion breakpoint sequences, whereas six of the nine contained highly repetitive elements. In 3 breakpoint flanking sequences we observed long-interspersed nuclear elements (LINEs), in two other fragments short-interspersed nuclear elements (SINEs) were present and one sequence contained a simple tandem repeat (STR)(See Table 2). In light of the 12 previously reported breakpoints, in total 4 breakpoint flanking sequences contained repetitive elements including one breakpoint flanking sequence contained a SINE element, the other two contained STRs and one breakpoint flanking sequence contained a LCR.

Non-B-conformations genetic elements

Certain genetic elements in the genome can adopt non-B conformations such as triplexes, tetraplexes, cruciforms, slipped hairpin structures and left-handed Z-DNA. Those conformations can hamper DNA replication and as a consequence may result in a break [START_REF] Bacolla | The involvement of non-B DNA structures in gross chromosomal rearrangements[END_REF][START_REF] Wells | Non-B DNA conformations, mutagenesis and disease[END_REF]. To investigate the potential role of non-B structural breakpoint flanking sequence contained an inverted repeat potentially adopting cruciform conformations and is proposed to be responsible for genome rearrangements and/or gross deletions [START_REF] Wells | Non-B DNA conformations, mutagenesis and disease[END_REF]. 12 other previously reported simple terminal deletions were analyzed in parallel with those on chromosome 4p. We observed direct, inverted and mirror repeats from at least 8 bp in respectively 4, 3 and 2 breakpoint flanking sequences. To investigate whether the finding of a certain element is significant or accidental, similar in silico analysis was performed for randomly chosen sequences extracted from the human genome and subsequently compared to the combined number at terminal breakpoints both at chromosome 4p and others. An overrepresentation of individual structural elements such as mirror repeats (p=0.106) and alternating purine/pyrimidine sequences (p=0.118) leading to respectively triplex structures and left-handed Z-DNA were identified (Table 1 andFig and 26 bp (Table 1 and Fig. 2).

Known sequence motifs

Besides the complex architectural features and repeats inducing altered secondary structures, certain sequence motifs are also known to predispose to breakage.

Overrepresentation of certain sequence motifs are found in the vicinity of the converted region which can promote recombination not by their sequence per se but because their potential to form non-B conformations leading to a break [START_REF] Chuzhanova | Gene conversion causing human inherited disease: evidence for involvement of non-B-DNA-forming sequences and recombination-promoting motifs in DNA breakage and repair[END_REF]. We evaluated the presence of known motifs within the breakpoint flanking sequences on both the 5' to 3' and the 3' to 5' strands. In total, 17/40 motifs were obtained from the nine breakpoint flanking sequences. Moreover, 3 of those motifs such as the deletion hotspot consensus (TGRRKM), the DNA polymerase arrest site (WGGAG) and the Murine parvovirus recombination hotspot (CTWTTY) were found to be frequent (> 50%), however not necessarily simultaneously present. Amongst the nine breakpoint flanking sequences, 5 contained such a deletion hotspot recombination hotspot motif (Table 1 1). Comparison using 500 sequences generated at random, not derived from the human genome sequence, showed statistically significant p-values for the DNA polymerase arrest site (p=0.026) and Ig heavy chain class switch repeat 5 (p=0,01).

However the same analysis with 500 randomly generated sequences from the human genome, showed that these particular motifs such as the DNA polymerase arrest site (p=0.119) and Ig heavy chain class switch repeats 5 and 3 (p=0.102 and 0.08) were overrepresented, although not statistically significant, in the group of in total 21 terminal breakpoints including 9 on chromosome 4p and 12 other terminal breakpoints previously reported.

Novel sequence motifs

Alternatively, the identification of novel common motifs was performed using Melina.

Among the breakpoint flanking sequences at chromosome 4p, one particular motif of 5 bp (KGGMA) was commonly found in all breakpoint flanking sequences. This novel motif was tested in the 12 previously reported breakpoint junctions. In 9/12 breakpoint flanking sequences, 19 hits were obtained in both sense and antisense direction. Investigation of this novel motif at 500 randomly generated sequences of the human genome, revealed no statistical overrepresentation or an enrichment within the breakpoint flanking sequences.

Discussion

To unravel the mechanisms leading up to terminal deletion syndromes, we mapped and sequenced the breakpoints of nine terminal 4p deletions. We demonstrate that all broken chromosomes contain pure TTAGGG repeats flanking chromosome specific sequence.

Healing of broken chromosomes

Broken chromosomes can acquire a telomere sequence either by capturing the telomere or by telomerase mediated telomere healing. There are two lines of evidence that imply that all chromosome 4p telomeres analyzed were healed by telomerase [START_REF] Ballif | The clinical utility of enhanced subtelomeric coverage in array CGH[END_REF]. In vitro studies have demonstrated that telomerase preferably adds a segment of the telomeric hexamer repeat unit onto a DNA fragment at its 3' end. A microhomology in frame with the telomere repeat of 2-5 bp with the telomerase template in 7/9 (77.7%) breakpoints was demonstrated. This result is concordant with previous published terminal breakpoints at other chromosomes in which this type of microhomology is detected at 9/12 (75%) breakpoints [Flint et al., 1994;Wong et al., 1997;Varley et al., 2000;[START_REF] Yatsenko | Molecular mechanisms for subtelomeric rearrangements associated with the 9q34.3 microdeletion syndrome[END_REF]. When a telomere is captured from another chromosome, usually not only the telomere but also subtelomeric sequences are captured [START_REF] Meltzer | Telomere capture stabilizes chromosome breakage[END_REF]. No such subtelomeric sequences were detected. In addition, telomeric repeats flanking the unique euchromatic sequence have been replicated during multiple cell divisions and thus acquire at the most proximal location many permutations of the hexameric TTAGGG sequence such as e.g. TTGGG, TGAGGG F o r P e e r R e v i e w 16 and TAGGG. Hence, when a telomere sequence is captured from another chromosome, those variants of the TTAGGG sequence would be present. On the other hand, when the telomere is acquired via telomerase only exact TTAGGG hexamers will be present. In all our junctions fragments exact TTAGGG repeats were flanking the breakpoint junction. Hence, all 4p broken chromosomes are likely stabilized via telomerase promoted repair and thus suggest a telomere healing event.

Indirectly, we can deduce when during development breaks arise. First, it is known that telomerase activity is low in mature oocytes and spermatozoa, but highly expressed from the blastocyst stage onwards [START_REF] Wright | Telomerase activity in human germline and embryonic tissues and cells[END_REF][START_REF] Bekaert | Telomere biology in mammalian germ cells and during development[END_REF][START_REF] Liu | Telomere lengthening early in development[END_REF]. If a breakage occurs in the oocyte, spermatocyte or in the zygote other mechanisms of DSB repair will be activated rather than repair via telomerase leading possibly to other types of rearrangements [START_REF] Ballif | Translocation breakpoint mapping and sequence analysis in three monosomy 1p36 subjects with der(1)t(1;1)(p36;q44) suggest mechanisms for telomere capture in stabilizing de novo terminal rearrangements[END_REF]. Otherwise, when a break occurs in early development, telomerase is present and will interfere with the DSB repair processes such as non allelic homologous recombination (NAHR) and non homologous end joining (NHEJ). Moreover, it has recently been shown that such a de novo addition via telomerase is negatively regulated by DNA damage signaling preventing aberrant healing of broken DNA ends by telomerase. This shows that healing via telomerase is subordinate to both non allelic homologous recombination (NAHR) and non homologous end joining (NHEJ) [Makovets and Blackburn, 2009]. Hence, telomeric deletions are often stabilized as derivative chromosomes rather than pure deletions [START_REF] South | Comprehensive analysis of Wolf-Hirschhorn syndrome using array CGH indicates a high prevalence of translocations[END_REF]. Secondly, our results show an extensive microhomology at the terminal breakpoints, suggesting an important role in telomerase dependent repair. Finally, it was recently shown that genomes of early cleavage stage embryos are prone to chromosome instability and that a high number of blastomeres carry chromosomal deletions [START_REF] Vanneste | Chromosome instability is common in human cleavage-stage embryos[END_REF]. Since chromosome instability is high in early cleavage stage embryos and since telomerase is most active during early development, it seems likely that terminal deletions Besides that approximately 76% of the terminal deletion breakpoints contained microhomology with the RNA template of telomerase, a small proportion of the breakpoints did not contain microhomology. It is known that binding and elongation of the telomerase enzyme involves base pairing with the telomerase RNA template and the substrate [START_REF] Morin | Recognition of a chromosome truncation site associated with alphathalassaemia by human telomerase[END_REF][START_REF] Harrington | Telomerase primer specificity and chromosome healing[END_REF]. In vitro studies have recently shown that binding of telomerase to a substrate also depends on the G-rich character at certain positions in the substrate [START_REF] Sealey | The Nterminus of hTERT contains a DNA-binding domain and is required for telomerase activity and cellular immortalization[END_REF]. According to our data, no obvious G-rich position nearby the breakpoint could be defined in a 16 bp sequence fragment flanking the terminal breakpoints without any microhomology. This suggests that in vivo telomerase can elongate broken chromosomes without having a complement to the RNA template nor a certain enrichment for a nucleotide near the healing site.

Mechanism of chromosomal breakage

Sequences such as repeats, non-B structural elements or certain motifs can predispose to DNA breakage [START_REF] Vilenchik | Endogenous DNA double-strand breaks: production, fidelity of repair, and induction of cancer[END_REF][START_REF] Abeysinghe | Translocation and gross deletion breakpoints in human inherited disease and cancer I: Nucleotide composition and recombination-associated motifs[END_REF][START_REF] Wang | Non-B DNA structure-induced genetic instability[END_REF][START_REF] Chuzhanova | Gene conversion causing human inherited disease: evidence for involvement of non-B-DNA-forming sequences and recombination-promoting motifs in DNA breakage and repair[END_REF][START_REF] Branzei | Maintaining genome stability at the replication fork[END_REF]. Here, the enrichment of particular motifs was observed at terminal breakpoints. A DNA polymerase arrest site was identified to impede the progress of DNA polymerase alfa at phiX174 DNA resulting in stalling of the replication fork [ [START_REF] Weaver | Specific sequences in native DNA that arrest synthesis by DNA polymerase alpha[END_REF]. Replication arrest actually stimulates slippage and can trigger recombination and thus the finding of a replication arrest motif can be a key factor in genome instability [START_REF] Stary | Molecular analysis of DNA junctions produced by illegitimate recombination in human cells[END_REF][START_REF] Hyrien | Mechanisms and consequences of replication fork arrest[END_REF][START_REF] Michel | Rescue of arrested replication forks by homologous recombination[END_REF]. In E.coli a mechanistic link between a replication arrest site and homologous [START_REF] Seigneur | RuvAB acts at arrested replication forks[END_REF]]. In about 50% of the breakpoint flanking sequences such an arrest site was present. Likewise, an overrepresentation of DNA polymerase arrest motifs (p=0.01) are found at sequences flanking the gene conversion region [START_REF] Chuzhanova | Gene conversion causing human inherited disease: evidence for involvement of non-B-DNA-forming sequences and recombination-promoting motifs in DNA breakage and repair[END_REF]. Furthermore, both Ig heavy chain switch class repeats 3 and 5 were enriched at our terminal breakpoints. These repeats mediate a process of nonhomologous recombination that introduces variation in the constant region of an immunoglobulin heavy chain gene. Those switch repeats have been noted at translocation breakpoints associated with a number of hematological malignancies [START_REF] Fenton | Follicular lymphoma with a novel t(14;18) breakpoint involving the immunoglobulin heavy chain switch mu region indicates an origin from germinal center B cells[END_REF]. Moreover, repeat 5 was also shown to be significantly overrepresented within close proximity at translocation breakpoints (p=0.05) [START_REF] Abeysinghe | Translocation and gross deletion breakpoints in human inherited disease and cancer I: Nucleotide composition and recombination-associated motifs[END_REF].

Besides the specific motifs, repetitive sequences inducing altered DNA conformations are frequent within the surroundings of the rearrangement breakpoints [START_REF] Bacolla | The involvement of non-B DNA structures in gross chromosomal rearrangements[END_REF]. Such elements can elevate the rate of single-strand lesions, and hence contribute to their conversion to double-strand breaks [START_REF] Vilenchik | Endogenous DNA double-strand breaks: production, fidelity of repair, and induction of cancer[END_REF][START_REF] Patel | PKD1 intron 21: triplex DNA formation and effect on replication[END_REF]. Within the terminal breakpoint flanking sequences of both chromosome 4p and others, direct, mirror, inverted and complementary repeats were found even as puridine/pyrimidine and oligo(G)n tracts. Interestingly, the fragments were likely to induce more left-handed Z-DNA (p=0.118) and triplex (p=0.106) structures than what we would expect from the random genome. Other studies with similar findings conclude that both structures are significantly associated with the site of DNA breakage at translocation junctions [START_REF] Abeysinghe | Translocation and gross deletion breakpoints in human inherited disease and cancer I: Nucleotide composition and recombination-associated motifs[END_REF][START_REF] Bacolla | The involvement of non-B DNA structures in gross chromosomal rearrangements[END_REF].

In addition, the presence of repetitive elements including high and low copy repeats might play a role in stimulating the formation of non-recurrent breakpoints [START_REF] Gu | Mechanisms for human genomic rearrangements[END_REF].

No low copy repeats were identified at or near the breakpoints whereas highly repetitive elements were abundant. 6/9 breakpoints are embedded within a repetitive element most often 2). In addition, 4/12 pure terminal breakpoints extracted from the literature are within segmental duplications, simple tandem repeats or SINEs (Table 1). Overall, the repetitive content of the breakpoint flanking sequences is heterogeneous and the fragments do not per se contain LCR nor repetitive elements. Hence, in our dataset, 50% of the breakpoints associate with LCR or highly repetitive sequences which are comparable with that of the human genome and thus there is not strong evidence of the contribution of those elements in the generation of breakages.

The relatively high frequency of DNA polymerase arrest sites at the breakpoint flanking sequences suggests one plausible mechanism of DNA breakage following DNA replication stalling. The presence of other structures rendering the DNA prone to DNA breakage might be witnesses from other breakage mechanisms. It is well known that dicentric chromosomes can be formed (either during meiosis or during mitosis). Following mitosis, the dicentric chromosome would break leading to two daughter cells having either a deleted 4p or an inv dup del (4p) and it can be expected that the breaks would occur at the "weakest" loci [START_REF] Zuffardi | Inverted duplications deletions: underdiagnosed rearrangements??[END_REF]. No inv del dupl (4p) were detected during our study. Another mechanism that might induce breakages is homologous recombination intermediates. Larger groups of terminal deletions need to be sequenced to determine the involvement of specific motifs in the breakage process and to discriminate whether the occurrence of different motifs might be relics of different mechanisms. *Not included in the analysis, because of similar locus as described previously by Wong and colleagues [START_REF] Wong | Molecular characterization of a 130-kb terminal microdeletion at 22q in a child with mild mental retardation[END_REF]. Hum Mol Genet. 2009;18(11):1924-36

F o r P e e r R e v i e w

  Despite their clinical importance, it remains largely unknown when during development those terminal deletions arise, what the mechanisms are causing the chromosomal breakages leading up to those terminal deletion syndromes or which mechanisms rescue the broken chromosomes.

  mechanisms leading to the chromosome breakage and subsequent rescue.

6 the

 6 SegMNT algorithm. A 10x average window was used for breakpoint determination and the visualization was done via SignalMap V1.9 (Roche NimbleGen systems).

  (LTRs) and simple tandem repeats (STRs). Non-B structural elements including left-handed DNA and tetraplex structures were detected by respectively ZHUNT

  the sequence surrounding the terminal breakpoint, different applications such as REPuter were used to investigate the presence of direct, inverted, complementary, mirror repeats and ZHUNT was used to search for left handed Z-DNA conformations. In total, 6/9 (67%) breakpoint flanking sequences on chromosome 4p16 contained one or more non-B structural elements. Moreover, direct repeats can potentially form slipped hairpin structures which in turn may trigger the rearrangement. Multiple direct repeats of minimal 8 bp were found within one breakpoint flanking sequence. Mirror repeats of minimal 8 bp, potentially leading to triplex structures, were detected in 2 breakpoint flanking sequences and complementary repeats were identified in 3 breakpoint flanking sequences. Finally, 1

  . 2). Although those elements were often seen in breakpoint flanking sequences, their p-values did not reach statistical significance (p<0.05) upon frequency determination.Furthermore, the local flipping of small regions of right-handed B-DNA to left-handed Z-DNA in the sequence requires negative super coiling making the sequence prone to breakage. This B to Z transition is strongly favored in alternating purine/pyrimidine sequences and can determined. ZHUNT analysis predicted at chromosome 4p in one breakpoint flanking sequence n° 4 a 15 bp sequence (TGGTGCGTGCCCGTA) 21 bp away from the breakpoint inducing a left-handed structure. Equally, within the group of 12 previously reported breakpoint flanking sequences, one contained a 15 bp sequence (AAGTGTGCACGCTCG) 56 bp proximal to the breakpoint (Fig.2). Finally, oligo(G)n tracts were identified which may form tetraplex structures potentially inducing breakage. Among nine breakpoint flanking sequences on chromosome 4p, 4 different oligo(G) n tracts of 15 to 29 bp were identified in three breakpoint flanking sequences, whereas in the 12 other breakpoint flanking sequences 7 contained at least one oligo(G)n repeat with a size varying between 11

  previously reported for 12 pure terminal breakpoints at other chromosome arms. Microhomology with the TTAGGG repeat was found in 16/21 (76.2%) terminal breakpoints. Furthermore, we analyzed the breakpoint flanking region for the presence of repeats, potential genomic architectural elements and sequence motifs that might ultimately lead to chromosome breakage. Herein, we demonstrate an enrichment of DNA polymerase arrest sites and non-B structural elements that might impede the progression of DNA polymerase during DNA replication. Hence, our data suggest that DNA polymerase pausing may lead to chromosomal breaks and those breaks are subsequently healed by telomerase.

  human development. This implies that the origin of generation and stabilization occurs during mitosis rather than meiosis and thus, it seems likely that the broken 4p chromosomes are generated and stabilized during early embryonic development.

Figure 1 .

 1 Figure 1. Overview of 9 terminal breakpoints at chromosome 4p16. A) Results of a targeted

Figure 2 .

 2 Figure 2. Overview of all the elements potentially leading towards a non-B conformation or

  Figure 1 254x190mm (96 x 96 DPI)

  A., Brundage E. K., Roney E. K., Cheung S. W., Chinault A. C. and Lupski J. R. Molecular mechanisms for subtelomeric rearrangements associated with the 9q34.3 microdeletion syndrome.

  3.1 Cycle Sequencing Kit (Applied Biosystems).

  ). All of the three motifs are scattered throughout the 150 bp. Whereas among the 12 previously reported breakpoint flanking sequences, in 9/12 Ig heavy chain class switch repeats 3 were found (Table

	one or more DNA polymerase arrest site(s), in 7/12 Ig heavy chain class switch repeats 5 and
	in 4/12 a
	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	14

1 Table 1 . Overview of the non-B structural elements and motifs tested at chromosome 4p and others

 11

	Ig Heavy chain class	GGGCT		2	2	22.2		4	7	33.3	6	9	28.6	130	171	26	123	138	0.80
	switch repeat 2
	consensus scaffold	WADAWAYAWW		2	4p-2	22.2		0	Ohter chr 0	0.0	2	Combined 2	9.5	Human sequences 105 151 21	Random 8 8	p-value 0.27
	attachment region 3																sequences
	consensus scaffold	TWWTDTTWWW		n° seq all hits 2 2	% 22.2	n° seq all hits 1 11	% 8.3	n° seq all hits 3 13	% 14.3	n° seq all hits 113 289	% 22.6	n° seq all hits 11 14	COM/ 0.59
	attachment region 4																	HUMAN
	DNA polymerase a	CTGGCG		n=9 1	n=9 1	11.1	n=12 0	n=12 0	0.0	n=21 1	n=21 1	4.8	n=500 n=500 6 6	1.2	n=500 n=500 32 34	0.25
	MICROHOMOLOGY frameshift hotspot 2 DNA polymerase b	ACCCWR		7 1	-1	77.8 11.1		9 6	-9	75.0 50.0	16 7	-10	76.2 33.3	-149	168	29.8	125	142	0.81
	NON B-frameshift hotspot 1	zDNA		1	1	11.1		1	1	8.3	2	2	9.5	12	14			0.12
	STRUCTURES DNA polymerase a/b	ACCCCA		1	1	11.1		1	1	8.3	2	2	9.5	40	43	8	39	42	0.68
	frameshift hotspot 1	Oligo(G)n repeats		3	4	33.3		7	14	58.3	10	18	47.6	181	220	36.2		0.50
	Human minisatelites	Direct repeats GCWGGWGG		1 1	3 1	11.1 11.1		4 1	19 1	33.3 8.3	5 2	22 2	23.8 9.5	6 18	6 19	3.6	9	9	1.00 0.19
	conserved seq/X-like element	Mirror repeat		2	3	22.2		2	4	16.7	4	7	19.0	0	0			0.11
	Vaccinia Topoisomerase	Complementary repeats YCCTT	3 1	3 1	33.3 11.1		0 7	0 10	58.3	3 8	3 11	14.3 38.1	2 246	3 364	49.2	244	336	1.00 0.38
	I consensus	Inverted repeats		1	1	11.1		3	3	25.0	4	4	19.0	2	2			0.66
	MOTIFS DNA polymerase arrest site deletion hotspot consensus DNA polymerase a frameshift hotspot 1 translin binding site 1 X-element E.coli Ade6-M26	WGGAG TGRRKM TCCCCC ATGCAG GCTGGTGG ATGACGT F o r	n=9 6 5 0 0 0 0	10 11 0 0 0 0	66.7 55.6 0 0 0 0	n=12 9 9 3 1 0 0	20 18 4 2 0 0	75.0 75.0 25.0 8.3 0 0	n=21 15 14 3 1 0 0	30 29 4 2 0 0	71.4 66.7 14.3 4.8 0 0	n=500 n=500 266 450 374 688 30 32 51 52 4 4 1 1	53.2 74.8 6 10.2 0.8 0.2	n=500 n=500 231 322 341 581 41 43 38 38 2 2 13 14	0.12 0.44 0.14 0.71
	Murine parvovirus recombination hotspot Ig Heavy chain class switch repeat 5 DNA polymerase a/b frameshift hotspot 2 ARS consensus S cerevisiae ARS consensus S Pombe Ig Heavy chain class switch repeat 3 consensus scaffold attachment region 1 consensus scaffold attachment region 2	CTWTTY TGAGC TGGNGT WTTTATRTTTW WRTTTATTTAW GGGGT AATAAAYAAA TTWTWTTWTT	5 4 3 0 0 3 0 0 e P e r	5 5 3 0 0 3 0 0	55.6 44.4 33.3 0 0 33.3 0 0		3 7 4 0 0 4 0 1	3 10 5 0 0 16 0 8	25.0 58.3 33.3 0 0 33.3 0 8.3	8 11 7 0 0 7 0 1	8 15 8 0 0 19 0 8	38.1 52.4 33.3 0 0 33.3 0 4.8	204 171 137 1 9 90 8 40	298 208 168 1 19 106 17 144	40.8 34.2 27.4 0.2 1.8 18 1.6 8	127 127 138 1 2 145 0 0	140 144 154 1 2 165 0 0	1.00 0.10 0.62 0.09 1.00
	Ig Heavy chain class switch repeat 4 translin binding site 2 Ig Heavy chain class switch repeat 1 Drosophila topoisomerase 2 consensus Heptamer recombination signal	TGGGG GCCCWSSW GAGCT GTNWAYATTVATNNR CACAGTG	3 2 2 0 R 5 2 2 0 0 0 e v 33.3 22.2 22.2 0 0 i e	6 2 3 0 2	11 2 7 0 2	50.0 16.7 25.0 0 16.7	9 4 5 0 2	16 4 9 0 2	42.9 19.0 23.8 0 9.5	135 56 124 0 19	179 67 160 0 20	27 11.2 24.8 0 3.8	138 28 120 0 7	156 30 157 0 7	0.13 0.29 1.00 0.21
						w

Table 2 . Overview of terminal breakpoints at chromosome 4p

 2 STR: simple tandem repeats, LINE: long interspersed nuclear elements, SINE: short interspersed nuclear elements.

	N° Genbank	Breakpoint	Repeat at	Micro-homology with
		ID	(bp)	breakpoint	TTAGGG (bp)
	1	HQ167740	1.691.685	No	2
	2	HQ167741	2.161.732	LINE	2
	3	HQ167742	2.168.865	LINE	0
	4	HQ167743	2.424.502	SINE	1
	5	HQ167744	2.615.988	No	3
	6	HQ167745	4.887.950	LINE	3
	7	HQ167746	5.581.455	STR	2
	8	HQ167747	5.981.420	No	5
	9	HQ167748	16.696.986	SINE	3
				6/9	7/9

Acknowledgments

We thank all families who participated in the study. This work was made possible in part by grants GOA/2006/12, and the SymBioSys Center of Excellence (Research Council, K.U.Leuven, EF/05/007) to J.R.V.