Abdelmadjid Cherifi
email: abdelmadjid.cherifi@univ-reims.fr

Kevin Guelton
email: kevin.guelton@univ-reims.fr

Valter Laurent Arcese
email: laurent.arcese@univ-reims.fr

J S Leite

Valter J S Leite
email: valter@ieee.org

Global non-quadratic D -stabilization of Takagi-Sugeno systems with piecewise continuous membership functions

Keywords: Takagi-Sugeno models, D-stability, Non-PDC controllers, Global non-quadratic design, Linear Matrix Inequalities

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Introduction

Since the pioneer work of Takagi and Sugeno [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF], dealing with the identification of nonlinear systems through the fuzzy formalism, Takagi-Sugeno (T-S) fuzzy model-based approaches have caught the attention of the control community. Indeed, thanks to their convex characters, these approaches may facilitate the analysis and control design for some classes of nonlinear systems by extending some linear results to the nonlinear framework [START_REF] Tanaka | Fuzzy control systems design and analysis: a linear matrix inequality approach[END_REF].

A T-S model is a set of linear models aggregated by convex Membership Functions (MFs). It may accurately match a nonlinear system in a compact set of its state space by using sector nonlinearity transformations [START_REF] Tanaka | Fuzzy control systems design and analysis: a linear matrix inequality approach[END_REF]. Their stability and stabilization are usually investigated via the well known direct Lyapunov methodology, aiming to obtain Linear Matrix Inequality (LMI) conditions [START_REF] Wang | An approach to fuzzy control of nonlinear systems: stability and design issues[END_REF][START_REF] Tanaka | A multiple Lyapunov function approach to stabilization of fuzzy control systems[END_REF][START_REF] Rhee | A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design[END_REF][START_REF] Mozelli | A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems[END_REF][START_REF] Dong | Control synthesis of T-S fuzzy systems based on a new control scheme[END_REF]. Indeed, LMI conditions may be solved efficiently via convex optimization tools [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF].

Controller design conditions for the stabilization of T-S models were firstly considered with common Quadratic Lyapunov Functions (QLF) and a Parallel Distributed Compensation (PDC) control scheme [START_REF] Tanaka | Fuzzy control systems design and analysis: a linear matrix inequality approach[END_REF][START_REF] Wang | An approach to fuzzy control of nonlinear systems: stability and design issues[END_REF]. However, they require to find common decision variables, solution of a set of LMI, which may lead to conservatism [START_REF] Sala | On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems[END_REF]. To reduce the conservatism, piecewise, switched or nonquadratic Lyapunov functions (NQLF) have been proposed with limitations [START_REF] Johansson | Piecewise quadratic stability of fuzzy systems[END_REF][START_REF] Tanaka | A multiple Lyapunov function approach to stabilization of fuzzy control systems[END_REF][START_REF] Guerra | LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form[END_REF][START_REF] Ohtake | Switching fuzzy controller design based on switching Lyapunov function for a class of nonlinear systems[END_REF][START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF][START_REF] Xie | Control synthesis of discrete-time ts fuzzy systems: reducing the conservatism whilst alleviating the computational burden[END_REF][START_REF] Xie | Relaxed control design of discrete-time takagi-sugeno fuzzy systems: An event-triggered real-time scheduling approach[END_REF]. Indeed, because of the MFs overlapping, piecewise or switched Lyapunov functions are inadequate for T-S models obtained from sector nonlinearity approaches. Moreover, when considering usual fuzzy NQLF in the continuous time case [START_REF] Tanaka | A multiple Lyapunov function approach to stabilization of fuzzy control systems[END_REF], the appearance of the MFs time derivatives in the stability conditions leads to local-based approaches, which require an estimation of the designed closed-loop domain of attraction [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF], and makes harder the obtention of LMIs or their practical application. In addition, since these local non-quadratic approaches require the finite bounds of the time derivatives of the MFs, they can't be applied to T-S models involving piecewise MFs. Note that in the discrete time case [START_REF] Guerra | LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form[END_REF][START_REF] Xie | Control synthesis of discrete-time ts fuzzy systems: reducing the conservatism whilst alleviating the computational burden[END_REF][START_REF] Xie | Relaxed control design of discrete-time takagi-sugeno fuzzy systems: An event-triggered real-time scheduling approach[END_REF], these derivatives are no longer occurring but these results are left out from the present study since it only focusses on continuous time Takagi-Sugeno models. Thus, to circumvent these drawbacks in the continuous time framework, another non-quadratic approach has been proposed with the consideration of Line-Integral Lyapunov Functions (LILF) [START_REF] Rhee | A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design[END_REF]. This approach provides global stability conditions but is restricted in stabilization to second order systems to obtain convex conditions [START_REF] Guelton | Comments on Fuzzy Control Systems Design via Fuzzy Lyapunov Functions[END_REF][START_REF] Márquez | Improvements on non-quadratic stabilization of Takagi-Sugeno models via line-integral Lyapunov functions[END_REF][START_REF] Guelton | Some refinements on stability analysis and stabilization of second order T-S models using line-integral Lyapunov functions[END_REF]. In [START_REF] Márquez | A non-quadratic Lyapunov functional for H ∞ control of nonlinear systems via Takagi-Sugeno models[END_REF], a new NQLF, which involves the mean values of the MFs on the interval [t -α, t] with α > 0, has been proposed. The major milestone of such NQLF is that the obtained closed-loop stability conditions are free of the time derivative of the MFs and also hold globally when there exist a solution.

In parallel to the previous mentioned studies, the improvement of the closedloop transient responses of complex nonlinear systems represented by T-S models remain an important challenge. In this context, the concept of D-stability can be considered. It is borrowed from linear systems to provide closed-loop performance while keeping the control signal suitable for real world applications. D-stability consists on placing the distribution of the eigenvalues of the closedloop system in a prescribed region of the complex plane. This can be achieved by adding constraints to the Lyapunov function [START_REF] Bachelier | Commande des systèmes linéaires incertains: placement de pôles robuste en D-stabilité[END_REF][START_REF] Chilali | Robust pole placement in LMI regions[END_REF], which lead to obtain LMI conditions that ensure both the closed-loop stability and some desired performances. It was firstly proposed for linear systems [START_REF] Chilali | H ∞ design with pole placement constraints: an LMI approach[END_REF], then extended to polytopic uncertain linear systems [START_REF] Peaucelle | A new robust Dstability condition for real convex polytopic uncertainty[END_REF][START_REF] Leite | An improved LMI condition for robust Dstability of uncertain polytopic systems[END_REF]. Based on these previous works, some quadratic D-stabilizing controller design conditions have been recently proposed for T-S systems [START_REF] Toulotte | Vehicle spacing control using robust fuzzy control with pole placement in LMI region[END_REF][START_REF] Assawinchaichote | Further results on robust fuzzy dynamic systems with LMI D-stability constraints[END_REF][START_REF] Cherifi | Quadratic design of D-stabilizing non-PDC controllers for quasi-LPV/T-S models[END_REF][START_REF] Cherifi | Uncertain TS model-based robust controller design with D-stability constraints-A simulation study of quadrotor attitude stabilization[END_REF]. Furthermore, it is to be pointed out that the less the conservatism is, better transient performances can be achieved. Thus, to reduce the conservatism, D-stable LMI-based conditions for non-PDC controller design have been proposed in the non-quadratic framework [START_REF] Cherifi | Local D-stabilization of uncertain TS fuzzy models via fuzzy Luapunov functions[END_REF]. Nevertheless, these non-quadratic conditions are only locally suitable since they suffer from the occurrence of the time derivatives of the MFs, where the optimization of the closed-loop domain of attraction remains a tricky challenge due to the presence of the D-stability constraints. Alternatively, some attempts have been done to propose global closed-loop D-stability conditions in the non-quadratic framework via LILF [START_REF] Bai | Fuzzy regional pole placement based on fuzzy Lyapunov functions[END_REF][START_REF] Lu | Pole placement with LMI constraint of fuzzy descriptor system[END_REF], but these results are unfortunately unsuitable as it is detailed in Appendix A.

According to the above mentioned concerns, it appears that, from previous literature, suitable global non-quadratic approach for the D-stabilization of T-S models are not available. The present study aims at filling this gap by considering a NQLF involving the mean values of the MFs instead of the LILF considered in [START_REF] Bai | Fuzzy regional pole placement based on fuzzy Lyapunov functions[END_REF][START_REF] Lu | Pole placement with LMI constraint of fuzzy descriptor system[END_REF]. Moreover, in this work we ought to show that, unlike previous local non-quadratic approaches [START_REF] Tanaka | A multiple Lyapunov function approach to stabilization of fuzzy control systems[END_REF][START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF][START_REF] Cherifi | Local D-stabilization of uncertain TS fuzzy models via fuzzy Luapunov functions[END_REF] Notations. Along this paper, one denotes the subset of integers I r = {1, ..., r}, r ≥ 1. Moreover, the symbol ( * ) in a matrix denotes a transpose quantity. In mathematical expressions, I denotes identity matrices with appropriate dimensions. M > 0 (resp. < 0) means that M is a positive definite matrix (resp. negative definite). ⊗ denotes the Kronecker product. For any matrices M with appropriate dimensions, one denotes H(M ) = M + M T . Finally, the time t is omitted in the proofs of the proposed theorems when there is no ambiguity.

Preliminaries and problem statement

Consider the class of Takagi-Sugeno systems described by the following polytopic state space representation [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF]:

ẋ(t) = r i=1 h i (z(t))(A i x(t) + B i u(t)) (1)
where on the time interval [t -α, t] with α > 0, will be considered to develop the main results proposed in the next Section. Hence, the following assumptions are considered in the present study.

x(t) = [x 1 (t) . . . x n (t)] T ∈ R n is the state vector, u(t) = [u 1 (t) . . . u m (t)] T ∈ R m is the input vector, z(t) = [z 1 (t) . . . z p (t)] ∈ R p is
Assumption 1. For a given scalar α > 0 and ∀t ∈ [-α, 0), h i (z(t)) = h i (z(0)).

Assumption 2. For all t ∈ [-α, +∞), each h i (t) is (at least) piecewise continuous.

According to assumptions 1 and 2, ∀t > 0 the piecewise MFs h i (z(t)) are integrable on intervals [t -α, t]. Hence, we can define their mean values on these intervals as:

v i (z(t)) = α -1 t t-α h i (z(τ ))dτ, ∀i ∈ I r (2)
As shown in [START_REF] Márquez | A non-quadratic Lyapunov functional for H ∞ control of nonlinear systems via Takagi-Sugeno models[END_REF], ∀t > 0 and ∀i ∈ I r , the mean values (2) exhibit the convex sum properties. Indeed, since h i (z(t)) ≥ 0 and r i=1 h i (z(t)) = 1, it is obvious that v i (z(t)) ≥ 0 and:

m i=1 v i (z(t)) = m i=1 α -1 t t-α h i (z(τ ))dτ = α -1 t t-α m i=1 h i (z(τ ))dτ = α -1 t t-α 1dτ = 1 (3)
Note that, if the membership functions h i (z(t)) are piecewise continuous, their derivatives ḣi (z(t)) are unbounded and previous classical and local nonquadratic approaches (see e.g. [START_REF] Tanaka | A multiple Lyapunov function approach to stabilization of fuzzy control systems[END_REF][START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF][START_REF] Cherifi | Local D-stabilization of uncertain TS fuzzy models via fuzzy Luapunov functions[END_REF]) can't apply, whereas:

vi (z(t)) = α -1 (h i (z(t)) -h i (z(t -α))) , ∀i ∈ I r (4)
are always finite and bounded for all t ∈ [-α, +∞).

Moreover, it is to be noticed that if α → 0, then vi (z(t)) → ḣi (z(t)). As a consequence, for small finite and strictly positive values of α, v i (z(t)) is continuous and can be view as a smoothed approximation of h i (z(t)) (see for example Summarizing, the mean values v i (z(t)) being continuous with bounded derivatives make them relevant (instead of h i (z(t))) as weighting functions for a NQLF candidate, presented in the next section, for the global stabilization of T-S systems (1) with piecewise MFs.

Remark 1. In [START_REF] Márquez | A non-quadratic Lyapunov functional for H ∞ control of nonlinear systems via Takagi-Sugeno models[END_REF], the focus was only put on the stabilization of T-S models with strictely continuous MFs and no mention was made on the applicability of the results with piecewise continuous MFs. If the authors of [START_REF] Márquez | A non-quadratic Lyapunov functional for H ∞ control of nonlinear systems via Takagi-Sugeno models[END_REF] miss this fact, the present study enable this oversight to be highlighted.

In the sequel, to lighten the mathematical expressions, for any sets of real matrix M i , i ∈ I r , we will denote:

M h = r i=1 h i (z(t))M i , M h = r i=1 h i (z(t -α))M i , M v = r i=1 v i (z(t))M i (5)
or for any other convex combinations (multiple fuzzy sums):

M hh hv = r i=1 r j=1 r k=1 r l=1 h i (z(t))h j (z(t))h k (z(t -α))v l (z(t))M ijkl (6)
Moreover, from (4) and the above defined notations, we have:

Ṁv = r i=1 vi (z(t))M i = α -1 M h -M h (7)
To stabilize (1), let us now consider the following non-PDC control law adapted from [START_REF] Jaadari | New controllers and new designs for continuous-time Takagi-Sugeno models[END_REF] and [START_REF] Márquez | A non-quadratic Lyapunov functional for H ∞ control of nonlinear systems via Takagi-Sugeno models[END_REF]:

u(t) = K h hv H -1 (.) x(t) (8)
where K h hv ∈ R m×n and H (.) ∈ R n×n (to be further detailed) are gain scheduled matrices to be synthesized.

Substituting ( 8) in ( 1), the closed-loop dynamics is expressed by:

ẋ(t) = A h + B h K h hv H -1 (.) x(t) (9)
Problem statement. Providing that only few local non-quadratic approaches 125 for the D-stabilization of T-S models involving continuous MFs are available from previous literature ( see e.g. [START_REF] Cherifi | Quadratic design of D-stabilizing non-PDC controllers for quasi-LPV/T-S models[END_REF][START_REF] Cherifi | Local D-stabilization of uncertain TS fuzzy models via fuzzy Luapunov functions[END_REF]), the goal of this paper is to propose new non-quadratic LMI conditions for the design of the gain scheduled matrices K h hv and H (.) such that ( 9) is globally non-quadratically D-stable; i.e such that the non-PDC controller (8) globally non-quadratically D-stabilizes the T-S fuzzy model ( 1) with piecewise MFs.

To deal with the D-stability concept, let us now recall the following definition of a LMI region [START_REF] Chilali | Robust pole placement in LMI regions[END_REF].

Definition 1. [START_REF] Chilali | Robust pole placement in LMI regions[END_REF]. A LMI region is a subset D of the complex plane, defined by the matrices

R 1 = R T 1 ∈ R δ×δ and R 2 ∈ R δ×δ such that: D = {λ ∈ C : R 1 + λR 2 + λR T 2 < 0} ( 10
)
where δ is the LMI region order.

Remark 2. By extension of the pioneer work [START_REF] Chilali | Robust pole placement in LMI regions[END_REF], where the D-stability concept has been defined for linear systems with constant uncertainties, it is used in the present study for nonlinear systems (i.e. the considered class of T-S fuzzy models) in the sense that, for each instant t, the distribution of the eigenvalues of ( 9) lies in a subset D of the complex plane. Moreover, because the proposed conditions are in the LMI form (see the main results in the next section), when the particular subset D recovers the left half-plane, the asymptotical stability of the nonlinear system can be concluded following similar steps as those used in Theorem 1 in [START_REF] Daafouz | Parameter dependent lyapunov functions for discrete-time systems with time-varying parametric uncertainties[END_REF].

For control purpose, the LMI region depicted in Figure 1 is usually considered. It is the intersection of three elementary regions of the complex plane:

the left half-plane defined by Re(λ) < -β, a conic sector defined by its apex at (γ, 0) with an inner angle π/2 -θ and a circle centered at (q, 0) with a radius s.

-Place Figure 1 here -According to Definition 1, this LMI region is defined by the following matrices [START_REF] Chilali | Robust pole placement in LMI regions[END_REF]:

R 1 =            2β 0 0 0 0 0 -2γ cos θ 0 0 0 0 0 -2γ cos θ 0 0 0 0 0 -s -q 0 0 0 -q -s            (11)
and

R 2 =            1 0 0 0 0 0 cos θ sin θ 0 0 0 -sin θ cos θ 0 0 0 0 0 0 1 0 0 0 0 0            (12)
See e.g. [START_REF] Bachelier | Commande des systèmes linéaires incertains: placement de pôles robuste en D-stabilité[END_REF][START_REF] Chilali | Robust pole placement in LMI regions[END_REF] for several LMI region examples illustrating how to set the matrices R 1 and R 2 .

With regards to the Lyapunov theory, the following definition is adapted from [START_REF] Chilali | Robust pole placement in LMI regions[END_REF] as basic D-stability conditions for nonlinear systems [START_REF] Nguang | Robust H ∞ output feedback control design for fuzzy dynamic systems with quadratic D-stability constraints: An LMI approach[END_REF].

Definition 2. [START_REF] Nguang | Robust H ∞ output feedback control design for fuzzy dynamic systems with quadratic D-stability constraints: An LMI approach[END_REF]. Given an LMI region defined by ( 10), a nonlinear system ẋ = f (x)x is said to be D-stable if there exists a Lyapunov function V(x(t))

satisfying 1 2 V(x(t)) V (x(t)) ∈ D, i.e.: R 1 ⊗ V(x(t)) + R 2 ⊗ 1 2 V(x(t)) + R T 2 ⊗ 1 2 V(x(t)) < 0 ( 13
)
where ⊗ denotes the Kronecker product. Let us now recall some useful properties of the Kronecker product.

Property 1. [START_REF] Brewer | Kronecker products and matrix calculus in system theory[END_REF] For any scalars η and matrices X, Y and Z with appropriate dimensions, the following properties of the Kronecker product holds:

(X ⊗ Y ) T = X T ⊗ Y T (X + ηY ) ⊗ Z = (X ⊗ Z) + η(Y ⊗ Z) X ⊗ (Y + ηZ) = (X ⊗ Y ) + η(X ⊗ Z)
In addition, to further relax the LMI conditions proposed in the next section, the following lemma [START_REF] Tuan | Parameterized linear matrix inequality techniques in fuzzy control system design[END_REF], usual in the context of T-S models, is adopted in the sequel since it is wellknown as a good compromise between computational complexity and conservatism improvement [START_REF] Sala | On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems[END_REF].

Lemma 1. (Tuan's lemma [START_REF] Tuan | Parameterized linear matrix inequality techniques in fuzzy control system design[END_REF]): Let Γ kl ij , for (i, j, k, l) ∈ I 4 r , be matrices of appropriate dimensions. Γ hv hh < 0 is satisfied if both the following conditions hold:

Γ kl ii < 0, ∀(i, k, l) ∈ I 3 r (14) 2 r -1 Γ kl ii + Γ kl ij + Γ kl ji < 0, ∀(i, j, k, l) ∈ I 4 r / i = j (15)
To conclude this preliminary section, let us recall that from Definition 2, LMI-based quadratic D-stability conditions and relaxed quadratic ones have been proposed in [START_REF] Nguang | Robust H ∞ output feedback control design for fuzzy dynamic systems with quadratic D-stability constraints: An LMI approach[END_REF][START_REF] Toulotte | Vehicle spacing control using robust fuzzy control with pole placement in LMI region[END_REF][START_REF] Assawinchaichote | Further results on robust fuzzy dynamic systems with LMI D-stability constraints[END_REF][START_REF] Cherifi | Quadratic design of D-stabilizing non-PDC controllers for quasi-LPV/T-S models[END_REF][START_REF] Cherifi | Uncertain TS model-based robust controller design with D-stability constraints-A simulation study of quadrotor attitude stabilization[END_REF]. To further relax these LMI conditions, local non-quadratic LMI-based conditions have been proposed in [START_REF] Cherifi | Local D-stabilization of uncertain TS fuzzy models via fuzzy Luapunov functions[END_REF]. Nevertheless, these previous local non-quadratic approach suffers from the occurrence of the time derivatives of the MFs and are not suitable when T-S systems involve piecewise continuous MFs (since their time derivatives are unbounded). Moreover, some attemps to provide global non-quadratic D-stabilization approaches, based on the consideration of a LILF, have been proposed in [START_REF] Bai | Fuzzy regional pole placement based on fuzzy Lyapunov functions[END_REF][START_REF] Lu | Pole placement with LMI constraint of fuzzy descriptor system[END_REF]. Unfortunately, as detailed in Appendix A, these previous LILF-based results are not correct. Therefore, in the next section, we aim at proposing new relaxed LMIbased non-quadratic conditions for the design of non-PDC controllers [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF], which globally non-quadratically D-stabilizes (1), even when they involve piecewise continuous MFs.

Main results

In this section, three theorems are proposed with new non-quadratic LMIbased conditions for the global non-quadratic D-stabilization of T-S model (1), which may have piecewise MFs, under the non-PDC control law [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. In this context, let us consider the following NQLF candidate [START_REF] Márquez | A non-quadratic Lyapunov functional for H ∞ control of nonlinear systems via Takagi-Sugeno models[END_REF], which involves the mean values of the MFs (2), under assumptions 1 and 2, as scheduling variables:

V(x(t)) = x T (t)P -1 v x(t) (16)
Remark 4. Note that Assumption 1 is similarly to what is done for time-delay systems to ensure the existence of a valued function for initial conditions, see e.g. [START_REF] Bourahala | Relaxed controller design conditions for takagi-sugeno systems with state time-varying delays[END_REF]). Moreover, Assumption 2 is the only requirement on the MFs h i for (16) to be continuous, making it relevant for the stability analysis of T-S models involving piecewise MFs without needing any dwell time or switching instants assumptions.

The following theorem summarizes the first proposed global non-quadratic conditions.

Theorem 1. Let R 1 and R 2 be two matrices defining a prescribed LMI region D in the left half-plane of the complex plane. The T-S model (1) is globally and non-quadratically D-stabilized by the non-PDC control law (8) if there exists a scalar α > 0 and, for (j, k, l) ∈ I 3 r , the matrices P k = P T k > 0, K jkl , such that the LMI conditions ( 14) and (15) are verified with:

Γ kl ij = R 1 ⊗ P l + H R 2 ⊗ A i P l + B i K jkl - 1 2α (P j -P k ) (17)
In that case, H (.) = P v in the non-PDC controller (8).

185

Proof. Consider the NQLF [START_REF] Guelton | Comments on Fuzzy Control Systems Design via Fuzzy Lyapunov Functions[END_REF]. From the convex properties of the MFs (2), ( 16) is obviously positive if P k = P T k > 0, ∀k ∈ I r . Moreover, from the closedloop dynamics [START_REF] Sala | On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems[END_REF], the time derivative of ( 16) can be expressed as:

V(x) = ẋT P -1 v x + x T P -1 v ẋ + x T Ṗ -1 v x = 2x T P -1 v A h + B h K h hv H -1 (.) + 1 2 Ṗ -1 v x (18)
Thus, from ( 16), [START_REF] Guelton | Some refinements on stability analysis and stabilization of second order T-S models using line-integral Lyapunov functions[END_REF] and according to Definition 2, the closed-loop dynamics ( 9) is D-stable if:

R 1 ⊗ x T P -1 v x + H R 2 ⊗ x T P -1 v A h + B h K h hv H -1 (.) + 1 2 Ṗ -1 v x < 0 (19)
Now, according to Property 1, with x = (I ⊗ x) we can rewrite [START_REF] Márquez | A non-quadratic Lyapunov functional for H ∞ control of nonlinear systems via Takagi-Sugeno models[END_REF] as:

xT

R 1 ⊗ P -1 v + H R 2 ⊗ P -1 v A h + B h K h hv H -1 (.) + 1 2 Ṗ -1 v x < 0 (20)
which holds ∀x if:

R 1 ⊗ P -1 v + H R 2 ⊗ P -1 v A h + B h K h hv H -1 (.) + 1 2 Ṗ -1 v < 0 (21)
Now, multiplying [START_REF] Chilali | Robust pole placement in LMI regions[END_REF] left and right by (I ⊗ P v ) and since

P v Ṗ -1 v P v = -Ṗv , (21) is equivalent to: R 1 ⊗ P v + H R 2 ⊗ A h P v + B h K h hv H -1 (.) P v - 1 2 Ṗv < 0 (22)
Moreover, from (4) yields:

R 1 ⊗ P -1 v + H R 2 ⊗ A h P v + B h K h hv H -1 (.) P v - 1 2α P h -P h < 0 (23)
Finally, by choosing H (.) = P v , then applying Lemma 1, ( 23) is satisfied if the LMI conditions expressed in Theorem 1 hold.

Remark 5. From Theorem 1, by taking P j = P common and K jkl = K j , the basic quadratic D-stabilizing controller design conditions proposed as Theorem 7 in [START_REF] Cherifi | Quadratic design of D-stabilizing non-PDC controllers for quasi-LPV/T-S models[END_REF] are recovered. Thus, the non-quadratic conditions of Theorem 1 are obviously less conservative than the quadratic ones (including those presented in [START_REF] Assawinchaichote | Further results on robust fuzzy dynamic systems with LMI D-stability constraints[END_REF]).

To further relax the global non-quadratic conditions proposed in Theorem 1, we propose two alternatives. The first one is inspired by the way borrowed in [START_REF] Peaucelle | A new robust Dstability condition for real convex polytopic uncertainty[END_REF] for linear systems with polytopic uncertainties and the second one can be considered, in the global non-quadratic D-stabilization context of this study, as an extension of recent stabilization results (see e.g. [START_REF] Oliveira | Robust state feedback LMI methods for continuous-time linear systems: Discussions, extensions and numerical comparisons[END_REF][START_REF] Jaadari | New controllers and new designs for continuous-time Takagi-Sugeno models[END_REF][START_REF] Guelton | Some refinements on stability analysis and stabilization of second order T-S models using line-integral Lyapunov functions[END_REF][START_REF] Cherifi | Uncertain TS model-based robust controller design with D-stability constraints-A simulation study of quadrotor attitude stabilization[END_REF]), obtained via the application of the Finsler's lemma [START_REF] Skelton | A Unified Algebraic Approach to Linear Control Design[END_REF].

The following theorem summarizes the first alternative.

Theorem 2. Let R 1 and R 2 be two matrices defining a prescribed LMI region D in the left half-plane of the complex plane. The T-S model (1) is globally and non-quadratically D-stabilized by the non-PDC control law (8) if there exists a scalar α > 0 and, for (j, k, l) ∈ I 3 r , the matrices P k = P T k > 0, K jkl , G jkl and Q jkl , such that the LMI conditions (14) and (15) are verified with:

Γ kl ij =   Φ kl ij ( * ) R 2 ⊗ (P l -Q jkl ) + I ⊗ G T jkl A T i -I ⊗ H (G jkl )   (24)
and:

Φ kl ij = R 1 ⊗ P l + H R 2 ⊗ A i Q jkl + B i K jkl - 1 2α (P j -P k ) (25)
In that case, H (.) = P v in the non-PDC controller (8).

Proof. If the conditions of Theorem 2 hold, from Lemma 1, they provide that:

  Φ hv hh ( * ) R 2 ⊗ (P l -Q jkl ) + I ⊗ G T jkl A T i -I ⊗ H (G jkl )   < 0 (26) with Φ hv hh = R 1 ⊗ P v + H R 2 ⊗ A h Q h hv + B h K h hv -1 2α P h -P h .
Pre-and post-multiplying [START_REF] Assawinchaichote | Further results on robust fuzzy dynamic systems with LMI D-stability constraints[END_REF] respectively by I I ⊗ A i and its transpose, after expending and simplifying the obtained mathematical expression, yields the inequality [START_REF] Peaucelle | A new robust Dstability condition for real convex polytopic uncertainty[END_REF] with H (.) = P v . This guarantees that the closed-loop dynamics ( 9) is globally non-quadratically D-stable.

205

The second alternative to relax the basic global non-quadratic conditions of Theorem 1 is proposed by the following theorem. Theorem 3. Let R 1 and R 2 be two matrices defining a prescribed LMI region D in the left half-plane of the complex plane. The T-S model (1) is globally and non-quadratically D-stabilized by the non-PDC control law (8) if there exists the scalars α > 0 and ε > 0, and, for (j, k, l) ∈ I 3 r , the matrices P k = P T k > 0, K jkl and H jkl , such that the conditions (14) and (15) are verified with:

Γ kl ij =   Ψ kl ij ( * ) R 2 ⊗ (P l -H jkl ) + εI ⊗ (H T jkl A T i + K T jkl B T i ) -εI ⊗ H(H jkl )   (27)
and:

Ψ kl ij = R 1 ⊗ P l + H R 2 ⊗ A i H jkl + B i K jkl - 1 2α (P j -P k ) (28)
In that case, H (.) = H h hv in the non-PDC controller (8).

Proof. If the conditions of Theorem 3 hold, from Lemma 1, they provide that:

  Ψ hv hh ( * ) R 2 ⊗ (P v -H h hv ) + εI ⊗ (H T h hv A T h + K T h hv B T h ) -εI ⊗ H(H h hv )   < 0 (29) with Ψ hv hh = R 1 ⊗ P v + H R 2 ⊗ A h H h hv + B i K h hv -1 2α P h -P h .

Pre-and post-multiplying (29) respectively by I I

⊗ A z + B z K z H -1 z 210
and its transpose, after expending and simplifying the obtained mathematical expression, yields the inequality [START_REF] Peaucelle | A new robust Dstability condition for real convex polytopic uncertainty[END_REF] with H (.) = H h hv . This guarantees that the closed-loop dynamics ( 9) is globally non-quadratically D-stable. Remark 6. Note that with Theorem 3, the non-PDC scheduled gain matrix H h hv is not required to be symmetric. Nevertheless, its invertibility is guaranteed if the conditions of Theorem 3 hold since it yields from blocks (2,2) of ( 27) that

H h hv + H T h hv > 0.
Remark 7. : Similarly to previous stabilization results without D-stability contraints based on the Finsler's lemma (see e.g. [START_REF] Oliveira | Robust state feedback LMI methods for continuous-time linear systems: Discussions, extensions and numerical comparisons[END_REF][START_REF] Jaadari | New controllers and new designs for continuous-time Takagi-Sugeno models[END_REF][START_REF] Guelton | Some refinements on stability analysis and stabilization of second order T-S models using line-integral Lyapunov functions[END_REF]), the conditions of Theorem 3 involve a prefixed scalar parameter ε > 0 and so are not strictly LMIs.

Note that this scalar is an additional slack decision variable as it can be easily shown that it results from the introduction of a null term by taking the converse way to prove Theorem 3 from Theorem 1. As quote in [START_REF] Jaadari | New controllers and new designs for continuous-time Takagi-Sugeno models[END_REF], this parameters is usually prefixed or optimized in practice by linear programming inside a logarithmically spaced family of values such like ε ∈ {10 -6 , 10 -5 , . . . , 10 6 } × ρ, with ρ ∈ {1, 2, . . . , 9}, in order to avoid an exhaustive linear search. Moreover, in [START_REF] Oliveira | Robust state feedback LMI methods for continuous-time linear systems: Discussions, extensions and numerical comparisons[END_REF], it has been shown that the introduction of such additional parameter was outperforming previous results in a large way for thousands of LPV models and comparing with numerous results (classical Q approach, Finslers application, and several other variations).

Remark 8. Note that, from their respective proofs, it is obvious that the conditions of theorems 2 and 3 are less conservative than the conditions of Theorem 1. Indeed, both these proofs show that from the LMI-based conditions of theorems 2 and 3, the LMI conditions of Theorem 1 can be recovered by simple mathematical manipulation (congruences). Moreover, theorems 2 and 3 involve slack decisions variables, which give more degree of freedom to their respective LMI problems than the one of Theorem 1. Therefore, Theorem 1 is a special case of both theorems 2 and 3, which in turns are obviously less conservative than Theorem 1. Moreover, due to the absence of the terms K T jkl B T i in the blocks (2,1) of ( 24) and their presence in the blocks (2,1) of ( 27), we cannot to prove that the Theorem 2 is a special case of Theorem 3. Nevertheless, we will show through a numerical example given in the next section that we may expect less conservative results from Theorem 3 than Theorem 2, despite Theorem 3 is not strictly LMI (see Remark 7). We assume that the conservatism improvement of Theorem 3 comes from the more flexibility given by the terms including the control gains as decision variables to relax the anti-diagonal blocks of [START_REF] Cherifi | Quadratic design of D-stabilizing non-PDC controllers for quasi-LPV/T-S models[END_REF].

Remark 9. Note that P h = lim α→0 P v and Ṗh = lim α→0 Ṗv . Thus, taking to that limit and assuming | ḣk | < φ k , theorems 1 and 3 lead respectively to the local non-quadratic results proposed as theorems 1 and 2 in [START_REF] Cherifi | Quadratic design of D-stabilizing non-PDC controllers for quasi-LPV/T-S models[END_REF]. In addition to being local (see also [START_REF] Cherifi | Local D-stabilization of uncertain TS fuzzy models via fuzzy Luapunov functions[END_REF]), these non-quadratic D-stabilization results are not suitable for T-S models with piecewise continuous. Consequently, these local non-quadratic results are left out in this paper.

Numerical example

In this section, an academic example is propose to show the effectiveness of theorems 1, 2 and 3, as well as to compare their conservatism regarding to recently proposed quadratic results [START_REF] Assawinchaichote | Further results on robust fuzzy dynamic systems with LMI D-stability constraints[END_REF][START_REF] Cherifi | Quadratic design of D-stabilizing non-PDC controllers for quasi-LPV/T-S models[END_REF].

Let us consider a T-S model with two vertices involving piecewise continuous

MFs and given by:

ẋ(t) = 2 i=1 h i (z(t))(A i x(t) + B i u(t)) (30)
with :

A 1 =         -1 -10 5 a 0 1 0 1 1 1 -2 3 3 2 1 1         , A 2 =         -1 -10 -5 4 0 -1 0 1 -1 2 -4 0 3 1 2 -1         , B 1 =         2 2 1 1         , B 2 =         1 b 2 1        
, and the piecewise continuous MFs :

h 1 (x) = 1 -sign(cos(x 4 ))sin(x 1 ) 2 , h 2 (x) = sign(cos(x 4 ))sin(x 1 ) + 1 2
Remark 10. The considered MFs are piecewise continuous (i.e. discontinuous for all x 4 = π/2 + κπ, κ ∈ Z). Assuming that the discontinuities are left limits and right continuous, the MFs are integrable ∀x ∈ R 2 . Thus, the recent local non-quadratic D-stabilizing controller design conditions, proposed in [START_REF] Cherifi | Local D-stabilization of uncertain TS fuzzy models via fuzzy Luapunov functions[END_REF], are unsuitable since they are dependent of the bounds φ k of | ḣi |, which cannot be globally defined in this example (see also Remark 9).

In [START_REF] Bai | Fuzzy regional pole placement based on fuzzy Lyapunov functions[END_REF], the parameters a and b are dedicated to compare the conservatism of the considered LMI-based conditions. Thus, Figure 2 shows the obtained feasibility fields, computed using YALMIP [START_REF] Löfberg | YALMIP: A toolbox for modeling and optimization in MAT-LAB[END_REF] and SeDuMi [START_REF] Sturm | Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones[END_REF], of theorems 1, 2, and 3, with α = 0.1 and an LMI region defined by ( 11) and ( 12) (see also

Figure 1) with s = 21, q = -17, β = 0.5, θ = 3π/10, γ = 9. Figure 2(a) shows a comparison between Theorem 1 and the quadratic conditions of Theorem 7 in [START_REF] Cherifi | Quadratic design of D-stabilizing non-PDC controllers for quasi-LPV/T-S models[END_REF] and Theorem 2 in [START_REF] Assawinchaichote | Further results on robust fuzzy dynamic systems with LMI D-stability constraints[END_REF]. Remark 11. Note that, as quote in Remark 7, the parameter ε has been searched by linear programming in an exponential family space. To save computational time, our LMI algorithm is stopped as soon as a solution is obtained and, for the -Place Figure 5 here -Remark 13. Note that in [START_REF] Márquez | A non-quadratic Lyapunov functional for H ∞ control of nonlinear systems via Takagi-Sugeno models[END_REF], D-stability constraints have not been considered.

        , H112 =         0.
Hence, it can be easily shown that theorems 1 and 2 in [START_REF] Márquez | A non-quadratic Lyapunov functional for H ∞ control of nonlinear systems via Takagi-Sugeno models[END_REF] are respectively special cases of theorems 1 and 2 by setting the left half-plane as D region (see remarks 2 and 3). Thus, in this case, checking their respective conservatism leads to obtain the same feasability regions (plots are left out since it is straightforward). However, there is no theorem in [START_REF] Márquez | A non-quadratic Lyapunov functional for H ∞ control of nonlinear systems via Takagi-Sugeno models[END_REF] which is a special case of Theorem 3 and, since it has been shown below that it improves the results obtained with theorems 2 and 3, we can expect less conservative results regarding to the results from both theorems 1 and 2 in [START_REF] Márquez | A non-quadratic Lyapunov functional for H ∞ control of nonlinear systems via Takagi-Sugeno models[END_REF].

To conclude this numerical example, let us recall that, for this example, no quadratic conditions exist with (a, b) = (-4, 17) from the previous results [START_REF] Cherifi | Quadratic design of D-stabilizing non-PDC controllers for quasi-LPV/T-S models[END_REF][START_REF] Cherifi | Uncertain TS model-based robust controller design with D-stability constraints-A simulation study of quadrotor attitude stabilization[END_REF]. Moreover, due to the piecewise character of the membership functions, the previous non-quadratic D-stabilization approaches are unsuitable [START_REF] Cherifi | Local D-stabilization of uncertain TS fuzzy models via fuzzy Luapunov functions[END_REF]. Moreover, when only the standard stabilization is considered (left half-plane set as LMI region), Theorem 3 has been shown less conservative than the results in [START_REF] Márquez | A non-quadratic Lyapunov functional for H ∞ control of nonlinear systems via Takagi-Sugeno models[END_REF]. Hence, this clearly shows the interest of the theoretical procedure proposed in this paper since it is not only improving the conservatism but also enlarging the considered class of T-S systems, which may now involve piecewise MFs.

Conclusion

In this paper, the problem of the global non-quadratic D-stabilization of T-S systems with piecewise continuous membership functions has been considered. This has been confirmed through a numerical example, where the conservatism improvements regarding to quadratic approaches have been illustrated.

The main advantages of the proposed global non-quadratic D-stabilization approaches for T-S models are multiple regarding to previous tools. First, they allow to provide less conservative conditions than quadratic ones (see e.g. [START_REF] Toulotte | Vehicle spacing control using robust fuzzy control with pole placement in LMI region[END_REF][START_REF] Assawinchaichote | Further results on robust fuzzy dynamic systems with LMI D-stability constraints[END_REF][START_REF] Cherifi | Quadratic design of D-stabilizing non-PDC controllers for quasi-LPV/T-S models[END_REF][START_REF] Cherifi | Uncertain TS model-based robust controller design with D-stability constraints-A simulation study of quadrotor attitude stabilization[END_REF]), which are also global. Secondly, most of the previous non-quadratic approaches were only local ones. Indeed, they require the estimation of the closed-loop Domain of Attraction (DA), which can be challenging when considering performance specifications such like D-stability constraints (leading to small DA), but also can't guarantee the closed-loop stability in the whole state space (initial conditions outside the DA can diverge), e.g. [START_REF] Cherifi | Local D-stabilization of uncertain TS fuzzy models via fuzzy Luapunov functions[END_REF]. Then, note that these local non-quadratic approaches are unsuitable for T-S models with piecewise MFs. Finally, let us also recall that some other previous attempts have been done to provide global non-quadratic D-stabilization approaches with the use of LILF [START_REF] Bai | Fuzzy regional pole placement based on fuzzy Lyapunov functions[END_REF][START_REF] Lu | Pole placement with LMI constraint of fuzzy descriptor system[END_REF] then it means that the κ th fuzzy set among θ l -based fuzzy sets is used in the i th fuzzy rule, then 1 α il r l for any i.

        , P i =         p α il 1,1 0 . . . 0 0 p α il 0

375

Note that from the LILF (A.1), the stability conditions are free of the time derivatives of the MFs. However in [START_REF] Bai | Fuzzy regional pole placement based on fuzzy Lyapunov functions[END_REF][START_REF] Lu | Pole placement with LMI constraint of fuzzy descriptor system[END_REF], to apply the D-stability conditions given as Definition 2, it has been assumed that the LILF (A.1) can be rewritten as:

V(x(t)) = x T (t) P 0 + r i=1 h i (x(t))P i x(t)

(A.3)
Unfortunately, such way of doing is not correct since it is obvious that (A. These concerns make unsuitable the LMI-based design for global non-quadratic D-stabilization of T-S models proposed in [START_REF] Bai | Fuzzy regional pole placement based on fuzzy Lyapunov functions[END_REF][START_REF] Lu | Pole placement with LMI constraint of fuzzy descriptor system[END_REF], while the procedures proposed above as Theorem 1, 2 and 3 provide alternative solutions.

Figures:

505 x 2 (t)

x 1 (t)

x 3 (t)

x 4 (t)

  , considering a NQLF which involves the mean values of the MFs applies globally, not only for T-S models with smooth (continuous) MFs, but also for T-S models involving piecewise continuous MFs. This will make the proposed LMI conditions relevant for the global non-quadratic D-stabilization of a larger class of nonlinear systems with piecewise continuous nonlinearities (e.g. a class of switching nonlinear systems) which is, until now and from the authors' knowledge, an open problem that hasn't been addressed in previous literature. The outline of this paper is given as follows: In Section 2, some preliminaries on T-S fuzzy models with useful definitions and lemmas for their D-stabilization are recalled. Then, the main results are presented in Section 3, where three theorems, summarizing the proposed non-quadratic LMI conditions for the design of D-stabilizing non-PDC controllers for T-S models involving piecewise MFs, are given. Finally, the effectiveness and the relative degree of conservatism of the different results are illustrated and compared to suitable quadratic results through an academic example.

Figure 4

 4 Figure 4 in section 4).

Remark 3 .

 3 When R 1 = 0 and R 2 = 1, this definition reduces to the standard asymptotical stability conditions of the considered nonlinear system (i.e. the 155 LMI region D reduces to the left half-plane, see also Remark 2).

Figure 2 (

 2 b) shows a comparison between Theorem 2 and the quadratic conditions of Theorem 1 in[START_REF] Cherifi | Uncertain TS model-based robust controller design with D-stability constraints-A simulation study of quadrotor attitude stabilization[END_REF] (without parametric uncertainties). Figure2(c) shows a comparison between Theorem 3 and the quadratic conditions of Theorem 10 in[START_REF] Cherifi | Quadratic design of D-stabilizing non-PDC controllers for quasi-LPV/T-S models[END_REF]. As expected, we observe that the proposed non-quadratic results are less conservative than their respective quadratic particular cases (see Remark 3) as well as outperforming another recent related quadratic D-stabilization result[START_REF] Assawinchaichote | Further results on robust fuzzy dynamic systems with LMI D-stability constraints[END_REF]. Moreover, Figure2(d) summarizes the feasibility fields obtained from theorems 1, 2 and 3. We observe that Theorem 2 provide slightly less conservative results than Theorem 1. Moreover, Theorem 3 provide significantly less conservative results than Theorem 2. This shows the conservatism improvements of the global non-quadratic stabilization conditions proposed in this paper.-Place Figure2here -Let us now consider the particular point (a, b) = (-4, 17), where a solution has only been found with Theorem 3. This solution is summarized by the following non-PDC controller (8) gain matrices (the other decision variables, completing the solution of Theorem 3, are left out for space reasons): F111 = -0.0319 -0.0292 -0.0236 -0.0096 , F112 = -0.0259 -0.0263 -0.0218 -0.0133 , F121 = -0.0184 -0.0231 -0.0241 -0.0098 , F122 = -0.0163 -0.0217 -0.0231 -0.0113 , F211 = -0.0272 -0.0189 -0.0127 -0.0012 , F212 = -0.0256 -0.0180 -0.0118 -0.0015 , F221 = -0.0180 -0.0170 -0.0184 -0.0040 , F222 = -0.0139 -0.0152 -0.0183 -0.0061 ,

  particular point (a, b) = (-4, 17) we obtained ε = 0.0006. Nevertheless, let us notice that other values of this parameter can also lead to feasible LMIs. Hence, in the case of a more stringent D region specification, such linear programming search on this parameter can be helpful to find a solution.

Figure 3 1 T.-

 31 Figure 3 shows the obtained closed-loop state trajectories and the migration of the closed-loop eigenvalues for the initial state x(0) = π 2 -2 -1 1

Figure 5 (

 5 Figure 5(b) shows the distribution of the eigenvalues when only the decay rate is considered. Compared to Figure 3(b), we can observe that the distribution of the eigenvalues goes far on the left and have larger imaginary parts, increasing the transient control cost when only a decay rate is considered.

First, usual definitions

  of the D-stability concept have been recalled. Then, from the choice of a non-quadratic Lyapunov function involving the mean values of the membership functions over a time interval [t -α, t] with α > 0, three LMIbased theorems have been proposed for the design of non-PDC controllers for the D-stabilization of T-S systems. It is highlighted that, thanks to the choice of such a Lyapunov function candidate, the non-quadratic design conditions are no longer depending on the time derivative of the membership functions, yielding to global asymptotical stabilization, also suitable for T-S models involving piecewise membership functions (i.e. for a class of switched nonlinear systems).

  where α il specifies which θ l -based fuzzy set is used in the i th fuzzy rule, r l is the number of θ l -based fuzzy sets, i.e. r = p l=1 r l . For example, if α il = κ

1 )R 1 ⊗ 2 Γ

 112 cannot be rewritten as (A.3) to cope with global asymptotical stabilization.Moreover, let us highlight that in the LILF framework, from Definition 2 and considering (A.1) instead of (A.3), the D-stability condition (13) becomes: (0,x(t))f T (ψ).dψ + 2R 2 ⊗ x T (t) P 0 + r i=1 h i (x(t))P i ẋ(t) < 0 (A.4) which convexification remains an open and strong problem due to the presence of the line-integral term.

Figure 1 :

 1 Figure 1: Usual LMI region for control purpose.

Figure 2 :

 2 Figure 2: Comparison of the feasibility fields on parameter space a × b.

λ 2 (

 2 x(t))

  Distribution of the eigenvalues (a) Closed-loop state responses

Figure 3 :

 3 Figure 3: Closed-loop state responses and distribution of the closed-loop eigenvalues.

Figure 4 : 5 Time

 45 Figure 4: Time evolution of the membership functions.

λ 3 (

 3 x(t))

Figure 5 :

 5 Figure 5: (a) Comparison of the control signals with the whole considered LMI region vs with only a decay rate β = 0.5; (b) distribution of the closed-loop eigenvalues with only a decay rate β = 0.5.

D-stabilization of T-S models, and so a suitable alternative to [START_REF] Bai | Fuzzy regional pole placement based on fuzzy Lyapunov functions[END_REF][START_REF] Lu | Pole placement with LMI constraint of fuzzy descriptor system[END_REF].

Several perspectives of this study are foreseen. First, since the D-stabilization allows to improve the closed-loop performances while keeping the energetic control cost to reasonable values (i.e. avoiding the distribution of the eigenvalues to goes far left in the complex plane), a study showing the links with optimal control design would be interesting. In the same way, analyzing the effects of associating input constraints such like anti-windup techniques with D-stability constraints can be appealing for practical applications. Finally, since the considered mean values of the MFs hold filter properties, employing them as PDC gain scheduling variables may help to smoother the control signal and to handle external disturbances attenuation. Therefore, our future prospect will be to propose a rigorous robustness analysis of such control plant design with regards to H ∞ performances.

Appendix A.

In recent studies [START_REF] Bai | Fuzzy regional pole placement based on fuzzy Lyapunov functions[END_REF][START_REF] Lu | Pole placement with LMI constraint of fuzzy descriptor system[END_REF], some attempts have been done to propose new global non-quadratic LMI conditions for the design of PDC controllers with D-stability constraints for T-S models via LILF, following the pioneer work of Rhee and Won [START_REF] Rhee | A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design[END_REF]. Such LILF is given by:

where Γ(0, x(t)) is the path from the origin 0 to the current state x(t), (.) stands for the inner product of vectors, ψ ∈ R n is a dummy vector for the integral and dψ is an infinitesimal displacement. Note that such approach requires the following structure of f (x(t)) to guarantee the path-independency of the LILF (A.1) [START_REF] Rhee | A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design[END_REF][START_REF] Guelton | Comments on Fuzzy Control Systems Design via Fuzzy Lyapunov Functions[END_REF]:

f (x(t)) = P 0 + r i=1 h i (x(t))P i x(t) (A.2)


S Bouheraoua

B Richard

F Renou
email: renou@coria.fr

Grisch

P Malbois

E Salaün

B Rossow

G Cabot

L Bouheraoua

S Richard

B Renou

F Grisch

OH-PLIF measurements under high-pressure conditions

Keywords: Gas turbine model combustor, Lean-Premixed Combustion, Quantitative kerosene-PLIF, Flame structure

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

Quantitative measurements of fuel distribution and flame structure in a lean-premixed aero-engine injection system by kerosene/OH-PLIF measurements under high-pressure conditions

Introduction

Achieving the forthcoming ACARE standards for pollutant emissions from aircraft engines will only be possible by pushing the combustion technologies towards their limits regarding their ability of pollutant emissions reduction and engine operability preservation. Among key parameters likely to fulfill these goals, the fuel injection system is one of them where substantial improvements can still be achieved. Indeed, fuel atomization, evaporation and mixing must be efficient whatever the operating conditions. Pressure is also an important parameter to consider on the evaporation process since it impacts the atomization process [START_REF] Lefebvre | Gas turbine Combustion[END_REF] which is determinant for the flame location and the flame shape [START_REF] Chterev | [END_REF]. Consequently, it is mandatory to investigate as finely as possible the physical and chemical mechanisms that govern turbu-lent combustion in relation with the architecture of the combustion system in which the combustion takes place.

In optical combustion chambers, this can be achieved by simultaneous 2D measurements of flame structure and fuel concentration by combining OH-PLIF with fuel-PLIF techniques. However, the study of industrial injection systems must be made as close as possible of real operating conditions, implying the use of liquid commercial-type fuels to be representative of their real operating conditions. Indeed, using fuel substitute may lead to significant changes on fuel distribution and flame shape in the combustion chamber [3]. Kerosene-PLIF technique was originally developed by [START_REF] Löfström | Proceedings of the Combustion Institute[END_REF][START_REF] Hicks | 33rd Joint Propulsion Conference AIAA[END_REF][START_REF] Allouis | [END_REF] and the low accuracy of the results justified the need to establish kerosene spectroscopy. This was done by [7,8] who thoroughly investigated the photophysics of kerosene and its components for various conditions of temperature, pressure and oxygen molar fraction, with a 266 nm excitation wavelength. ) with an intensifier gate width of 500 ns. All the details of the measurements can be found in [7].

HERON combustion facility

The High prEssure facility for aeRO-eNgines (HERON) was used to fully characterize aeronautical fuel injectors operating in high pressure and high temperature conditions, with mass flow rates relevant to injector operating conditions [START_REF] Malbois | Proceedings of ASME Turbo Expo[END_REF]. A full optical access of the flame is possible from three sides over the full width of the visualizing sector. UV-silica optical windows are made large enough (80 × 100 mm) to allow laser based measurements close to the wall chamber. The pressure chamber is regulated by an adaptive nozzle from 0.41 to 1.8 MPa. Air mass flow is controlled up to 300 g/s, and preheated to 670 K. Liquid kerosene (Jet-A1) is injected in the Lean Premixed injection system developed by SAFRAN HE, with a mass flow rate in the range of 0-5 g/s, leading to mass Fuel Air Ratio (FAR) from 3.0% to 4.2%. These operating points have been identified as representative of various helicopter engines flight conditions.

Simultaneous OH and Kerosene-PLIF

The global flame structure is investigated by OH-PLIF imaging at low repetition rate (10 Hz). A Nd-YAG-laser operating at 532 nm is used to pump a tunable dye laser (SIRAH Precision Scan) supplied with Rhodamine 590 dye. The resultant output pulse energy is 25 mJ per shot in the probe volume. The excitation wavelength is tuned to the Q 1 (5) transition line of

the A 2 Σ + (v = 1) ← X 2 Π(v =

Kerosene PLIF methodology

Principle

The species composition of a commercial JET A1 fuel indicates the presence of aliphatics, single-and two-rings aromatic compounds [START_REF] Edwards | [END_REF]. Considering a 266 nm laser wavelength excitation, the global kerosene fluorescence signal obtained in the high-pressure cell presents two spectrally distinct broadbands of fluorescence signal (Fig. 2) on the spectral ranges of 270-310 nm and 310-400 nm associated with the fluorescence of mono-and di-aromatics compounds, respectively. The intensity of the fluorescence signal decreases faster with the temperature increase for the high-frequency band than for the low-frequency, and this difference will be used as the principle to estimate both the kerosene mole fraction and temperature by a dual-wavelength detection strategy. For two different set of filters (Filters A and B), the recorded fluorescence signals can be expressed by : In order to combine these two fluorescence signals, it is necessary to consider the effect of different fuel volatilities along the kerosene droplet evaporation process. Indeed, it is assumed that the proportion k i of mono-aromatics and di-aromatics contained in the kerosene vapor remains constant during the experiments such as x i = k i • x kero , where x kero is the kerosene mole fraction. This can be justified by the high temperature environment (T≥ the same as the initial one even near the end of evaporation process [12].

S F,i = K i • P • x i • S 0,i (T, P, x O 2 ) T (1)
Consequently, this assumption will be considered as valid for temperatures greater than 650 K where all the components of kerosene are evaporated and up to 1100-1300 K, where kerosene pyrolysis may be observed. Equations [START_REF] Lefebvre | Gas turbine Combustion[END_REF] can then be expressed as a function of two equations, one for each camera, with two unknowns, x kero and T :

S F,i = Γ i • P • x kero • S 0,i (T, P, x O 2 ) T (2)
where Γ i = k i • K i . The constants Γ i will be defined by a calibration procedure, described in the next paragraph. S 0,i (T, P, x O 2 ) can be represented by the Stern-Volmer equation:

S 0,i (T, P, x O 2 ) = S 0,i (T, P, x 0 2 =0 ) 1 + α i (T ) • x 0 2 • 1 T (3)
The dependence of the filtered fluorescence kerosene spectra with the oxygen mole fraction was investigated for four temperatures (450 K, 550 K, 650 K and 750 K). Stern-Volmer plots exhibited a highly linear behavior which enables to determine the quenching influence using the linear regression coefficient α i (T ) [7]. In addition, it is shown in Fig. 3a of mono-and di-aromatics surrogates of kerosene. The pressure dependency on S 0,i (T, P, x O 2 ) is also reported in Fig. 3c for two different temperatures.

These results indicate that pressure effect are non-negligible especially for pressure under 1.0 MPa and must be taken into account. Finally, the experimental transmission constants Γ i are obtained by recording the fluorescence of a heated kerosene vapor/air jet at atmospheric pressure. A preheated kerosene vapor/air jet with different equivalence ratio is directly inserted inside the combustion chamber through the fourth side of the combustion chamber so that the optical set-up (excitation and collection) is unchanged.

Along this calibration procedure, pressure and kerosene/oxygen mole fraction are fixed and S 0,i can be extracted from the look-up table. The constants Γ i can then be obtained from the recorded fluorescence signals for each camera:

Γ i = T P • x kero • S 0,i (T, P, x O 2 ) S F,i (4)
The calibration procedure has been performed for two equivalence ratios = 0.1 and 0.05, and one temperature (T = 447 K), for 3 camera gains in order to check the linearity of the fluorescence signal. Consequently, 6 different values of Γ i have been determined and averaged. They will be kept constant along this work. From the knowledge of these constants, the non-linear system of equations (Eq. 1) is characterized by two unknowns (x kero and T ).

Usually, temperature is deduced first from the ratio of the two fluorescence signals [15], and the fuel concentration is then calculated. However, the reference fluorescence signal is slightly sensitive to the temperature within the investigated temperature range (700 -800 K). Consequently, the ratio of the two fluorescence signals will provide little accurate values of temperature, and would therefore impact directly the determination of x kero . The method selected in this study consists of solving iteratively this set of two equations, and the optimization is made on a couple of solutions (x kero and T ) by using least-square methods by Matlab routines.

Uncertainties quantification

The accuracy of the methodology was estimated from the images used for the calibration, by comparing the spatially averaged values of temperature and x kero calculated from Eq. ( 1) with the averaged constants Γ i , with the values imposed on the calibration procedure (Fig. 4). The accuracy of the mass flow-meters (respectively thermocouples) leads to an experimental uncertainties of 4% on fuel mole fraction (respectively 1% on temperature) which are reported on the x-axis of Fig. 4. The sensitivity of the calculated kerosene mole fraction and temperature with the temperature dependence of S 0,i (T, P, x O 2 ) is reported in the y-axis to validate the extrapolation curve in Fig. 3b. To estimate this latter, the reference fluorescence signals S 0,i (T, P, x O 2 ) were shifted from (±5%) (dotted lines in Figure 3b) and used along the equations solving. This value of (±5%) comes from the error estimation analysis made by [7]. It is worth noting that it mainly impacts the determination of the kerosene molar concentration. Indeed, kerosene temperature is less sensitive since the fluorescence signal present a low temperature dependency in the range 400 -450 K as well as in the range 700 -800 K.

The maximum difference between the measured values and calibration values reported in Fig. 4 is less than 12% on the kerosene molar concentration and 8% for the temperature (410 K instead of 447 K).

Results and Discussion

Instantaneous flow fields

Instantaneous results obtained on the HERON combustion chamber are presented in Fig. 5 The same part of the flow is displayed for both quantities for comparison.

Moreover, OH gradient intensity is reported to visualize qualitatively the reaction intensity (beige color map). Here, the smaller gradients are masked by Indeed, the position corresponding to the total consumption of kerosene vapor along the flow axis is the position in which the combustion process is ended. This behavior is not very sensitive to the OH threshold value presented previously. Measurements exhibit also temperature of kerosene vapor between 750 and 800 K. These results are consistent with the air inlet temperature of 670 K and the fuel heating by radiation in the injection system bowl. Increasing the pressure significantly reduces the kerosene mole fraction at the exit of the entrance of the combustion chamber indicating that the combustion is located deeper inside the nozzle of the injection system. This result is confirmed by the increase of the temperature of the kerosene/air mixture at the bottom part of the combustion chamber for higher pressure.

Mean fuel mass fraction and OH distributions

Conclusions

The work developed in this paper focused on the characterization of the quantitative fuel distribution and flame structure at the exit of an aeronautical LP injection system operating in nominal conditions. The flame was operated in the HERON gas turbine model combustion chamber with kerosene-air at high pressure at a thermal power up to 180 kW. Kerosene-PLIF was applied using a single excitation path and a two spectral channel fluorescence collection. Thanks to a detailed analysis of the kerosene fluorescence and its component and a calibration procedure, it was possible to estimate the kerosene mole fraction with less than 12% error and a 8% error on the temperature estimation in aeronautical operating conditions. Simultaneously, OH-PLIF was applied to obtain the instantaneous flame structure and the correlation with the fuel repartition inside the combustion chamber.

An accurate description of the mixing and burning processes resulting from an LP injection system has been obtained, thanks to two different lasers and cameras arrangements. The measurements show that the main part of kerosene is consumed inside the injection system and only a small part of the injected fuel is burning inside the combustion chamber. The results suggest that such a LP injection system can provide a fairly homogeneous repartition of kerosene vapor around the inner flame. Nevertheless, eight spots where kerosene concentration is higher were observed and are likely to correspond to the holes of multi-point fuel injection. However, regions with large gradients of kerosene vapor were then limited, which contributed to a better flame stabilization as well as a global reduction in pollutant emission.

2 .

 2 These results demonstrated that kerosene fluorescence spectrum exhibited two fluorescence bands in the UV-visible range, attributed to mono-and di-aromatics naturally present in kerosene. Based on the strong different fluorescence band dependences with temperature, various imaging strategies were suggested to estimate simultaneously vapor fuel concentration and temperature in kerosene/air sprays. A pioneer work demonstrated the feasibility of this technique[8] but few experimental data are available in literature for realistic operating conditions of industrial injection systems (e.g. P ≥ 1.0 MPa), and when available, only qualitative results are presented[START_REF] Chterev | [END_REF]8,9].The first part of this work is to extend the capacity of the kerosene-PLIF technique to perform quantitative measurements of fuel vapor mole fraction and temperature in a combustion chamber, in which an industrial Lean Premixed fuel injector operated at high pressure and high temperature. As part of this objective, the effort is on the quantification of the experimental uncertainty. In a third part, both the fuel mole fraction and its temperature deduced from PLIF measurements are analyzed jointly with the flame structure obtained by OH-PLIF in different axial and radial planes, and for a large range of operating conditions. These results bring new insights on pressure effects on turbulent flame structure and on fuel mixing directly applied to a real LP fuel injector under nominal operating conditions. These results will be also directly taken into consideration as a feedback by the manufacturer to improve the technology and to provide an extended database for Large Eddy Simulation (LES) improvement and validations. Experimental set-up 2.1. High-pressure cell facility A stainless steel test cell was used first to characterize the spectroscopic properties of vapor kerosene fluorescence at temperature between 450 and 900 K, and pressure from 0.1 to 3.0 MPa under laser excitation at 266 nm. The outer cell dimensions are 190×145×120 mm with 15 mm overall minimal wall thickness and its inner volume is 230 cm 3 in which the operating conditions are well-controlled. The fluorescence emission from the illuminated tracer vapor is collected via a telescope-like combination of two UV achromatic doublets (f 1 = 160 mm and f 2 = 100 mm) by a spectrograph (Jobin Yvon, SPEX 270M). The light dispersed by the grating is recorded with a 16-bit intensified CCD camera (Princeton Instruments, model HSICCD-576G/BT, 576 × 384 pixels 2

  0) band of OH at = 282.75 nm. The collection system consists of an EmICCD camera (PIMAX 4, Roper Scientific) with two colored glass filters (WG 295 and UG 5 from Schott) and an interferential filter centered at 310 nm (FWHM = 5.6 nm). The camera is tilted from the perpendicular axis and equipped with a Scheimpflug mount.Simultaneously, kerosene vapor fluorescence signal is obtained from a singleexcitation scheme using a frequency-quadrupled Nd:YAG laser at 266 nm with 60 mJ/pulse. Time delay between the two laser beams was fixed to 60 ns in order to avoid cross-talks between fluorescence signals of OH and kerosene. Both laser beams are superimposed with a prism and they are transformed into two superimposed collimated sheets using a unique set of cylindrical and spherical lenses. Kerosene fluorescence is recorded by two 16bit EmICCD cameras, with a 100 mm, f /2.8 UV-CERCO lens (Fig.1). The first camera is equipped with a combination of a colored glass filter (Schott WG 280) and a custom-made bandpass filter collecting the whole fluorescence signal of kerosene in the range 260 -400 nm (hereafter Filters A). The second camera uses a combination of a colored filter (Schott WG 305) and a custom-made bandpass filter collecting fluorescence in the range 300 -400 nm to collect mainly fluorescence from di-aromatics (hereafter Filters B).PLIF images are corrected by noise, laser sheet energy inhomogeneity, and distortion using DaVis 8.1 commercial software. These optical diagnostics were applied successively in the axial (z,y) and the radial (or cross stream) plane (x,y) of the burner. It is worth noting that very few and small droplets are present in the combustion chamber. Indeed, fuel is injected upstream the bowl of the injection system and consequently upstream the bottom part of the combustion chamber. The high temperature of reactants enables a fast evaporation of the droplets. Consequently, the recorded fluorescence signals arise mainly from the kerosene vapor.

Figure 1 :

 1 Figure 1: Schematic representation of the optical set-up on HERON combustion facility for axial measurements.

  with i = A or B. The constants K i are the experimental transmission constants for each set of camera and filters, and have to be determined by a calibration procedure. The pressure P is measured and assumed to be constant within the combustion chamber. x B is the kerosene mole fraction filtered by the optical filters B. Consequently, this fluorescence signal mainly comes from the di-aromatic molecules in kerosene. x A is the kerosene mole fraction filtered by the optical filters A. This fluorescence signal comes from the mono-and di-aromatic molecules in kerosene. S 0,i (T, P, x O 2 ) is a referenced fluorescence signal and represents the temperature, pressure and quenching dependency of the filtered fluorescence signals.

Figure 2 :

 2 Figure 2: Normalized kerosene fluorescence spectra under 266 nm excitation. Thick black line: Kerosene spectrum at 450 K. Thick dark gray solid (dotted) lines: Filtered A (Filtered B) Kerosene spectrum at 700 K.

  that for oxygen concentration above 19% and temperature above 650 K, effect of O 2 quenching on fluorescence signal intensity becomes nearly constant and independent of x O 2 . During the experiments, x O 2 will be assumed constant and equal to 21% in the combustion chamber. This assumption is confirmed a posteriori by the low values of x kero in the combustion chamber and the absence of OH fluorescence signal where x kero is measured. The temperature dependencies of the two fluorescence signals (Filters A and B) at constant pressure andx O 2 = 21% are plotted in Fig.3b. Experimental points are reported and interpolations on the experimental data are used on the full temperature range (450 -750 K) to provide a look-up table for each given pressure. The shapes of the interpolated curves are in agreement with the fluorescence models of 1,2,4 Trimethylbenzene[13] and Naphthalene[14] which are representative

Figure 3 :

 3 Figure 3: (a) Effect of x O2 and T on the reference fluorescence signal S 0,A (T, P, x O2 ), at P = 0.1 MPa. Similar results are found for Filter B. (b) Temperature dependency on the reference fluorescence signals S 0,i (T, P, x O2 ), at P = 0.1 MPa and x O2 = 0.21. (c) Pressure dependency on the reference fluorescence signals S 0,i (T, P, x O2 ) for T = 450 K and 750 K, and for x O2 = 0.21.

  , for 0.83 MPa and 1.80 MPa with an air inlet temperature of 670 K. Results are presented both for axial and radial configurations. OH fluorescence signals (grey to white levels) are superimposed with kerosene mole fraction (jet color map) on the top half side and with the temperature of the kerosene air mixture on the bottom half side (red to yellow color map).

Figure 4 :

 4 Figure 4: Experimental measurement of kerosene mole fraction (x kero,m ) and temperature (T m ) from the calibration images. Imposed equivalence ratios are fixed to 0.05 and 0.1, and the temperature is set to 447 K. Imposed (subscript "theo") and measured values (subscript "m") are normalized by T max = 670 K and x kero,max = 0.00139.

Figure 5 :

 5 Figure 5: Instantaneous kerosene mole fraction (top), OH-PLIF signals and temperature (bottom) at 0.83 MPa in axial configuration (a) and in radial configuration at z/D = 0.75 (b), at 1.80 MPa in axial configuration (c), with T airinlet = 670 K.

  selecting a threshold value (15% of the maximum value, arbitrarily chosen) and displaying only the gradients above this threshold value indicating the most reactive flame fronts. For both pressure conditions, the flame presents a "tulip shape" with high levels of OH radicals mainly concentrated along the centerline and decreasing further downstream. Intense flame front wrinkling induced by strong velocity gradient between the ejected fresh gases and the recirculation zones can be also identified. The kerosene/air mixture is located at the edge of the injection system with a hollow cone shape. The origin of this fuel distribution arises from the architecture of the LP injector that spreads the liquid fuel towards the walls of the injector bowl located upstream the field of view (z ≤ 0). The resulting thin liquid fuel film produced along the edge of the bowl is then atomized at the outlet of the injector in a small crown-shaped region which progressively grows with the distance from the injector. At the upper part of the images, low values of kerosene mole fraction are detected with typical values close to 0.002 and decrease further downstream to become negligible at z/D = 1. It is also noticed that the instantaneous distribution of kerosene vapor is relatively heterogeneous due to the presence of local evaporation of small individual droplets initially produced by the liquid fuel film atomization and then spreading into the combustor. These results suggest that a large part of the fuel is consumed before entering in the combustion chamber. Combustion occurs mainly inside the injection system as shown by the large quantities of OH radicals at the nozzle exit. It is noticeable that the full consumption of kerosene fuel along the flow axis is also well correlated with the net depletion of the axial OH gradient.

Figure 6 :

 6 Figure 6: Mean kerosene mole fraction and associated mean OH distribution along the combustion chamber and across the combustion chamber at z/D = 0.75 for various operating conditions. Green lines: flame length (axial plane) and mean position of fresh/burnt gas interface (radial plane).

Acknowledgments

The authors are grateful for the financial support of the French aircraft motorist SAFRAN and the French National Research Agency (Industrial Chair PERCEVAL). We also thank Benjamin Quevreux, Gilles Godard and Felix Frindt for their technical support.