Anh-Tu Nguyen
email: nguyen.trananhtu@gmail.com.

Raymundo Márquez

Thierry-Marie Guerra

Antoine Dequidt

A.-T Nguyen

T.-M Guerra

Improved LMI Conditions for Local Quadratic Stabilization of Constrained Takagi-Sugeno Fuzzy Systems

Keywords: Takagi-Sugeno fuzzy systems, Local quadratic stabilization, Actuator saturation, State constraints, Descriptor redundancy representation, Linear matrix inequality (LMI)

This paper is devoted to the control synthesis of constrained Takagi-Sugeno (T-S) fuzzy systems. These systems are subject to multiple constraints, namely actuator saturation, system state constraints and also L 2 -disturbances. Using descriptor redundancy approach, new design conditions guaranteeing the existence of an input-saturated parallel distributed state feedback compensator are formulated as an LMI optimization problem with extra matrix decision variables. Differently from most of results dealing with the same local control context of constrained T-S systems, a generalized sector condition is used to handle effectively the nonlinearity of the control input. The use of a descriptor redundancy representation in conjunction with a sector condition for actuator saturation allows to reduce significantly the design conservatism while keeping a simple control structure. The effectiveness of the proposed method compared to many recent works is clearly demonstrated by means of numerical examples.

Introduction

Over the past two decades, model-based fuzzy control has been an issue of great interest in control theory and its applications [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF][START_REF] Su | A novel approach to output feedback control of fuzzy stochastic systems[END_REF][START_REF] Nguyen | Simultaneous design of parallel distributed output feedback and anti-windup compensators for constrained Takagi-Sugeno fuzzy systems[END_REF][START_REF] Lam | Stability analysis of sampled-data fuzzymodel-based control systems[END_REF][START_REF] Chen | Robust T-S fuzzyneural control of uncertain active suspension systems[END_REF][START_REF] Nguyen | Robust H ∞ control design for switching uncertain system: Application for turbocharged gasoline air system control[END_REF]. In particular, control technique based on Takagi-Sugeno (T-S) fuzzy models [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF] has become one of the most popular and promising approaches for dealing with complex nonlinear systems, see for instance [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF][START_REF] Su | A novel control design on discrete-time Takagi-Sugeno fuzzy systems with time-varying delays[END_REF][START_REF] Nguyen | LMI-based control synthesis of constrained Takagi-Sugeno fuzzy systems subject to L 2 and L ∞ disturbances[END_REF][START_REF] Pan | Nonquadratic stabilization of continuous T-S fuzzy models: LMI solution for a local approach[END_REF][START_REF] Wu | Dissipativity analysis and synthesis for discrete-time T-S fuzzy stochastic systemswith time-varying delay[END_REF] and references therein. Indeed, thanks to the polytopic structure of T-S models, T-S fuzzy model-based approaches provide a systematic framework for the studies of stability and stabilization of nonlinear systems [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. Moreover, the concept of parallel distributed compensation (PDC) offers an effective control design tool. Direct Lyapunov method has been widely applied in the control context of T-S fuzzy systems, leading to the conditions for checking the stability and control design which are usually expressed in terms of linear matrix inequalities (LMIs) [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. Without a doubt, quadratic approaches based on a common Lyapunov function still remain the most employed in fuzzy control framework and also in practical situations. This comes from the fact that they can provide a stabilizing controller with simple structure in many cases with reasonable computational cost [START_REF] Montagner | Convergent LMI relaxations for quadratic stabilizability and control of Takagi-Sugeno fuzzy systems[END_REF]. Sufficient quadratic conditions proving the existence of PDC controllers can be found in [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF][START_REF] Tuan | Parameterized linear matrix inequality techniques in fuzzy control system design[END_REF][START_REF] Nguyen | Lyapunov-based robust control design for a class of switching non-linear systems subject to input saturation: application to engine control[END_REF][START_REF] Ksantini | On the global exponential stabilization of Takagi-Sugeno fuzzy uncertain systems[END_REF]. Recent results dealing with the non-quadratic stabilization can be also mentioned such as [START_REF] Pan | Nonquadratic stabilization of continuous T-S fuzzy models: LMI solution for a local approach[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] Lam | LMI-based stability and performance conditions for continuous-time nonlinear systems in Takagi-Sugeno's form[END_REF][START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF][START_REF] Márquez | Asymptotically necessary and sufficient conditions for Takagi-Sugeno models using generalized non-quadratic parameter-dependent controller design[END_REF] for continuous-time T-S fuzzy systems and [START_REF] Guerra | LMI-based relaxed non-quadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form[END_REF][START_REF] Lee | Improvement on non-quadratic stabilization of discrete-time Takagi-Sugeno fuzzy systems: Multipleparameterization approach[END_REF] for discrete-time cases.

The present work is motivated by practical control issues. Concretely, for real-time applications, control signals and closed-loop system trajectories are usually limited in given ranges due to physical and/or safety reasons. Notice also that state constraints appear ubiquitously when the nonlinear sector-decomposition approach is used to obtain the T-S fuzzy representation of nonlinear systems [START_REF] Nguyen | Lyapunov-based robust control design for a class of switching non-linear systems subject to input saturation: application to engine control[END_REF]. Considering explicitly these limitations allows preventing destabilizing initial conditions [START_REF] Nguyen | Anti-windup based dynamic output feedback controller design with performance consideration for constrained Takagi-Sugeno systems[END_REF]. In particular, this consideration becomes crucial in the case where disturbance signals are involved in the system dynamics. However, state constraints are often neglected in T-S fuzzy model-based control design [START_REF] Nguyen | Lyapunov-based robust control design for a class of switching non-linear systems subject to input saturation: application to engine control[END_REF]. Control design of input-saturated systems is another challenging task in control theory and its applica-tions. It is worth noting that actuator saturation is unavoidable for most of physical systems, especially high performance applications. Neglecting this phenomenon in the control design procedure may lead to the degradation of closedloop performance, or even unstable behaviors [START_REF] Nguyen | Lyapunov-based robust control design for a class of switching non-linear systems subject to input saturation: application to engine control[END_REF][START_REF] Nguyen | Anti-windup based dynamic output feedback controller design with performance consideration for constrained Takagi-Sugeno systems[END_REF][START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]. Over the years, the control synthesis of input-saturated systems has been subject to extensive investigation, see for instance [START_REF] Nguyen | LMI-based control synthesis of constrained Takagi-Sugeno fuzzy systems subject to L 2 and L ∞ disturbances[END_REF][START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF][START_REF] Gassara | Control of time delay polynomial fuzzy model subject to actuator saturation[END_REF][START_REF] Du | Fuzzy control for nonlinear uncertain electrohydraulic active suspensions with input constraint[END_REF] and references therein. In T-S fuzzy control framework, notable works can be mentioned as follows. Norm-bounded approach was proposed in [START_REF] Du | Fuzzy control for nonlinear uncertain electrohydraulic active suspensions with input constraint[END_REF][START_REF] Tseng | H ∞ fuzzy control design for nonlinear systems subject to actuator saturation[END_REF] to deal with control input constraints. This method may lead to conservative results [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF][START_REF] Cao | Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation[END_REF]. Moreover, the resulting unsaturated PDC controllers generally offer poor closed-loop performance. Polytopic representation of the saturation nonlinearity [START_REF] Cao | Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation[END_REF] has been exploited in [START_REF] Ting | A robust fuzzy control approach to stabilization of nonlinear time-delay systems with saturating inputs[END_REF] for time-delay T-S systems, in [START_REF] Kim | H ∞ state-feedback control for fuzzy systems with input saturation via fuzzy weighting-dependent Lyapunov functions[END_REF] for discrete-time T-S systems, and in [START_REF] Nguyen | Lyapunov-based robust control design for a class of switching non-linear systems subject to input saturation: application to engine control[END_REF] for a class of switching T-S systems. In [START_REF] Gassara | Control of time delay polynomial fuzzy model subject to actuator saturation[END_REF], the authors proposed a polynomial fuzzy controller for a class of polynomial fuzzy systems subject to both state delay and actuator saturation using polytopic model of the input saturation. In particular, it has been shown in [START_REF] Nguyen | Lyapunov-based robust control design for a class of switching non-linear systems subject to input saturation: application to engine control[END_REF] that this polytopic representation approach leads to much less conservative results than those of the input-unsaturated approaches. Very recently, descriptor redundancy approach [START_REF] Tanaka | A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions[END_REF][START_REF] Guelton | Robust dynamic output feedback fuzzy lyapunov stabilization of Takagi-Sugeno systems -a descriptor redundancy approach[END_REF] has been proposed to deal with input-saturated T-S systems in [START_REF] Bezzaoucha | Stabilization of nonlinear systems subject to uncertainties and actuator saturation[END_REF]. It is important to stress that the results in [START_REF] Bezzaoucha | Stabilization of nonlinear systems subject to uncertainties and actuator saturation[END_REF] can be only applied to a restrictive class of T-S fuzzy systems with all linear subsystems that are open-loop stable. Moreover, this method cannot guarantee the asymptotic stability even exogenous disturbances are not involved in the dynamics of the closed-loop systems. Notice also that state constraints were not considered in most of these above works (except for [START_REF] Nguyen | Lyapunov-based robust control design for a class of switching non-linear systems subject to input saturation: application to engine control[END_REF]).

This paper presents a new LMI-based control method for continuous-time T-S fuzzy systems subject to both state and control input constraints and in presence of energy-bounded disturbances. The novelty and contributions of the proposed method can be summarized as follows.

(i) The closed-loop T-S fuzzy systems will be rewritten in an equivalent descriptor redundancy form which is compatible with the use of a new parameter-dependent Lyapunov function. This latter allows to avoid the unknown time-derivatives of the membership functions involved in the theoretical developments. In particular, a special matrix transformation is judiciously exploited to introduce slack variable matrices for the purpose of conservatism reduction. (ii) Differently from existing works in the control context of constrained T-S fuzzy systems, an extended parameterdependent version of the generalized sector condition [START_REF] Nguyen | LMI-based control synthesis of constrained Takagi-Sugeno fuzzy systems subject to L 2 and L ∞ disturbances[END_REF] is applied to deal with the actuator saturation. As a consequence, we can derive conveniently LMI-based design conditions with less design conservatism. The proposed control method is able to deal with open-loop unstable T-S fuzzy systems which is not the case of [START_REF] Bezzaoucha | Stabilization of nonlinear systems subject to uncertainties and actuator saturation[END_REF]. Therefore, it can be applied to a larger class of nonlinear constrained systems. (iii) By means of numerical experiments, it is demonstrated that in many cases the new method may lead to much less conservative results than recent works in quadratic [START_REF] Nguyen | Lyapunov-based robust control design for a class of switching non-linear systems subject to input saturation: application to engine control[END_REF][START_REF] Cao | Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation[END_REF] and non-quadratic [START_REF] Pan | Nonquadratic stabilization of continuous T-S fuzzy models: LMI solution for a local approach[END_REF][START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF][START_REF] Márquez | Asymptotically necessary and sufficient conditions for Takagi-Sugeno models using generalized non-quadratic parameter-dependent controller design[END_REF] control frameworks dealing with similar local control issue. It is also worth noting that for the proposed method, the control structure is very simple and the computational complexity remains reasonable for real-world applications.

The paper is organized as follows. Section 2 defines the considered control problem, and then some useful lemmas for the theoretical developments are also presented. Section 3 discusses on the derivation procedure of the proposed LMIbased design conditions. Numerical examples are given in Section 4 to demonstrate the effectiveness of the new method.

Finally, Section 5 provides some concluding remarks.

Notation. For an integer number r, Ω r denotes the set of integer numbers {1, 2, . . . , r}. I denotes the identity matrix of appropriate dimensions. For any square matrix X, He (X) = X + X . X > 0 means that the matrix X is symmetric and positive definite. The ith element of a vector u is denoted u (i) and X (i) denotes the ith row of matrix X. ( * ) stands for matrix blocks that can be deduced by symmetry. For a matrix P > 0 and a positive scalar ρ, we denote E (P, ρ) = x : x P x ≤ ρ and for brevity E (P ) ≡ E (P, 1). The scalar functions η 1 , . . . , η r are said to verify the convex sum property if η i (θ) ≥ 0, r i=1 η i = 1, ∀i ∈ Ω r . For such functions with any argument θ, and for matrices Y i and Z ij of appropriate dimensions, the following notation will be used for brevity:

Y θ = r i=1 η i (θ) Y i , Y -1 θ = r i=1 η i (θ) Y i -1 , Z θθ = r i=1 r j=1 η i (θ) η j (θ) Z ij . (1)
2 Problem Formulation

System description

Consider the following continuous-time T-S system [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]:

           ẋ = r i=1 η i (θ) (A i x + B u i v + B w i w) z = r i=1 η i (θ) (C i x + D w i w) (2)
where x ∈ R nx is the state, v ∈ R nu is the control input, w ∈ R nw is the system disturbance, and z ∈ R nz is the performance output. All components of the premise variable vector θ ∈ R np are supposed to be measured. The real constant matrices A i , B u i , B w i , C i , D w i , i ∈ Ω r are of adequate dimensions, and the scalar membership functions (MFs) η i (θ) satisfy the convex sum property. In this paper, the input vector v is bounded in amplitude:

-u max(l) ≤ v (l) ≤ u max(l) , l ∈ Ω nu , (3)
where the control bounds u max(l) > 0, l ∈ Ω nu are given. Moreover, we assume that the disturbance signal w in ( 2) is bounded in energy, i.e.

w ∈ W δ = w : R + → R nw , ∞ 0 w(τ ) w(τ )dτ ≤ δ ,
where the bound δ > 0 is given. In order to respect the control input limitations (3), the following saturated PDC control law is considered:

v = sat (u) , u = r i=1 η i (θ) K i x = K θ x, (4)
where sat (u) ∈ R nu and sat (u

) (l) = sign u (l) min u (l) , u max(l) , for l ∈ Ω nu . Let us denote x = x ẋ , ψ (u) = u -sat (u) . (5)
From ( 2) and ( 4), the closed-loop system can be equivalently represented in the following descriptor form:

E ẋ = A θθ x -B ψ θ ψ (u) + B w θ w, (6)
where

E = I 0 0 0 , A θθ = A θ + B u θ K θ , K θ = K θ 0 , A θ = 0 I A θ -I , B u θ = B ψ θ = 0 B u θ , B w θ = 0 B w θ . (7)

Problem definition

In this paper, we search for a PDC control law (4) for the constrained T-S fuzzy system (2) such that the following closed-loop properties are satisfied:

(i) State constraints: Given vectors h k ∈ R nx , k ∈ Ω q ,
there exist a matrix P > 0 and a positive scalar ρ such that, for any x (0) ∈ E (P, ρ) and for any w ∈ W δ , the closed-loop trajectory of (2) remains inside the following polyhedral set which defines the state constraints of the T-S fuzzy system (2):

P x = x ∈ R nx : h k x ≤ 1, k ∈ Ω q .
(ii) Regional quadratic stability and L 2 -gain performance:

In the absence of disturbances, i.e. when w = 0, any closed-loop trajectory initialized inside the set E (P ) converges exponentially to the origin. When w = 0 and for ∀w ∈ W δ , if x (0) ∈ E (P, ρ), then there exists a positive scalar γ such that the performance output z of the closed-loop system (2) is in L 2 and its norm satisfies the following condition:

∞ 0 z(τ ) z(τ )dτ ≤ γ ∞ 0 w(τ ) w(τ )dτ + ρ.

Useful lemmas

In what follows, some useful preliminaries needed for the control design in Section 3 are presented.

Lemma 1 Given matrices K i ∈ R nu×nx and G i ∈ R nu×nx , i ∈ Ω r .
Let us define the following set:

P u = x ∈ R nx : r i=1 η i K i(l) -G i(l) x ≤ u max(l)
for l ∈ Ω nu . Consider the PDC control law u of the form (4). If x ∈ P u , then the following inequality on the deadzone nonlinearity ψ (u):

ψ (u) r i=1 η i S i -1 ψ (u) - r i=1 η i G i x ≤ 0,
holds for any positive diagonal matrices S i ∈ R nu×nu , and for any scalar functions η i , i ∈ Ω r , satisfying the convex sum property.

Lemma 1 presents an extension of the generalized sector condition for control input nonlinearity proposed in [START_REF] Nguyen | LMI-based control synthesis of constrained Takagi-Sugeno fuzzy systems subject to L 2 and L ∞ disturbances[END_REF]. Notice that for this extended version, all involved matrices are parameter-dependent which helps to reduce the conservatism of the results.

Lemma 2 [START_REF] Peaucelle | A new robust D-stability condition for real convex polytopic uncertainty[END_REF] Let A, R, L, P and Q be matrices of appropriate dimensions. The following statements are equivalent:

a) A P + P A + Q < 0, b) ∃R, L : A L + L A + Q P -L + A R P -L + R A -R -R < 0.
Lemma 3 [START_REF] Tuan | Parameterized linear matrix inequality techniques in fuzzy control system design[END_REF] Let Υ ij , i, j ∈ Ω r be symmetric matrices of appropriate dimensions and η 1 , . . . , η r be a family of scalar functions satisfying the convex sum property. The following condition:

r i=1 r j=1 η i η j Υ ij < 0 (8) is verified if    Υ ii < 0, i ∈ Ω r 2 r -1 Υ ii + Υ ij + Υ ji < 0, i, j ∈ Ω r , i = j (9)
Notice that more efficient relaxation results than (9) can be found in [START_REF] Sala | Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya's theorem[END_REF] at the expense of high computational cost. At last, the results presented in this paper are independent to the relaxation methods in the sense that different relaxations can be straightforwardly applied to convert the parameterdependent condition (8) to a finite set of LMIs.

Main Results

The following theorem provides LMI conditions that allow the design of a PDC controller (4) solving the control problem defined in Section 2.

Theorem 1 Given a T-S fuzzy system (2) and a scalar δ > 0 where w ∈ W δ . Assume there exist positive definite matrix

X ∈ R nx×nx , positive diagonal matrices S i ∈ R nu×nu , regular matrices X ij 22 ∈ R nx×nx , matrices X ij 21 ∈ R nx×nx , H j 11 ∈ R nx×nx , H j 12 ∈ R nx×nx , H ij 21 ∈ R nx×nx , H ij 22 ∈ R nx×nx , R j 11 ∈ R nx×nx , R j 12 ∈ R nx×nx , R ij 21 ∈ R nx×nx , R ij 22 ∈ R nx×nx , V i ∈ R nu×nx , W i ∈ R nu×nx for i, j
∈ Ω r , and positive scalars ρ, γ such that

ρ + γδ ≤ 1, (10)
X * V i(l) -W i(l) u 2 max(l) ≥ 0, i ∈ Ω r , l ∈ Ω nu , (11) X Xh k * 1 ≥ 0, k ∈ Ω q , (12)
Ψ ii < 0, i ∈ Ω r , ( 13
) 2 r -1 Ψ ii + Ψ ij + Ψ ji < 0, i, j ∈ Ω r and i = j, (14)
where the quantities Ψ ij are defined in [START_REF] Nguyen | Lyapunov-based robust control design for a class of switching non-linear systems subject to input saturation: application to engine control[END_REF]. Then, the PDC controller (4) with the feedback gains given by

K i = V i X -1 , i ∈ Ω r ,
solves the control design problem stated in Section 2.

Proof Let us introduce the following parameter-dependent matrix:

X θθ = r i=1 r i=1 η i (θ) η j (θ) X 0 X ij 21 X ij 22 .
Since X = X > 0, it follows that

X θθ E = EX θθ ≥ 0. ( 16
)
Since X > 0 and the matrices X ij 22 are regular, for i, j ∈ Ω r , then the parameter-dependent matrix X θθ is also regular. Denoting P (θ) = X -1 θθ , it follows from ( 16) that

E P (θ) = P (θ) E ≥ 0, (17)
where the matrix P (θ) is of the following form:

P (θ) = r i=1 r i=1 η i (θ) η j (θ) X 0 X ij 21 X ij 22 -1 . ( 18
)
Considering the Lyapunov function

V (x) = x E P (θ) x.
From the definitions of x in ( 5), E in [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF], and P (θ) in ( 18), it can be easily observed that

V (x) = x P x,
where P = X -1 . It follows from Lemma 1 that the following inequality:

Ξ = V (x) + z z -γw w -2ψ (u) S -1 θ [ψ (u) -G θ x] = He ẋ E P (θ) x -ψ (u) S -1 θ [ψ (u) -G θ x] + z z -γw w < 0, (19)
implies clearly that

V (x) + z z -γw w < 0, ∀x ∈ P u . (20)
Moreover, with the expression of E ẋ defined in (6), using the property [START_REF] Tanaka | A multiple Lyapunov function approach to stabilization of fuzzy control systems[END_REF] and the notations in (1), the quantity Ξ in ( 19) can be rewritten in the following form:

Ξ =   x ψ (u) w   Φ   x ψ (u) w   < 0, (21)
Ψ ij = He                     H ij 21 H ij 22 0 0 0 X + R ij 21 R ij 22 A i H j 11 + B u i V j -H ij 21 A i H j 12 -H ij 22 -B u i S j B w i 0 A i R j 11 -R ij 21 A i R j 12 -R ij 22 W j 0 -S j 0 0 0 0 0 0 0 -γI/2 0 0 0 C i X 0 0 D w i -I/2 0 0 -H j 11 -H j 12 0 0 0 -R j 11 -R j 12 X ij 21 -H ij 21 X ij 22 -H ij 22 0 0 0 -R ij 21 -R ij 22                     (15)
where

Φ =   A θθ P (θ) + P (θ) A θθ -P (θ) B ψ θ + G θ S -1 θ P (θ) B w θ * -2S -1 θ 0 * * -γI   +   C θ 0 D w θ   C θ 0 D w θ ,
and

C θ = C θ 0 , G θ = G θ 0 , A θθ = A θ + B u θ K θ .
It can be observed that the condition (21) holds if Φ < 0. By the well-known Schur complement lemma [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF], this latter can be proved to be equivalent to the following inequality:

    He A θθ P (θ) -P (θ) B ψ θ + G θ S -1 θ P (θ) B w θ C θ * -2S -1 θ 0 0 * * -γI D w θ * * * -I     < 0 (22)
Pre-and post-multiplying [START_REF] Guerra | LMI-based relaxed non-quadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form[END_REF] with diag X θθ , S θ , I, I and its transpose, this inequality is found to be equivalent to

Υ =     Υ 11 -B ψ θ S θ + X θθ G θ S θ B w θ X θθ C θ * -2S θ 0 0 * * -γI D w θ * * * -I     < 0, (23)
where

Υ 11 = He (A θ X θθ + B u θ K θ X θθ ). Applying Lemma 2 with P = X θθ , A = A θ , L = H θθ , R = R θθ and Q = He (B u θ K θ X θθ )
, to the block-matrix entry Υ 11 of Υ defined in [START_REF] Lee | Improvement on non-quadratic stabilization of discrete-time Takagi-Sugeno fuzzy systems: Multipleparameterization approach[END_REF], we can show that this inequality is equivalent to the condition [START_REF] Nguyen | Anti-windup based dynamic output feedback controller design with performance consideration for constrained Takagi-Sugeno systems[END_REF]. Now, let us denote

H θθ = H θ 11 H θ 12 H θθ 21 H θθ 22 , R θθ = R θ 11 R θ 12 R θθ 21 R θθ 22 , W i = G i X. (25)
Then, the inequality ( 24) becomes ( 26) after some algebraic manipulations. Notice that the use of full parameter dependent matrix H θθ defined in [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF] with extra degrees of freedom instead of X 0 X θθ 21 X θθ

22

explains the relaxation idea under this new control scheme. By the relaxation result of Lemma 3, we can prove that if x ∈ P u , then the satisfaction of the conditions ( 13)- [START_REF] Tuan | Parameterized linear matrix inequality techniques in fuzzy control system design[END_REF] with Ψ ij , i, j ∈ Ω r , defined in [START_REF] Nguyen | Lyapunov-based robust control design for a class of switching non-linear systems subject to input saturation: application to engine control[END_REF] implies clearly the condition [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF]. Moreover, by Schur complement lemma, ( 11) is found to be equivalent to

X - V i(l) -W i(l) V i(l) -W i(l) u 2 max(l) ≥ 0, (27)
for i ∈ Ω r , l ∈ Ω nu . Pre-and post-multiplying [START_REF] Du | Fuzzy control for nonlinear uncertain electrohydraulic active suspensions with input constraint[END_REF] with the matrix P yields

P - K i(l) -G i(l) K i(l) -G i(l) u 2 max(l) ≥ 0, (28)
for i ∈ Ω r and l ∈ Ω nu . Since the positive scalar functions η i , i ∈ Ω r , satisfy the convex sum property, it is then deduced that the satisfaction of ( 28) implies the inclusion E (P ) ⊆ P u . Similarly, the inequality [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] implies the inclusion E (P ) ⊆ P x . Now, we distinguish two following cases:

(i) If w = 0, it follows from (20) that V (x) < 0, x ∈ E (P ) ,
which guarantees the regional quadratic internal stability of the closed-loop system (2).

(ii) If w = 0 and w ∈ W δ , integrating both sides of (20) from 0 to any time T > 0 yields

V (x (T )) < V (x (0)) + γ T 0 w(τ ) w(τ )dτ (29)
- T 0 z(τ ) z(τ )dτ.
Then, it follows from ( 10) and (29) that

V (x (T )) ≤ ρ + γδ ≤ 1, ∀x (0) ∈ E (P, ρ) ,
which means that the corresponding closed-loop trajectory remains inside the set E (P ) ⊆ P x . This fact proves that state constraints are satisfied. Furthermore, from (29) and considering the limit case T → ∞, we obtain

∞ 0 z(τ ) z(τ )dτ ≤ γ ∞ 0 w(τ ) w(τ )dτ + V (x (0)) ,
which implies the L 2 -gain performance of the closed-loop system. Thus, the proof can be now concluded.

      He (A θ H θθ + B u θ K θ X θθ ) -B ψ θ S θ + X θθ G θ S θ B w θ X θθ C θ X θθ -H θθ + A θ R θθ * -2S θ 0 0 0 * * -γI D w θ 0 * * * -I 0 * * * * -R θθ -R θθ       < 0. ( 24
) He                   H θθ 21 H θθ 22 0 0 0 X + R θθ 21 R θθ 22 A θ H θ 11 + B u θ V θ -H θθ 21 A θ H θ 12 -H θθ 22 -B u θ S θ B w θ 0 A θ R θ 11 -R θθ 21 A θ R θ 12 -R θθ 22 W j 0 -S θ 0 0 0 0 0 0 0 -γI/2 0 0 0 C θ X 0 0 D w θ -I/2 0 0 -H θ 11 -H θ 12 0 0 0 -R θ 11 -R θ 12 X θθ 21 -H θθ 21 X θθ 22 -H θθ 22 0 0 0 -R θθ 21 -R θθ 22                   < 0. ( 26
)
Remark 1 From the results of Theorem 1, it is easily to formulate the LMI optimization that can search for a PDC controller solving the control problem defined in Section 2 while minimizing the estimated bound of the system L 2 -gain. Such an LMI optimization can be stated as follows:

min γ, subject to: γ > 0 and the LMI conditions in Theorem 1.

Remark 2 Since local control context is considered here, in many cases it is desirable to achieve the largest estimate of the domain of attraction of the closed-loop system. Notice that the ellipsoid E (P ) represents the domain of attraction of (2) in the absence of system disturbances, i.e. w = 0. Note also that the size of E (P ) can be characterized by trace (P ) [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. Then, the maximization of E (P ) can be formulated as follows:

min trace (R) , (30)
subject to LMI conditions in Theorem 1 and

R I I X > 0, (31)
where R > 0 is a slack variable. By Schur complement lemma, it follows from (31) that R > P = X -1 , thus trace(R) > trace(P ). Then, the minimization of trace(R) implies clearly the minimization of trace(P ).

Remark 3 Notice that all matrices X θθ , H θθ , R θθ , used in the theoretical development of Theorem 1 depend on the MFs η i (θ), i ∈ Ω r , in a polynomial form. These matrices can be chosen in the following general forms defined in [START_REF] Tanaka | A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions[END_REF]. In the expressions [START_REF] Tanaka | A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions[END_REF], N can be viewed as the degree of the polynomials in function of MFs of the each parameter-dependent matrix. Then, less conservative results can be achieved by increasing N . For a trade-off between relaxation and computational cost, the design conditions of Theorem 1 are obtained with N = 1. However the results for N ≥ 2 can be easily generalized. Notice also that in the case of high-dimensional parameter-dependent summations (i.e. with a large value of N ), the relaxation result in [START_REF] Sala | Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya's theorem[END_REF] would be more interesting than that of Lemma 3.

Remark 4

The conservatism of the proposed method comes mainly from the fact that matrix X in ( 32) is not parameter dependent. This is done on purpose to avoid the unknown time-derivatives of the MFs ηi (θ), i ∈ Ω r , involved in the theoretical developments. Up two now, this inherent control issue is still very challenging in the framework of continuous-time T-S fuzzy systems, see [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] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF][START_REF] Mozelli | Reducing conservativeness in recent stability conditions of T-S fuzzy systems[END_REF] and references therein. In [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF], local design conditions based on fuzzy Lyapunov functions for continuous-time T-S systems have been proposed. An interesting feature of this method consists in the fact that implicit bounds on ηi (θ), i ∈ Ω r , are not required for the design conditions as in many other works, for instance [START_REF] Tanaka | A multiple Lyapunov function approach to stabilization of fuzzy control systems[END_REF][START_REF] Mozelli | Reducing conservativeness in recent stability conditions of T-S fuzzy systems[END_REF]. It is noteworthy that although this requirement is understandable for stability results, it is more questionable for stabilization due to the explicit presence of the to-be-designed control actions u in the expressions of ηi = ∂ηi ∂x ẋ and ẋ. As a consequence, the validity region of the obtained controller must be checked a posteriori. In this context, the method in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF] requires the knowledge on the control bounds to obtain LMI formulation. Therefore, even the goals pursued in the present work and the one in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF] are definitively different, they can be however compared objectively which is not the case for other nonquadratic methods previously cited. Note also that the control method in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF] already allows outperforming many wellknown results existing in the literature of T-S control systems, see for instance [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].

Remark 5

We note that non-quadratic Lyapunov based methods in the local control framework of continuous-time T-S systems usually involve complex design conditions [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF]. The computational complexity of LMI-based optimization problems is related to the number of scalar decision variables N var and the number of rows N row of all considered LMIs. These numbers corresponding to the LMIs of different results are respectively given as follows.

(i) For the design conditions of Theorem 3 in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF]:

         N var = 1 + r (n x (n x + 1) /2 + n u n x + n p n u n x ) + n p n 2 x r 2 (n x + n u ) (n x + n u + 1 + (n x + 1) r) N row = 1 + rn x + r 3 n x + n z + n w + n p n 2 x n u + r 2 n u n x + (n x + n u ) n p n 2 x X (θ) = X 0 X 21 (θ) X 22 (θ) = r i 0 =1 r i 1 =1 • • • r i N =1   N j=0 η i j (θ)   X 0 X i 0 i 1 ...i N 21 X i 0 i 1 ...i N 22 , H (θ) = H 11 (θ) H 12 (θ) H 21 (θ) H 22 (θ) = r i 0 =1 r i 1 =1 • • • r i N =1   N j=0 η i j (θ)   H i 1 ...i N 11 H i 1 ...i N 12 H i 0 i 1 ...i N 21 H i 0 i 1 ...i N 22 , R (θ) = R 11 (θ) R 12 (θ) R 21 (θ) R 22 (θ) = r i 0 =1 r i 1 =1 • • • r i N =1   N j=0 η i j (θ)   R i 1 ...i N 11 R i 1 ...i N 12 R i 0 i 1 ...i N 21 R i 0 i 1 ...i N 22 . (32)
(ii) For the design conditions in [START_REF] Nguyen | Lyapunov-based robust control design for a class of switching non-linear systems subject to input saturation: application to engine control[END_REF][START_REF] Cao | Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation[END_REF]:

     N var = 3 + n x (n x + 1) /2 + 2rsn u n x N row = 4 + q (n x + 1) + 3n x + rsn u (n x + 1) + 2 nu r 2 s (n x + n z + n w )
with s = 1 (i.e. one switching region). (iii) For the design conditions of Theorem 1 in this work:

         N var = n x (n x + 1) /2 + r n u + 4n 2 x + 2n u n x + 6r 2 n 2 x + 2 N row = 1 + q (n x + 1) + rn u (n x + 1) + r 2 (4n x + n u + n z + n w )
In order to illustrate the computational burden of different control methods, the numbers of decision variables and LMI rows corresponding to the design conditions of Theorem 1 in [START_REF] Nguyen | Lyapunov-based robust control design for a class of switching non-linear systems subject to input saturation: application to engine control[END_REF], Theorem 3 in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF], and Theorem 1 in this work for different values of system dimensions are given in Table 1. We can observe that the numerical complexity of the new control method is reasonable compared to available results in the literature. Notice also that the number of LMI rows of conditions in [START_REF] Nguyen | Lyapunov-based robust control design for a class of switching non-linear systems subject to input saturation: application to engine control[END_REF] depends strongly on the numbers of control inputs (n u ) and switching regions (s).

Another advantage of the proposed method with respect to the non-quadratic one in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF] is that the controller structure ( 4) is very simple compared to the non-PDC control law in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF] since it does not require online parameter-dependent matrix inversion for real-time implementation. Therefore, the new control method seems to be more appropriate for real-world applications, especially for T-S fuzzy systems of high dimension and/or with an important number of subsystems. At last, from the viewpoint of input-saturated control systems, in many cases the proposed method can lead to less conservative results than those of [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF]. These facts will be clearly illustrated in Section 4.

Numerical Examples

In this section, the effectiveness of the proposed method will be demonstrated by means of two numerical examples. The first one deals with a physically motivated nonlinear system. The second example aims for a study on the conservatism of the proposed method compared to several related works in the literature. All LMI problems are solved with the help of Matlab R2013b using YALMIP toolbox [START_REF] Löfberg | YALMIP: A toolbox for modeling and optimization in MATLAB[END_REF].

Example 1 Consider the well-known chaotic Lorenz system described by

     ẋ1 = -µ 1 x 1 + µ 1 x 2 ẋ2 = µ 2 x 1 -x 2 -x 1 x 3 ẋ3 = x 1 x 2 -µ 3 x 3 (33)
The T-S fuzzy representation of ( 33) is given as follows [START_REF] Lee | Robust fuzzy control of nonlinear systems with parametric uncertainties[END_REF]:

Plant rules: Rule 1: If x 1 is about η 1 , then x = A 1 x Rule 2: If x 1 is about η 2 , then x = A 2 x (34)
where

A 1 =   -µ 1 µ 1 0 µ 2 -1 20 0 -20 -µ 3   , A 2 =   -µ 1 µ 1 0 µ 2 -1 -30 0 30 -µ 3   , η 1 (x 1 ) = (30 -x 1 ) /50, η 2 (x 1 ) = 1 -η 1 (x 1 ) . ( 35
)
Table 1 Illustration of the computational complexity for different control methods (green-good, black-intermediate, red-worst)

System parameters

Method in [START_REF] Nguyen | Lyapunov-based robust control design for a class of switching non-linear systems subject to input saturation: application to engine control[END_REF] Method in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF] Theorem 1

N var N row N var N row N var N row r = n x = 2, q = 4, n u = n w = n z = s = n p = 1 14 60 255 125 143 65 r = q = 4, n x = n u = n p = 2, n w = n z = s = 1 22 162 1565 745 473 203 r = q = 4, n x = s = 2, n u = n w = n z = n p = 1 22 302 1565 745 473 203 r = q = 4, n x = s = 2, n u = 3, n w = n z = n p = 1 54 1118 2941 1449 513 259
After introducing the control inputs, the system disturbance and the performance outputs into (34)- [START_REF] Peaucelle | A new robust D-stability condition for real convex polytopic uncertainty[END_REF], we obtain the T-S system (2) with the following data:

B u 1 =   µ 1 0 0   , B u 2 =   0 µ 2 0   , C 1 = 1 0 0 , B w 1 =   0.1µ 1 0.1µ 2 0.5   , B w 2 =   -0.1µ 1 0.1µ 2 0.5   , C 2 = 1 0 0 , D w 1 = 0.1, D w 2 = -0.1, µ 1 = 10, µ 2 = 28, µ 3 = 2.67.
It should be stressed that in order to guarantee the validity of the membership functions η 1 (x 1 ) and η 2 (x 1 ) in ( 35), the state constraints P x = x ∈ R 2 : -20 ≤ x 1 ≤ 30 must be explicitly considered into the design procedure. Moreover, it is assumed that the control input is limited in amplitude with u max = 3 and the disturbance belongs to a class of L 2 functions, i.e. w ∈ W δ with δ = 2.8. Notice that all these system constraints have not been considered in previous works. Now, solving LMI conditions of Theorem 1 leads to the following results:

P =   0.
Fig. 1 shows the closed-loop behaviors corresponding to the initial condition x 0 = -1 0 1 for the energy-bounded disturbance defined as follows:

w (t) = 9, if 1 ≤ t ≤ 1.1 0, otherwise (37)
The trajectory of the performance output z and the L 2 -gain

ratio ∞ 0 z (t) z (t) dt/ ∞ 0 w (t) w (t) dt + ρ correspond-
ing to the results given in Fig. 1 are depicted in Fig. 2. It can be clearly observed that the effect of the disturbance w is well attenuated with the proposed controller and the L 2 -gain performance of the closed-loop T-S system is respected under the considered simulation condition. Fig. 3 shows the comparison between the closed-loop responses obtained with Theorem 1 and those obtained with the design conditions in [1, Chapter 3]. We observe that the explicit consideration of system constraints into the proposed design procedure can help to improve significantly the transient behaviors of the closed-loop T-S fuzzy system. Moreover, note also that the control method in [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] cannot provide any guarantee on the closed-loop stability in the presence of system constraints. Fig. 1 Closed-loop behaviors corresponding to the initial condition x 0 = [-1 0 1] and the energy-bounded disturbance in [START_REF] Mozelli | Reducing conservativeness in recent stability conditions of T-S fuzzy systems[END_REF] with respect to the feedback gains given in [START_REF] Sala | Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya's theorem[END_REF]. Fig. 3 Closed-loop behaviors corresponding to the initial condition x 0 = [-17 7 30] and the energy-bounded disturbance in (37) obtained with Theorem 1 and the control method proposed in [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF].

Example 2 Consider a family of T-S systems (2) with the following system data:

where (a, b) ∈ [-4, 4]. The membership functions in [START_REF] Löfberg | YALMIP: A toolbox for modeling and optimization in MATLAB[END_REF] are given by

η 1 (x 1 ) = 1 -sin (x 1 ) 2 , η 2 (x 1 ) = 1 + sin (x 1 ) 2 .
The constraints of the T-S system [START_REF] Löfberg | YALMIP: A toolbox for modeling and optimization in MATLAB[END_REF] are characterized by δ = 0.1, u max = 5 and

P x = x ∈ R 2 : |x 1 | ≤ 1, |x 2 | ≤ 1 .
We note that all T-S systems given in [START_REF] Löfberg | YALMIP: A toolbox for modeling and optimization in MATLAB[END_REF] are open-loop unstable. Now, let us check the existence of a stabilizing controller of [START_REF] Löfberg | YALMIP: A toolbox for modeling and optimization in MATLAB[END_REF] for all the points of a 17-by-17 square grid on the parameter space (a, b) by using different approaches dealing with similar local control context. The feasibility spaces obtained with conditions of Theorem 1 and those of the quadratic approach using polytopic representation of the saturation function in [START_REF] Nguyen | Lyapunov-based robust control design for a class of switching non-linear systems subject to input saturation: application to engine control[END_REF][START_REF] Cao | Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation[END_REF] are depicted in Fig. 4. It can be easily observed that the proposed method provides significantly less conservative results than those of [START_REF] Nguyen | Lyapunov-based robust control design for a class of switching non-linear systems subject to input saturation: application to engine control[END_REF][START_REF] Cao | Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation[END_REF]. It is important to note also that the following design conditions based on: (i) fuzzy Lyapunov functions using a non-parallel distributed control law proposed in [START_REF] Pan | Nonquadratic stabilization of continuous T-S fuzzy models: LMI solution for a local approach[END_REF][START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF][START_REF] Márquez | Asymptotically necessary and sufficient conditions for Takagi-Sugeno models using generalized non-quadratic parameter-dependent controller design[END_REF], and (ii) a descriptor redundancy representation of the closed-loop system using a polytopic representation of the saturation function proposed in [START_REF] Bezzaoucha | Stabilization of nonlinear systems subject to uncertainties and actuator saturation[END_REF], are all unfeasible for any point of the considered parameter grid. We note that for an objective comparison, the system constraints and the induced L 2 -gain performance have been considered in the design procedure of [START_REF] Pan | Nonquadratic stabilization of continuous T-S fuzzy models: LMI solution for a local approach[END_REF][START_REF] Márquez | Asymptotically necessary and sufficient conditions for Takagi-Sugeno models using generalized non-quadratic parameter-dependent controller design[END_REF] as the same way in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF]. Notice also that the results proposed in [START_REF] Pan | Nonquadratic stabilization of continuous T-S fuzzy models: LMI solution for a local approach[END_REF][START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF][START_REF] Márquez | Asymptotically necessary and sufficient conditions for Takagi-Sugeno models using generalized non-quadratic parameter-dependent controller design[END_REF] have already outperformed (in terms of conservatism reduction) many other outstanding results in nonquadratic T-S control framework, for instance [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]. For this example, the numbers N var and N row characterizing the computational complexity of LMI conditions (see Remark 5) for different methods are shown in Table 2. We can see that both N var and N row of the new control method are more important than those of [START_REF] Nguyen | Lyapunov-based robust control design for a class of switching non-linear systems subject to input saturation: application to engine control[END_REF][START_REF] Cao | Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation[END_REF]. This is due to the use of Lemma 2 to introduce slack variables for the purpose of conservatism reduction. However, the computational burden of the proposed method is significantly lower than that of the non-quadratic approach in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF] even though this latter cannot provide any solution for the considered example. At a first glance, it seems very surprising that the new quadratic method could lead to much less conservative results than the fuzzy Lyapunov based methods proposed in [START_REF] Pan | Nonquadratic stabilization of continuous T-S fuzzy models: LMI solution for a local approach[END_REF][START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF][START_REF] Márquez | Asymptotically necessary and sufficient conditions for Takagi-Sugeno models using generalized non-quadratic parameter-dependent controller design[END_REF]. However, from the viewpoint of constrained T-S fuzzy systems, the design conservatism comes not only from the choice of Lyapunov functions but also the methods to handle the system constraints. Indeed, the control method in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF] made use of an unsaturated control law (due to its way to deal with the time-derivative of the membership functions involved in the theoretical developments) which can lead to over-conservative results as shown in [START_REF] Nguyen | Lyapunov-based robust control design for a class of switching non-linear systems subject to input saturation: application to engine control[END_REF]. Moreover, that method depends strongly on the value of control bound limits, i.e. u max . Indeed, feasible solutions can be generally obtained with [START_REF] Pan | Nonquadratic stabilization of continuous T-S fuzzy models: LMI solution for a local approach[END_REF][START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF][START_REF] Márquez | Asymptotically necessary and sufficient conditions for Takagi-Sugeno models using generalized non-quadratic parameter-dependent controller design[END_REF] only if the limitations on the control input amplitude become arbitrarily large, for example u max ≥ 24 for the LMIs of [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF], u max ≥ 19.1 for the LMIs of [START_REF] Pan | Nonquadratic stabilization of continuous T-S fuzzy models: LMI solution for a local approach[END_REF], and u max ≥ 15.5 (with = 5) for the LMIs of [START_REF] Márquez | Asymptotically necessary and sufficient conditions for Takagi-Sugeno models using generalized non-quadratic parameter-dependent controller design[END_REF], in the case of the constrained T-S system [START_REF] Löfberg | YALMIP: A toolbox for modeling and optimization in MATLAB[END_REF]. Now, let us consider the constrained T-S system (38) with a = b = 0. It is noteworthy that none of the results of existing works in [START_REF] Pan | Nonquadratic stabilization of continuous T-S fuzzy models: LMI solution for a local approach[END_REF][START_REF] Nguyen | Lyapunov-based robust control design for a class of switching non-linear systems subject to input saturation: application to engine control[END_REF][START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF][START_REF] Márquez | Asymptotically necessary and sufficient conditions for Takagi-Sugeno models using generalized non-quadratic parameter-dependent controller design[END_REF][START_REF] Cao | Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation[END_REF][START_REF] Bezzaoucha | Stabilization of nonlinear systems subject to uncertainties and actuator saturation[END_REF] can provide a stabilizing controller for the corresponding T-S system. Solving the LMI optimization problem in [START_REF] Ting | A robust fuzzy control approach to stabilization of nonlinear time-delay systems with saturating inputs[END_REF] We can observe in Fig. 5 that the ellipsoid S 1 ≡ E(P ) is maximized along the polyhedral set S 2 ≡ P u ∩ P x which is, in turn, included inside the following sector set:

S 3 = x ∈ R nx : r i=1 η i K i(l) -G i(l) x ≤ u max(l) ,
for l ∈ Ω nu . Moreover, simulated trajectories of the closedloop T-S system starting from S 1 always remain inside this set. Therefore, the set inclusion property stated in Section 2 is clearly verified.

The results obtained with the above examples confirm the strong interest of the proposed control method compared to existing results in the local control framework of constrained T-S fuzzy systems.

Conclusions

In this paper, a new control method is proposed to improve the quadratic design conditions of constrained T-S fuzzy systems. The controller design amounts to solving a set of LMIs with some available numerical tools. By exploiting the descriptor redundancy representation of the closed-loop system as well as a special matrix transformation, slack decision variables are judiciously introduced to reduce effectively the design conservatism. We also propose to use a parameter-dependent version of generalized sector condition for dealing with control input nonlinearity. By means of numerical experiments, it has been demonstrated that these facts allows for an outperformance in terms of relaxation of the proposed method compared to several recent works dealing with similar local control issue in both quadratic and non-quadratic T-S control frameworks. Moreover, the control structure of the PDC control law is simple and the computational complexity of the proposed design conditions is reasonable for real-time applications. Future works focus on the application of the new method to intelligent vehicle systems [START_REF] Nguyen | Shared lateral control with online adaptation of the automation degree for driver steering assist system: A weighting design approach[END_REF]. Moreover, extensions of the proposed approach to T-S output feedback control contexts [START_REF] Su | A novel approach to output feedback control of fuzzy stochastic systems[END_REF][START_REF] Su | A novel control design on discrete-time Takagi-Sugeno fuzzy systems with time-varying delays[END_REF] would be also interesting.

Fig. 2

 2 Fig. 2 Trajectory of the performance output z and the ratio ∞ 0

1 =

 1 C 4 = -0.1 -0.4 , C 2 = 0.1 -0.1 , D w i = 0.15, i ∈ Ω 4 , C 3 = 0.2 0.1 ,

Fig. 4

 4 Fig. 4 Feasibility spaces obtained with Theorem 1 (+) and control methods proposed in [15, 29] ( ).

1 =

 1 -6.93 -6.60 , γ = 10.0, K 2 = -3.97 -9.80 .

3 Fig. 5

 35 Fig.5Set inclusion property and simulated system trajectories.

Table 2

 2

	Comparison of numerical complexity between different con-
	trol design methods
	Complexity Theorem 1 Method in [20] Method in [15, 29]
	N row	203	745	162
	N var	473	1565	22

Acknowledgements This work was supported in part by the Program "Vehicle for the Future" via ADEME (French Environment and Energy Management Agency), the European Community (through the FEDER European Funds for Regional Development), the region Hautsde-France in the framework of the CERVIFER project; in part by the International Campus on Safety and Intermodality in Transportation, the Délégation Régionale à la Recherche et à la Technologie, the Ministère de l'Enseignement Supérieur et de la Recherche and the Centre National de la Recherche Scientifique (CNRS); in part by the Mexican Agency PRODEP via Project No. DSA/103.5/16/10200. The authors gratefully acknowledge the support of these institutions.