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

Raymundo Marquez

Antoine Dequidt

Anh-Tu Nguyen

Raymundo Márquez

A.-T Nguyen

Márquez

An Augmented System Approach for LMI-Based Control Synthesis of Constrained Takagi-Sugeno Systems

Keywords: Takagi-Sugeno fuzzy systems, constrained control, system augmentation approach, L 2 -gain performance, linear matrix inequality (LMI)

In this paper, we address the control design of a class of constrained Takagi-Sugeno fuzzy systems. These systems are subject to both control input and state constraints. Sufficient design conditions are derived via a judicious use of an equivalent augmentation form of the closed-loop system together with a new parameter-dependent Lyapunov function. The control input nonlinearity is dealt with via a generalized sector condition. These instrumental control tools allow reducing significantly the conservatism of the results while keeping a simple control structure with a low level of numerical complexity. In particular, it has been shown that in many cases the new method can provide better results (in terms of conservatism reduction) than some recent works developed in non-quadratic Lyapunov framework. Several numerical examples are given to illustrate the effectiveness of the proposed method.

Introduction

In recent years, control design 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 research topics in nonlinear model-based control, see [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] for quick overviews. This is due to the fact that T-S fuzzy models can approximate any smooth nonlinear systems with any degree of accuracy [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. In addition, the nonlinear sector decomposition approach provides a systematic way to derive an exact T-S representation of a nonlinear model in a compact set. Moreover, the control technique based on the Parallel Distributed Compensation (PDC) control law provides a systematic and effective design framework to deal with nonlinear systems. Up to now, numerous successful applications of the T-S fuzzy model-based control technique have been reported in the literature [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF][START_REF] Du | Fuzzy control for nonlinear uncertain electrohydraulic active suspensions with input constraint[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].

For real-world applications, the control input signals and the closed-loop trajectories are usually bounded for physical and/or safety constraints [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]. Notice that state constraints appear naturally when the nonlinear sector-decomposition approach is used to derive a T-S representation of a nonlinear system [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality 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]. Such type of system constraints must be explicitly taken into account in the control design procedure to obtain a sufficiently large domain of attraction [START_REF] Nguyen | Anti-windup based dynamic output feedback controller design with performance consideration for constrained Takagi-Sugeno systems[END_REF]. Control input saturation is ubiquitous in practice, especially for high performance applications; neglecting it may lead to a severe degradation of control performance, or even unstable behaviors of closed-loop systems [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]. The control problem of T-S fuzzy models with saturated inputs has recently received increasing attention [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] Kim | H∞ state-feedback control for fuzzy systems with input saturation via fuzzy weighting-dependent Lyapunov functions[END_REF][START_REF] Zhao | Robust stabilization of T-S fuzzy discrete systems with actuator saturation via PDC and non-PDC law[END_REF][START_REF] Ariño | Guaranteed cost control analysis and iterative design for constrained Takagi-Sugeno systems[END_REF]. The authors in [START_REF] Ariño | Guaranteed cost control analysis and iterative design for constrained Takagi-Sugeno systems[END_REF] have proposed an iterative control technique for constrained T-S fuzzy systems which is based on multi-parametric quadratic programming. In [START_REF] Du | Fuzzy control for nonlinear uncertain electrohydraulic active suspensions with input constraint[END_REF], a norm-bounded approach is used to handle the input constraints which results in non-saturating controllers with poor performance [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]. Descriptor representation approach [START_REF] Bouarar | Robust non-quadratic static output feedback controller design for Takagi-Sugeno systems using descriptor redundancy[END_REF] has been recently employed to deal with inputsaturated T-S systems together with a polytopic representation of the saturation function [START_REF] Bezzaoucha | Stabilization of nonlinear systems subject to uncertainties and actuator saturation[END_REF]. Note that the results in [START_REF] Bezzaoucha | Stabilization of nonlinear systems subject to uncertainties and actuator saturation[END_REF] are only applicable to a restrictive class of T-S fuzzy models with all linear subsystems that must be open-loop stable. Moreover, that method cannot guarantee the closed-loop asymptotic stability even if exogenous disturbances are not involved in the system dynamics. Another polytopic representation of the saturation function proposed in [START_REF] Cao | Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation[END_REF] has been employed 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][START_REF] Zhao | Robust stabilization of T-S fuzzy discrete systems with actuator saturation via PDC and non-PDC law[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. To reduce the design conservatism, homogeneous polynomial parameterdependent Lyapunov functions have been exploited in [START_REF] Delibaşı | L 2 control of LPV systems with saturating actuators: Pólya approach[END_REF] for control design of input-saturated linear parameter varying (LPV) systems. Unfortunately, the method in [START_REF] Delibaşı | L 2 control of LPV systems with saturating actuators: Pólya approach[END_REF] suffers a serious technical problem since the unknown time-derivative of the parameter-dependent Lyapunov matrix involved in the development has been completely ignored. Recently, the synthesis of antiwindup (AW) compensators for constrained T-S systems have been also considered. In [START_REF] Ting | Robust anti-windup controller design of timedelay fuzzy systems with actuator saturations[END_REF], a two-step AW approach has been employed to deal with time-delay constrained T-S systems. For that, dynamic output feedback controllers must be given a priori and the control goal is only to design AW compensators to minimize any adverse effect caused by the input saturation. Despite its simple design procedure, two-step AW approaches may lead to excessively small operating regions due to the separate design of nominal controllers and AW compensators. To overcome that, a new one-step AW approach for constrained T-S systems has been proposed in [START_REF] Nguyen | Anti-windup based dynamic output feedback controller design with performance consideration for constrained Takagi-Sugeno systems[END_REF] where dynamic controllers and AW compensators can be simultaneously designed. Notice that state constraints have not been considered in most of existing results in the T-S control framework (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][START_REF] Nguyen | Anti-windup based dynamic output feedback controller design with performance consideration for constrained Takagi-Sugeno systems[END_REF]).

Motivated by the above control issues, this paper presents a new LMI-based control method for constrained T-S fuzzy systems subject to L 2disturbances. The key features of the proposed method can be summarized as follows.

(i) An augmented system approach is used with a new parameter-dependent Lyapunov function for control design. By this way, slack decision variables can be introduced to reduce significantly the design conservatism.

(ii) The new method can lead to less conservative results than the most recent fuzzy Lyapunov based approach dealing with the same control context in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF]. Notice that this latter has already outperformed many remarkable results in non-quadratic Lyapunov control framework, e.g. [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]. In addition, our control structure is much simpler and does not require online inversion of parameter-dependent matrices for implementation. Moreover, the proposed design conditions are of lower numerical complexity compared to those in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF]. These outstanding features are of crucial importance for real-time applications.

(iii) Differently from [START_REF] Bezzaoucha | Stabilization of nonlinear systems subject to uncertainties and actuator saturation[END_REF], the proposed method is able to deal with open-loop unstable T-S systems. Therefore, it can be applied to a larger class of nonlinear constrained systems. Moreover, it can guarantee the asymptotic stability of the closed-loop system in the absence of disturbances.

The paper is organized as follows. Section 2 describes the control problem and presents useful results for control design. Section 3 is devoted to the theoretical derivation of the new design conditions. Examples illustrating the interests of the new method are given in Section 4. Finally, Section 5 provides some concluding remarks.

Notations. Ω r denotes the set {1, 2, . . . , r}. I denotes the identity matrix of appropriate dimension. For any square matrix X, He(X) = X + X , and X > 0 means that X is positive definite. 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 > 0, we denote E(P, ρ) = x : x P x ≤ ρ and E(P ) ≡ E(P, 1). The scalar functions η i , i ∈ Ω r verify the convex sum property if

η i ≥ 0 and
η i (θ)Y i , Z θθ = r i=1 r j=1 η i (θ)η j (θ)Z ij , Y -1 θ = r i=1 η i (θ)Y i -1
. The explicit time-dependence of the variables is omitted for convenience.

Problem Statement and Preliminaries

Control problem statement

Consider the following T-S fuzzy 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 sat(u) + B w i w) z = r i=1 η i (θ) (C i x + D w i w) (1)
where x ∈ R nx is the state, u ∈ R nu is the control input, w ∈ R nw is the disturbance, z ∈ R nz is the performance output, and θ ∈ R np is the vector of measured premise variables. The 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. The saturation function sat(u) : R nu → R nu in (1) is defined as

sat u (l) = sign u (l) min u (l) , u max(l) ,
where the bounds u max(l) > 0, l ∈ Ω nu , are given. The disturbance w in (1) belongs to the set of functions

W δ = w : R + → R nw , ∞ 0 w(τ ) w(τ )dτ ≤ δ ,
where the energy bound δ > 0 is given. In this paper, we consider the following PDC control law:

u = r i=1 η i (θ) K i x = K θ x. (2)
Let x = x z u and ψ (u) = u -sat (u). From ( 1) and ( 2), the closed-loop system can be strictly rewritten in the following augmented descriptor form:

E ẋ = A θ x + B w θ w -B ψ θ ψ (u) , (3)
where

E =   I 0 0 0 0 0 0 0 0   , A i =   A i 0 B u i C i -I 0 K i 0 -I   , B w i = B w i D w i 0 , B ψ i = B u i 0 0 . Remark 1.
There are different ways to choose the augmented state x of (3) for control design, for instance x = x u could be another possible option. Each choice leads to a corresponding set of design conditions which would be different to those of others. Here, we define x = x z u to explicitly consider z in (3). This conveniently allows the derivation of LMI conditions as shown later. However, the new method can be generalized to other options of x.

This paper presents a new LMI-based method to design a PDC controller (2) for T-S fuzzy system (1) that satisfies the three properties.

(P1) [State constraints]. Given vectors h k ∈ R nx , k ∈ Ω q ,
there exist a matrix P > 0 and a positive scalar ρ > 0 such that, for any x (0) ∈ E (P, ρ) and w ∈ W δ , the corresponding solution of (1) remains inside the following polyhedral set:

P x = x ∈ R nx : h k x ≤ 1, k ∈ Ω q . ( 4
) (P2) [Local stability].
Given a scalar α > 0, any closed-loop trajectory of the undisturbed system (i.e. w = 0) starting from E (P ) converges exponentially to the origin with a decay rate less than α/2.

(P3) [L 2 -gain performance].

There exists a scalar γ > 0 such that the system (1) has a restricted L 2 -gain less than √ γ.

That is, for x(0) = 0 and any w ∈ W δ compatible with

x(0) = 0, one has z(t) 2 < √ γ w(t) 2 .
Notice that the above closed-loop specifications have been also considered in [START_REF] Nguyen | Anti-windup based dynamic output feedback controller design with performance consideration for constrained Takagi-Sugeno systems[END_REF] for the control design of an anti-windup based dynamic output feedback controller of T-S systems subject to persistent disturbances.

Preliminaries

The following preliminary results are useful for the theoretical development in next section.

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

P u = r i=1 Φ (K i -G i ) ,
where

Φ (K i -G i ) = x ∈ R nx : K i(l) -G i(l) x ≤ u max(l) , for l ∈ Ω nu . If x ∈ P u , then it follows that ψ (u) r i=1 η i S i -1 ψ (u) - r i=1 η i G i x ≤ 0,
for any positive diagonal matrices S i , and any family of functions η i , i ∈ Ω r , satisfying the convex sum property. Lemma 1 presents an extension of the generalized sector condition to deal with the control input nonlinearity [START_REF] Nguyen | LMI-based control synthesis of constrained Takagi-Sugeno fuzzy systems subject to L 2 or L∞ disturbances[END_REF]. This extension allows all involved matrices being parameter-dependent which contributes to reduce the conservatism of the proposed method.

Lemma 2. [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 condition

r i=1 r j=1 η i η j Υ ij < 0 holds if Υ ii < 0, i ∈ Ω r , 2 r -1 Υ ii + Υ ij + Υ ji < 0, i, j ∈ Ω r , i = j.
Lemma 2 is used to convert a parameter-dependent LMI condition to a finite set of LMIs. Notice that more efficient relaxation results 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 complexity.

Main Results

The following theorem provides LMI conditions for designing a PDC controller (2) that solves the control problem considered in Section 2.

Theorem 1. Given positive scalars δ and α, assume there exist positive definite matrix

X ∈ R nx×nx , positive diagonal matri- ces S i ∈ R nu×nu , matrices X ij 21 ∈ R nz×nx , X ij 22 ∈ R nz×nz , X ij 23 ∈ R nz×nu , X i 31 ∈ R nu×nx , X i 32 ∈ R nu×nz , X i 33 ∈ R nu×nu , V i ∈ R nu×nx , W i ∈ R nu×nx , i, j ∈ Ω r , and pos- itive scalars ρ, γ such that X * V i(l) -W i(l) u 2 max(l) ≥ 0, i ∈ Ω r , l ∈ Ω nu (5) X Xh k * 1 ≥ 0, k ∈ Ω q (6) ρ + γδ ≤ 1 (7) Ψ ii < 0, i ∈ Ω r (8) 2 r -1 Ψ ii + Ψ ij + Ψ ji < 0, i, j ∈ Ω r , i = j (9)
where the quantity Ψ ij is defined in [START_REF] Bouarar | Robust non-quadratic static output feedback controller design for Takagi-Sugeno systems using descriptor redundancy[END_REF].

Ψ ij =         Ψ 11 ij Ψ 12 ij Ψ 13 ij Ψ 14 ij Ψ 15 ij Ψ 16 ij * Ψ 22 ij Ψ 23 ij 0 0 D w i * * Ψ 33 ij 0 0 0 * * * -I 0 0 * * * * -2S j 0 * * * * * Ψ 66 ij         , (10)
with

Ψ 11 ij = He(A i X + B u i X j 31 ) + αX, Ψ 12 ij = B u i X j 32 + XC i -X ij 21 , Ψ 13 ij = B u i X j 33 + V i -X j 31 , Ψ 14 ij = XC i , Ψ 15 ij = W i -B u i S j , Ψ 16 ij = B w i + XC i D w j , Ψ 22 ij = -He(X ij 22 ), Ψ 23 ij = -X ij 23 -X j 32 , Ψ 33 ij = -He(X j 33 ) and Ψ 66 ij = -γI + D w i D w j .
Then, the PDC controller (2) with the feedback gains given by

K i = V i X -1 , i ∈ Ω r , (11)
solves the design problem stated in Section 2 for the closedloop T-S fuzzy system (1).

Proof. Let us introduce the parameter-dependent matrix

X θθ = r i=1 r j=1 η i (θ) η j (θ)   X 0 0 X ij 21 X ij 22 X ij 23 X i 31 X i 32 X i 33   . ( 12
)
Since X > 0, it follows that

X θθ E = EX θθ =   X 0 0 0 0 0 0 0 0   ≥ 0. (13)
Notice that if [START_REF] Zhao | Robust stabilization of T-S fuzzy discrete systems with actuator saturation via PDC and non-PDC law[END_REF] and ( 9) hold, then by Lemma 2, it follows that Ψ θθ < 0 with Ψ ij given in [START_REF] Bouarar | Robust non-quadratic static output feedback controller design for Takagi-Sugeno systems using descriptor redundancy[END_REF]. This implies that

X θθ 22 X θθ 23 X θ 32 X θ 33
is nonsingular, and so is X θθ since X > 0. Let P (θ) = X -1 θθ . Pre-and post-multiplying [START_REF] Ting | A robust fuzzy control approach to stabilization of nonlinear time-delay systems with saturating inputs[END_REF] with P (θ) and its transpose, it follows that

E P (θ) = P (θ) E ≥ 0. ( 14
)
Consider a candidate Lyapunov function

V (x) = x E P (θ) x. (15)
Denote P = X -1 . From the definitions of x and E in (3), and the relation [START_REF] Delibaşı | L 2 control of LPV systems with saturating actuators: Pólya approach[END_REF], it follows that

V (x) ≡ V (x) = x P x. ( 16
)
Remark 2. The time-derivative of the Lyapunov function V (x) in ( 15) along the solution of ( 3) is given by

V (x) = ẋ E P (θ) x + xE P (θ) ẋ + x E Ṗ(θ) x.
Due to the definition of E and X θθ in [START_REF] Cao | Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation[END_REF], it follows that x E Ṗ(θ) x = 0. This allows to avoid the unknown timederivatives of the MFs, see also Remark 5.

Let G i = W i X -1 .
Using Schur complement lemma [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF], we can prove that the condition Ψ θθ < 0 is equivalent to

      Γ 11 (θ) Γ 12 (θ) Γ 13 (θ) Γ 14 (θ) Γ 15 (θ) * Γ 22 (θ) Γ 23 (θ) 0 D w θ * * Γ 33 (θ) 0 0 * * * -2S θ 0 * * * * Γ 55 (θ)       < 0, (17)
where

Γ 11 (θ) = Ψ θθ(1,1) + XC θ C θ X, Γ 12 (θ) = Ψ θθ(1,2) , Γ 13 (θ) = Ψ θθ(1,3) , Γ 14 (θ) = W θ -B u θ S θ , Γ 15 (θ) = B w θ + XC θ D w θ , Γ 22 (θ) = -He X θθ 22 , Γ 23 (θ) = -X θθ 23 -X θ 32 , Γ 33 (θ) = -He X θ
33 and Γ 55 (θ) = -γI + D w θ D w θ . Note that (17) can be rewritten in the form

  Υ (θ) -B ψ θ S θ + X θθ G * θ B w θ + X θθ D w θθ * -2S θ 0 * * -γI + D w θ D w θ   < 0( 18
)
where

Υ (θ) = He (A θ X θθ ) + X θθ C * θθ X θθ , G * θ = G θ 0 0 , C * θθ =   αP + C θ C θ 0 0 0 0 0 0 0 0   , D w θθ =   C θ D w θ 0 0   .
Pre-and post-multiplying [START_REF] Rhee | A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design[END_REF] with diag X - θθ , S -1 θ , I and its transpose yields

  Φ (θ) -P (θ) B ψ θ + G * θ S -1 θ P (θ) B w θ + D w θθ * -2S -1 θ 0 * * -γI + D w θ D w θ   < 0 (19)
with Φ (θ) = P (θ) A θ + A θ P (θ) + C * θθ . Pre-and postmultiplying ( 19) by x ψ (u) w and its transpose, after some simple manipulations we obtain:

V (x) + αx P x + z z -γw w -2ψ (u) S -1 θ [ψ (u) -G θ x] < 0. (20)
By Lemma 1, (20) implies clearly that

V (x) + αx P x + z z -γw w < 0, x ∈ P u . ( 21
)
Following similar line as in [START_REF] Nguyen | Anti-windup based dynamic output feedback controller design with performance consideration for constrained Takagi-Sugeno systems[END_REF], we can prove that (5) (respectively ( 6)) implies the inclusion E (P ) ⊆ P u (respectively E (P ) ⊆ P x ). Now, if w = 0, then it follows from (21) that V (x) < -αx P x, ∀x ∈ E (P ), which guarantees (P2). For w = 0 and w ∈ W δ , integrating both sides of ( 21) from 0 to

T f > 0 yields V (x (T f )) < V (x (0)) + γ T f 0 w(τ ) w(τ )dτ (22)
From ( 7) and ( 22), it follows that V (x (T f )) < ρ + γδ ≤ 1, ∀x (0) ∈ E (P, ρ). This means that the corresponding solution of ( 1) is confined in E (P ) ⊆ P x , which proves (P1). Moreover, from ( 22) and considering T f → ∞, we obtain

∞ 0 z(τ ) z(τ )dt < γ ∞ 0 w(τ ) w(τ )dτ , x (0)
= 0. This, in turn, proves (P3). Therefore, the proof of Theorem 1 can be now concluded.

The following remarks can be stated for Theorem 1.

Remark 3. The matrix X θθ in (12) related to the Lyapunov function [START_REF] Ting | Robust anti-windup controller design of timedelay fuzzy systems with actuator saturations[END_REF] depends on the MFs η i (θ), i ∈ Ω r , in a polynomial form. A general form of this matrix is given as follows:

X (θ) =   X 0 0 X 21(θ) X 22(θ) X 23(θ) X 31(θ) X 32(θ) X 33(θ)   (23) = r i0=1 • • • r i N =1 N j=0 η ij (θ)   X 0 0 X i0...i N 21 X i0...i N 22 X i0...i N 23 X i1...i N 31 X i1...i N 32 X i1...i N 33   .
Then, less conservative results can be achieved by increasing N . In this paper, a trade-off between relaxation and computational burden has been done by taking N = 1, but the results for N ≥ 2 can be easily generalized from Theorem 1. Then, the relaxation results in [START_REF] Sala | Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya's theorem[END_REF] can be also applied in these cases. Remark 4. Under the effect of the PDC controller (2), the form of the Lyapunov function (15) (respectively system (3)) and the quadratic Lyapunov function (16) (respectively system (1)) are strictly equivalent. However, the use of (15) associated with (3) allows introducing the parameter-dependent matrices X 21(θ) , X 22(θ) , X 23(θ) , X 31(θ) , X 32(θ) , X 33(θ) (see [START_REF] Tanaka | A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions[END_REF]) into the control design. This aims to reduce the conservatism of the results [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]. Corollary 1 presents the design conditions without using the augmented system approach.

Corollary 1. Given positive scalars δ and α, assume there exist positive definite matrix X ∈ R nx×nx , positive diagonal matrices S i ∈ R nu×nu , matrices V i ∈ R nu×nx , W i ∈ R nu×nx , i ∈ Ω r , and positive scalars ρ, γ satisfying ( 5), ( 6), [START_REF] Kim | H∞ state-feedback control for fuzzy systems with input saturation via fuzzy weighting-dependent Lyapunov functions[END_REF] and the following conditions:

Ξ ii < 0, i ∈ Ω r 2 r -1 Ξ ii + Ξ ij + Ξ ji < 0, i, j ∈ Ω r , i = j
where the quantity Ξ ij is defined by

Ξ ij =     Ξ ij(1,1) Ξ ij(1,2) B w i + XC i D w j XC i * -2S j 0 0 * * -γI + D w i D w j 0 * * * -I     , with Ξ ij(1,1) = He (A i X + B u i V j )+αX and Ξ ij(1,2) = W j - B u i S j .
Then, the PDC controller (2) with the feedback gains (11) solves the control problem in Section 2.

Proof. The result of Corollary 1 is derived from the proof of Theorem 1 by using respectively ( 1) and ( 16) in place of ( 3) and ( 15) in the development. Thus, this proof is omitted here for brevity.

Remark 5. The main source of conservatism of the proposed method comes from the fact that X is not parameter dependent, see [START_REF] Tanaka | A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions[END_REF]. Multiple Lyapunov functions [START_REF] Tanaka | A multiple Lyapunov function approach to stabilization of fuzzy control systems[END_REF] may be used to overcome this drawback. However, if this approach is successfully applied to discrete-time T-S systems [START_REF] Kim | H∞ state-feedback control for fuzzy systems with input saturation via fuzzy weighting-dependent Lyapunov functions[END_REF][START_REF] Zhao | Robust stabilization of T-S fuzzy discrete systems with actuator saturation via PDC and non-PDC law[END_REF], it is still a challenging research topic for continuous-time cases. This is basically due to the presence of a priori unknown time-derivatives of the MFs η i (θ), i ∈ Ω r , in the derivation of design conditions [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF]. The use of X (θ) in (23) (or [START_REF] Cao | Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation[END_REF] in the proof of Theorem 1) allows avoiding this major drawback. Remark 6. For local control purposes, two following convex optimization problems can be formulated [START_REF] Nguyen | Anti-windup based dynamic output feedback controller design with performance consideration for constrained Takagi-Sugeno systems[END_REF].

Optimization problem 1. To find a controller that satisfies the properties in Section 2 while minimizing the L 2 -gain, the following optimization can be performed: min γ, subject to γ > 0 and LMI conditions in Theorem 1.

Optimization problem 2. To find a controller that satisfies the properties in Section 2 while maximizing E (P ), the following optimization can be performed [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]: min trace (R) , subject to the LMIs in Theorem 1 and R I I X > 0. By Schur complement lemma, this latter leads to R > P , thus trace(R) > trace(P ). Then, the minimization of trace(R) implies clearly the minimization of trace(P ).

Remark 7. The control problems in local context based on non-quadratic Lyapunov approaches usually lead to complex design conditions [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF]. The numerical complexity of LMIbased optimizations can be evaluated by the number of scalar decision variables N var and the number of rows of all considered LMI conditions N row . For Theorem 3 in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF], these numbers are given as follows:

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 whereas these numbers corresponding to Theorem 1 are

N var =2 + n x (n x + 1) /2 + r 2 n z n x + n 2 z + n z n u + r n u + n u n z + n 2 u + 3n u n x N row =3 + rn u (n x + 1) + r 2 (n x + 2n u + 2n z + n w ) + q (n x + 1)
The numerical complexity of these methods is evaluated later. Note also that the control structure in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF] is much more complex than (2), which requires online inversion of a parameterdependent matrix for implementation. Thus, this approach does not seem suitable for real-time applications, especially for high dimensional T-S systems with an important number of linear subsystems. At last, from the viewpoint of input-saturated control systems, in many cases the new 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 are clearly illustrated in Section 4.

Illustrative Examples

In this section, three examples are given to illustrate the interests of the proposed method. The first example is a physically motivated application while the two others are used to demonstrate the capacity of the new control method in reducing the design conservatism. Note that since all systems considered here are open-loop unstable, the method in [START_REF] Bezzaoucha | Stabilization of nonlinear systems subject to uncertainties and actuator saturation[END_REF] cannot be applied to these cases. All LMI optimizations are solved with the help of YALMIP toolbox [START_REF] Löfberg | YALMIP: A toolbox for modeling and optimization in MAT-LAB[END_REF] in Matlab and SDPT3 solver.

Example 1. Let us the control problem of an inverted pendulum whose dynamics is given as follows [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]:

ẋ1 = x 2 (24) ẋ2 = g sin (x 1 ) -amlx 2 2 sin (2x 1 ) /2 -a cos x 1 u 4l/3 -aml cos 2 x 1 + w
where x 1 is the angle of the pendulum from the vertical, x 2 is the angular velocity, and u is the control force applied to the cart. Physically reasonable parameters for [START_REF] Guelton | Robust dynamic output feedback fuzzy lyapunov stabilization of Takagi-Sugeno systems -a descriptor redundancy approach[END_REF] are given by M

= 8 [kg], m = 2 [kg], 2l = 1 [m], g = 9.
8 [m/s 2 ] and a = 1/ (m + M ). For the performance output, we take z = x 1 .

The disturbance w represents a force exerted on the pendulum. Suppose that system (24) is constrained by u max = 300 [N],

h 1 = 3/π 0 , h 2 = -3/π 0 , h 3 = 1/5 0 , h 4 = -1/5 0 .
In the light of the results in [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF], the following T-S model is used for control design of nonlinear system (24):

A 1 = 0 1 17.294 0 , B u 1 = 0 -0.177 , B w 1 = 0 1 , A 2 = 0 1 12.631 0 , B u 2 = 0 -0.078 , B w 2 = 0 1 ,
with the membership functions

η 1 (x 1 ) = ω 2 (x 1 ) -1 ω 1 (x 1 ) ω 2 (x 1 ) , η 2 (x 1 ) = 1 -η 1 (x 1 ) , ω 1 (x 1 ) = 1 + exp (-7 (x 1 + π/4)) , ω 2 (x 1 ) = 1 + exp (-7 (x 1 -π/4)) .
Notice that the decay rate α in (P2) is related to the time performance of the closed-loop system [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. Solving the Optimization problem 1 with α = 0.1 leads to P = 1.78 0.30 0.30 0.13 , K 1 = 1155.4 444.81 , K 2 = 788.28 290.35 , and √ γ = 0.12. For simulation, the unknown disturbance w is selected as a constant of 25 happening at t = 2s with a duration of 0.3 [s]. Then, the closed-loop response corresponding to the initial condition x 0 = -π/3 5 is depicted in Figure 1.

Observe that the closed-loop system ( 24) is stabilized despite an important input saturation at the beginning and the disturbance is well attenuated. Note that the classical PDC controller (without any constraint consideration) in [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] induces unstable behaviors under the same simulation conditions, see Figure 2. This clearly shows the interest of considering the system constraints into the control design for real-world applications.

For the maximization of E(P ), the Optimization problem 2 is performed leading to

η i (K i -G i ) .
Example 2. Consider the following family of T-S fuzzy systems (1) taken from Example 1 in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF]: [START_REF] Tuan | Parameterized linear matrix inequality techniques in fuzzy control system design[END_REF][START_REF] Löfberg | YALMIP: A toolbox for modeling and optimization in MAT-LAB[END_REF] and b ∈ [0, 25]. The existence of stabilizing controllers for this example is checked using the new quadratic relaxation approach (QRA: Theorem 1) and the nonquadratic approach (NQA: Theorem 3 in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF]) for 806 points of a 31-by-26 parameter grid. For that, both approaches are applied with δ = 0.1, α = 0, u max = 10 and

A 1 = 2 -10 2 0 , B u 1 = 1 1 , B w 1 = 1 1 , D w 1 = 0 0 , A 2 = a -5 1 2 , B u 2 = b 2 , B w 2 = 1 1 , D w 2 = 0 0 , for a ∈ [-
P x = x ∈ R 2 : |x i | ≤ π/2, i = 1, 2 .
Note that the NQA in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF] are only for undisturbed T-S systems (w = 0) and no constraint guaranteeing an L 2 -gain has been incorporated therein. Hence, the comparison done in this example between two approaches is not objective. However, this can provide an idea on the conservatism of the new method. Figure 4 shows that the feasibility space given by the proposed method is distinctly larger than that of the NQA. Notice that the results of the NQA have already outperformed those based on a fuzzy Lyapunov function [START_REF] Tanaka | A multiple Lyapunov function approach to stabilization of fuzzy control systems[END_REF], and a line-integral Lyapunov function [START_REF] Rhee | A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design[END_REF]. For this example, the new method also leads to less computational burden compared to the NQA, see Table 1. The results in Figure 4 and Table 1 confirm Remark 7. The outperformance of the proposed method compared to [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF] in terms of relaxation is not surprising since in its present state of progress, the NQA in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF] is based on an unsaturated control law and depends strongly on the control bound limits (due to its way to deal with unknown time-derivatives of the MFs). This can make the results become over-conservative. [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]. The family of T-S systems (25) can represent complex nonlinear systems since the matrices of all four subsystems are significantly different. This example aims to show the potential of Theorem 1 in terms of relaxation with respect to different values of N in [START_REF] Tanaka | A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions[END_REF]. To this end, the existence of stabilizing controllers for [START_REF] Löfberg | YALMIP: A toolbox for modeling and optimization in MAT-LAB[END_REF] is checked for 289 points of a 17-by-17 square parameter grid. In this study, the following design data are considered for comparison purposes δ = 0.1, α = 0, u max = 10 and

A 1 = 1.59 -7.29 0.01 0 , B u 1 = 1 0 , C 1 = -0.1 -0.4 A 2 = 0.02 -4.64 0.35 0.21 , B u 2 = 8 0 , C 2 = 0.1 -0.1 A 3 = -a -4.33 0 0.05 , B u 3 = 6 -b -1 , C 3 = 0.2 0.1 A 4 = 0.89 -5.29 0.1 0 , B u 4 = 1 0 , C 4 = -0.1 -0.4 B w 1 = B w 3 = 0.1 0.1 , B w 2 = B w 4 = 0 0 (25)
D w i = -0.5, i ∈ Ω 4 and a, b ∈ [-4,
P x = x ∈ R 2 : |x 1 | ≤ 2, |x 2 | ≤ 2 .
The feasibility spaces obtained with the proposed method for three different cases N = 1, 2 and 3 are depicted in Figure 5. As indicated in Remark 3, less conservative results are obtained by increasing N . Moreover, the feasibility space obtained with the conditions in Corollary 1 is clearly included by that obtained with the new augmented system approach, see Remark 4. It is also important to note that the design conditions using: (1) a fuzzy Lyapunov function in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF]; (2) polytopic representations 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] Bezzaoucha | Stabilization of nonlinear systems subject to uncertainties and actuator saturation[END_REF]; (3) a one-step AW dynamic output feedback control scheme in [START_REF] Nguyen | Anti-windup based dynamic output feedback controller design with performance consideration for constrained Takagi-Sugeno systems[END_REF], are all unfeasible for any point of the considered parameter grid. The above facts strongly confirm the interest of the new method for its capacity to reduce the design conservatism compared to existing results.

Concluding Remarks

In this paper, a new method is proposed to design a saturated PDC control law for constrained T-S fuzzy systems subject to L 2 -disturbances. The control design is cast as an LMI optimization, and then the obtained controller can guarantee several closed-loop properties. The novel idea of this method consists in a judicious use of an equivalent augmented form of the closed-loop system associated with a parameter-dependent Lyapunov function for constrained T-S systems. Several examples are given to demonstrate the effectiveness of the proposed control method. It has been proved that in many cases the new method allows outperforming the most recent results based on non-quadratic approaches dealing with the same local control context. Notice also that the control structure in this work is much simpler than those used for comparison purposes. Therefore, this systematic control method is more suitable for practical uses. Future works focus on the real-time application of the proposed method to the challenging design issue of vehicle shared steering control [START_REF] Nguyen | Shared lateral control with online adaptation of the automation degree for driver steering assist system: A weighting design approach[END_REF].

r i=1 η i = 1 .

 1 For such functions with any argument θ, and matrices Y i and Z ij of appropriate dimensions, we denote Y θ = r i=1

Figure 1 :

 1 Figure 1: Closed-loop behaviors obtained with the proposed control method corresponding to the initial condition x 0 = [-π/3 5].

Figure 2 :

 2 Figure 2: Closed-loop behaviors obtained with the control method in [2] corresponding to the initial condition x 0 = [-π/3 5].

1

 1 = 405.03 366.47 , K 2 = 408.11 263.04 , and an L 2 -gain of √ γ = 0.183. Figure 3 shows that S 1 = E(P ) is maximized along S 2 = P u ∩ P x which is included inside S 3 = Φ r i=1η i (K i -G i ) . Thus, the set inclusion property is clearly verified.

Figure 3 :

 3 Figure 3: Set inclusion property: S 1 = E(P ), S 2 = Pu ∩ Px and S 3 = Φ

Figure 4 :

 4 Figure 4: Feasibility spaces obtained with QRA (+) and NQA (•).

Example 3 .

 3 Consider the following family of open-loop unstable T-S fuzzy systems in the form (1) with:

Figure 5 :

 5 Figure 5: Feasibility spaces obtained with Theorem 1 for N = 1 and Corollary 1 (•); N = 2 (• and * ); N = 3 (•, * and ×).

Table 1 :

 1 Comparison of complexity between different methods

	Complexity Theorem 1 Theorem 3 in [16]
	N row	57	137
	N var	65	255

Acknowledgement

We would like to thank Prof. Michel Dambrine, Prof. Thierry-Marie Guerra, the Associate Editor and one of the reviewers for valuable comments. This work was supported in part by the Program "Vehicle for the Future" via the French Environment and Energy Management Agency, the European Community, the region Hauts-de-France in the framework of the CERV-IFER 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.


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

Kazuo Tanaka
email: ktanaka@mce.uec.ac.jp

Antoine Dequidt
email: antoine.dequidt@univ-valenciennes.fr

Michel Dambrine
email: michel.dambrine@univ-valenciennes.fr

Static Output Feedback Design for a Class of Constrained Takagi-Sugeno Fuzzy Systems

Keywords: Takagi-Sugeno fuzzy systems, static output feedback control, actuator saturation, delayed fuzzy Lyapunov functions, domain of attraction, linear matrix inequality (LMI)

This paper addresses the static output feedback (SOF) control problem of Takagi-Sugeno fuzzy systems subject to both control input and state constraints. A new parameter-dependent sector condition is proposed to deal with the input saturation in SOF control context. Based on a judicious use of Finsler's lemma in fuzzy Lyapunov control framework, new design conditions guaranteeing the existence of a stabilizing controller are recast as an LMI-based optimization with additional slack variables. These extra variables offer more flexibility to reduce the design conservatism. In particular, the proposed method requires neither special constraints on the state-space system matrices nor linear matrix equalities, which are hard to be satisfied. Several numerical examples are given to demonstrate the interests of the new control method.

Introduction

Over the past two decades, nonlinear control based on Takagi-Sugeno (T-S) fuzzy systems [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF] has been extensively studied [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF]. The success of T-S fuzzy models comes from their outstanding capacity to approximate any smooth nonlinear system with any preciseness [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. Especially, if the system nonlinearities are bounded in a compact set, the sector nonlinearity approach [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] provides a systematic way to derive an equivalent T-S fuzzy representation of the original nonlinear system. As a consequence, a great number of results on stability analysis and control design of T-S fuzzy systems have been reported in the literature [START_REF] Nguyen | LMI-based control synthesis of constrained Takagi-Sugeno fuzzy systems subject to L 2 or L ∞ disturbances[END_REF][START_REF] Chadli | On stability and stabilization of singular uncertain Takagi-Sugeno fuzzy systems[END_REF][START_REF] Campos | New stability conditions based on piecewise fuzzy Lyapunov functions and tensor product transformations[END_REF][START_REF] Montagner | Convergent LMI relaxations for quadratic stabilizability and control of Takagi-Sugeno fuzzy systems[END_REF][START_REF] Gassara | Observer based (Q, V, R)-α-dissipative control for TS fuzzy descriptor systems with time delay[END_REF]. In addition, T-S fuzzy-model-based control techniques have been successfully applied to various engineering applications [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF][START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF][START_REF] Nguyen | Driver-automation cooperative approach for shared steering control under multiple system constraints: Design and experiments[END_REF]. It is noteworthy that most of T-S fuzzy control methods are based on state feedback control schemes [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF]. However, for numerous real-time applications, output feedback control must be used since only partial state information is accessible for control design and implementation [START_REF] Kau | Robust H ∞ fuzzy static output feedback control of T-S fuzzy systems with parametric uncertainties[END_REF][START_REF] Qiu | Static-output-feedback control of continuous-time T-S fuzzy affine systems via piecewise Lyapunov functions[END_REF]. Note also that state-feedback based methods are not easily adapted to output control designs which are more difficult and involved [START_REF] Nguyen | Simultaneous design of parallel distributed output feedback and anti-windup compensators for constrained Takagi-Sugeno fuzzy systems[END_REF]. For these reasons, output feedback control has recently received intensive research efforts [START_REF] Qiu | Static-output-feedback control of continuous-time T-S fuzzy affine systems via piecewise Lyapunov functions[END_REF][START_REF] Nguyen | Anti-windup based dynamic output feedback controller design with performance consideration for constrained Takagi-Sugeno systems[END_REF].

Among all output feedback schemes, static output feedback (SOF) represents the simplest control structure. Note that the design of dynamic controllers can be reformulated as a SOF control problem involving augmented plants in many cases [START_REF] Cao | Static output feedback stabilization: An ILMI approach[END_REF]. Despite its practical and theoretical importance, SOF control is still one of the most challenging control topics due to its inherent non-convex characterization [START_REF] Qiu | Static-output-feedback control of continuous-time T-S fuzzy affine systems via piecewise Lyapunov functions[END_REF][START_REF] Chadli | LMI solution for robust static output feedback control of discrete Takagi-Sugeno fuzzy models[END_REF][START_REF] Lee | Fuzzy static output feedback may be possible in LMI framework[END_REF][START_REF] Dong | Robust static output feedback control synthesis for linear continuous systems with polytopic uncertainties[END_REF]. Moreover, existing convex SOF design methods are often too restrictive [START_REF] Chadli | LMI solution for robust static output feedback control of discrete Takagi-Sugeno fuzzy models[END_REF][START_REF] Dong | Robust static output feedback control for linear discretetime systems with time-varying uncertainties[END_REF]. Up to now, various approaches dealing with SOF control have been proposed which can be classified into two research directions. The first one focuses on developing numerical algorithms to solve non-convex design conditions [START_REF] Ghaoui | A cone complementarity linearization algorithm for static output-feedback and related problems[END_REF][START_REF] Huang | Robust H ∞ static output feedback control of fuzzy systems: An ILMI approach[END_REF][START_REF] Wu | An ILMI approach to robust H 2 static output feedback fuzzy control for uncertain discrete-time nonlinear systems[END_REF]. The second direction consists in getting new sufficient convex conditions as less conservative/restrictive as possible that can be effectively solved with current numerical solvers [START_REF] Chadli | LMI solution for robust static output feedback control of discrete Takagi-Sugeno fuzzy models[END_REF][START_REF] Lee | Fuzzy static output feedback may be possible in LMI framework[END_REF][START_REF] Dong | Robust static output feedback control synthesis for linear continuous systems with polytopic uncertainties[END_REF][START_REF] Gershon | Static H 2 and H ∞ output-feedback of discrete-time LTI systems with state multiplicative noise[END_REF]. In this case, some notable works dealing with linear systems can be cited as follows. A state coordinate transformation approach has been proposed in [START_REF] Dong | Robust static output feedback control synthesis for linear continuous systems with polytopic uncertainties[END_REF] for linear polytopic uncertain systems. Parameter independent slack variables with a lower-triangular structure have been introduced in [START_REF] Dong | Robust static output feedback control for linear discretetime systems with time-varying uncertainties[END_REF][START_REF] Oliveira | Extended H 2 and H ∞ characterizations and controller parametrizations for discrete-time systems[END_REF] while a linear parameter dependent Lyapunov function has been employed in [START_REF] Gershon | Static H 2 and H ∞ output-feedback of discrete-time LTI systems with state multiplicative noise[END_REF]. In T-S fuzzy control framework, a special structure of the quadratic Lyapunov matrix together with a coordinate transformation have been used in [START_REF] Lo | Robust H ∞ nonlinear control via fuzzy static output feedback[END_REF]. Based on the P-problem and W-problem given in [START_REF] Crusius | Sufficient LMI conditions for output feedback control problems[END_REF], quadratic design conditions involving linear matrix equalities are presented in [START_REF] Kau | Robust H ∞ fuzzy static output feedback control of T-S fuzzy systems with parametric uncertainties[END_REF][START_REF] Lee | Fuzzy static output feedback may be possible in LMI framework[END_REF][START_REF] Fang | A new LMI-based approach to relaxed quadratic stabilization of T-S fuzzy control systems[END_REF]. Recently, to reduce the design conservatism, piecewise quadratic conditions have been proposed in [START_REF] Qiu | Static-output-feedback control of continuous-time T-S fuzzy affine systems via piecewise Lyapunov functions[END_REF] and [START_REF] Zhang | T-S fuzzy-model-based piecewise H ∞ output feedback controller design for networked nonlinear systems with medium access constraint[END_REF] for T-S fuzzy affine systems and networked T-S fuzzy systems, respectively. Fuzzy Lyapunov functions in conjunction with some convexification techniques have been also exploited in [START_REF] Chadli | LMI solution for robust static output feedback control of discrete Takagi-Sugeno fuzzy models[END_REF].

Actuator saturation is another important control issue [START_REF] Nguyen | LMI-based control synthesis of constrained Takagi-Sugeno fuzzy systems subject to L 2 or L ∞ disturbances[END_REF][START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[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] Zou | Stabilization via extended nonquadratic boundedness for constrained nonlinear systems in Takagi-Sugeno's form[END_REF]. This topic is motivated by the facts that control input saturation is ubiquitous in practice, especially for applications requiring a high performance level. The effects of input saturation on the system dynamics may be critical since it can seriously degrade the control performance, and in some cases may lead to system instability [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]. Although numerous methods are now available to handle saturation effects, most of them have only dealt with linear systems [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]. It should be stressed that this practical control issue has not been completely addressed for T-S fuzzy 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][START_REF] Zou | Stabilization via extended nonquadratic boundedness for constrained nonlinear systems in Takagi-Sugeno's form[END_REF]; especially in output feedback control framework [START_REF] Nguyen | Simultaneous design of parallel distributed output feedback and anti-windup compensators for constrained Takagi-Sugeno fuzzy systems[END_REF]. Note that besides the control input limitations, the system states are also usually bounded in engineering applications due to physical and/or safety reasons. Specifically, the validity domain of T-S fuzzy models obtained with the sector nonlinearity approach can be described by state constraints [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]. Hence, an explicit consideration of state constraints in the control design is crucial to guarantee the closed-loop stability [START_REF] Nguyen | LMI-based control synthesis of constrained Takagi-Sugeno fuzzy systems subject to L 2 or L ∞ disturbances[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].

Motivated by the practical and theoretical significance of both SOF scheme and constrained T-S fuzzy systems, this paper aims to develop a new input-saturated SOF formulation using LMI framework [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. The contributions of the proposed method can be summarized as follows.

1. The new method requires neither special structure of Lyapunov matrices as in [START_REF] Lo | Robust H ∞ nonlinear control via fuzzy static output feedback[END_REF] nor additional linear matrix equalities, which are hard to be satisfied, as in [START_REF] Kau | Robust H ∞ fuzzy static output feedback control of T-S fuzzy systems with parametric uncertainties[END_REF][START_REF] Lee | Fuzzy static output feedback may be possible in LMI framework[END_REF][START_REF] Crusius | Sufficient LMI conditions for output feedback control problems[END_REF][START_REF] Fang | A new LMI-based approach to relaxed quadratic stabilization of T-S fuzzy control systems[END_REF]. In particular, this method can handle T-S fuzzy systems with multiple output matrices of subsystems. Moreover, these matrices are not explicitly required to be of full row rank as in most of existing works [10, 20-22, 24, 26]. Hence, the proposed method can not only lead to less conservative results but also be applied to a larger class of T-S fuzzy systems. 2. A new parameter-dependent sector condition is proposed to deal with the input nonlinearity. This sector condition is especially appropriate for the considered SOF scheme. Together with a judicious use of Finsler's lemma, this allows for an LMI-based formulation. 3. A delayed fuzzy Lyapunov function (FLF) is exploited to derive the design conditions. This type of FLFs offers more flexibility to decrease the conservatism of the results via slack variables [START_REF] Guerra | An efficient Lyapunov function for discrete T-S models: Observer design[END_REF].

To the best of our knowledge, the proposed results on SOF design for constrained T-S fuzzy systems using fuzzy Lyapunov control framework have been not observed in any previous work. Furthermore, it will be shown through several examples that the new method provides less conservative results compared to recent related works dealing with unconstrained T-S fuzzy systems, i.e. systems without control input and/or state constraints. The paper is organized as follows. The control problem and preliminary results are presented in Section 2. The main results are stated in Section 3. The interests of the proposed method are illustrated via numerical examples in Section 4. Finally, Section 5 concludes the paper.

Notations. Ω r denotes the set {1, 2, . . . , r}. For a vector x, x i denotes its ith element. For a matrix X, X > 0 means X is positive definite, X (i) (respectively X (i,j) ) denotes its ith row (respectively its element in the ith row and jth column) and He (X) = X + X . I denotes the identity matrix of appropriate dimension. Denote Ξ = {ξ ∈ R r : r i=1 ξ i = 1, ξ i ≥ 0}. For a function η : R → Ξ and matrices X i , Y ij , Z ijk of appropriate dimensions, we denote

X η = r i=1 η i (t)X i , X η d = r i=1 η i (t -1)X i , Y η d η = r k=1 r j=1 η k (t -1)η j (t)Y kj , Z η d ηη = r k=1 r j=1 r i=1 η k (t -1)η j (t)η i (t)Z kji .
We assume that z (-1) = z (0) for any variable of interest z (t). Arguments will be omitted when their meaning is straightforward.

Problem Formulation and Preliminaries

System Description

Consider the following T-S fuzzy system with control input saturation:

x(t + 1) = r i=1 η i (t)(A i x(t) + B i sat(u(t))) y(t) = r i=1 η i (t)C i x(t) (1)
where

x(t) ∈ P x ⊂ R nx is the state, u(t) ∈ R nu is the control input, y(t) ∈ R ny
is the measured output. The real matrices of adequate dimensions A i , B i , C i are constant. The following assumptions are considered in this paper.

Assumption 1. The domain of validity P x of system (1) is described by the polyhedral set

P x = x ∈ R nx : M (m) x ≤ 1, m ∈ Ω q (2)
where the matrix M ∈ R q×nx is given.

Assumption 2. The scalar membership functions (MFs) η i (t) in (1), for i ∈ Ω r , are functions of measurable signals of interest such as system output, measured external disturbances and/or time. It is also assumed that the sector nonlinearity approach [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] is used to derive an exact T-S fuzzy representation (1) of a given nonlinear system, then η ∈ Ξ as long as x ∈ P x .

Assumption 3. The control input u(t) is subject to a componentwise saturation map whose lth component is defined as sat(u l ) = sign(u l ) min(|u l |, ūl ), l ∈ Ω nu , where ūl > 0 denotes the amplitude bound of the lth control input.

In order to motivate the above assumptions, we consider the following example. Motivating Example. Consider the nonlinear system borrowed from [START_REF] Chadli | LMI solution for robust static output feedback control of discrete Takagi-Sugeno fuzzy models[END_REF]:

x 1 (t + 1) = -x 2 (t)x 3 (t) -x 3 (t) + u 1 (t) x 2 (t + 1) = x 1 (t) x 3 (t + 1) = x 2 (t) + u 2 (t) y(t) = (2 + x 3 (t))x 2 (t) x 3 (t) (3)
where -1 ≤ x 3 ≤ 1. From (3), x 3 can be naturally chosen as the premise variable which is a component of the measured output y. Using the sector nonlinearity approach [3], the nonlinear system (3) can be exactly represented in the form (1) for

x ∈ P x = x ∈ R 3 : |x 3 | ≤ 1 . It should be stressed that, although |x 3 | ≤ 1
is the primary condition to obtain such a T-S fuzzy representation, this was not explicitly taken into account in [START_REF] Chadli | LMI solution for robust static output feedback control of discrete Takagi-Sugeno fuzzy models[END_REF] and other related works. Moreover, we also consider the limitations of the control inputs, i.e. |u 1 | ≤ 0.5 and |u 2 | ≤ 0.5. As will be shown in Section 4, an explicit consideration of these system constraints in the control design can increase significantly the performance of the closed-loop system, for instance in terms of the size of the guaranteed domain of attraction.

Control Problem

Let us consider a static output feedback controller of the form

u(t) = H -1 η d η K η d η y(t) (4)
where the matrices H kj ∈ R nu×nu and K kj ∈ R nu×ny , for k, j ∈ Ω r , have to be designed. For initializing the controller, we consider that η d (0) = η(0). Note that at each time instant, the computation of the current control input requires the inversion of the matrix H η d η . From ( 1) and ( 4), the closed-loop T-S fuzzy system can be rewritten as follows:

x(t + 1) = A η + B η H -1 η d η K η d η C η x(t) -B η ψ(u(t)) y(t) = C η x(t) (5)
where the dead-zone nonlinearity is defined as

ψ(u) = u -sat(u).
This paper aims to propose a constructive LMI-based control method to design a SOF control law (4) such that the closed-loop system (5) satisfies the following properties.

• Property 1. Given a matrix M ∈ R q×nx , see [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF]. There exist positive definite matrices P i ∈ R nx×nx , i ∈ Ω r , such that, for any initial state x 0 taken in a set E 0 depending on those matrices and specified later, the corresponding closed-loop trajectory of (5) remains inside the set P x .

• Property 2. The set E 0 is included in the domain of attraction of the closedloop system (5), namely any closed-loop trajectory starting from E 0 converges to the origin.

Preliminary Results

We present hereafter some useful lemmas for the design procedure.

Lemma 1. Given diagonal positive definite matrices S ijk ∈ R nu×nu , matrices

U ijk ∈ R nu×nx for i, j, k ∈ Ω r , two vectors ξ, η ∈ Ξ, and a vector x ∈ R nx . If x satisfies S -1 ξηη U ξηη (l) x ≤ ūl , ∀l ∈ Ω nu , (6)
then

ψ(u) S ξηη u -ψ(u) -S -1 ξηη U ξηη x ≥ 0, (7)
for any u ∈ R nu .

Proof. Note that the left-hand term of ( 7) is equal to

nu l=1 S ξη η(l,l) u l -sat(u l ) sat(u l ) -(S -1 ξηη U ξηη ) (l) x .
Since S ξη η(l,l) > 0, then it suffices to show that

u l -sat(u l ) sat(u l ) -(S -1 ξηη U ξηη ) (l) x ≥ 0. (8)
This is obvious if |u l | ≤ ūl since then u l -sat(u l ) = 0. Inequality (8) holds also if u l > ūl , since then u l -sat(u l ) > 0, and from ( 6)

sat(u l ) -S -1 ξηη U ξηη (l) x = ūl -S -1 ξηη U ξηη (l)
x ≥ 0.

The last possible case u l < -ū l can be treated similarly.

Remark 1. Lemma 1 presents a powerful tool to deal with the dead-zone nonlinearity ψ (u). It is motivated by the results on the generalized sector condition in [START_REF] Gomes Da Silva Jr | Anti-windup design with guaranteed regions of stability for discrete-time linear systems[END_REF]. However, since the feedback gain matrices H kji , K kji are not explicitly involved in ( 6) and [START_REF] Montagner | Convergent LMI relaxations for quadratic stabilizability and control of Takagi-Sugeno fuzzy systems[END_REF], this new version is especially appropriate for the SOF control context. Indeed, it seems hard to obtain an LMI formulation by applying the sector condition in [START_REF] Gomes Da Silva Jr | Anti-windup design with guaranteed regions of stability for discrete-time linear systems[END_REF] to the considered SOF control problem. As will be seen, this aims for a direct computation of the control gains without using any change of variable. Hence, all feedback gains are parameter-dependent (i.e. H η d η and K η d η ) to improve the closed-loop performance and to reduce the design conservatism.

Lemma 2 (Finsler's Lemma [START_REF] Skelton | A Unified Algebraic Approach to Linear Control Design[END_REF]). For two matrices Q = Q ∈ R n×n and R ∈ R m×n such that rank (R) < n, the following statements are equivalent:

i) x Qx < 0, ∀x ∈ {ξ ∈ R n : ξ = 0, Rξ = 0 m×1 } , ii) ∃F ∈ R n×m such that Q + FR + R F < 0.
Lemma 3. Let Υ kji , for i, j, k ∈ Ω r , be symmetric matrices of appropriate dimensions. The condition

r k=1 r j=1 r i=1 ξ k η j η i Υ kji < 0 holds for ξ, η ∈ Ξ if    Υ kii < 0, i, k ∈ Ω r 2 r -1 Υ kii + Υ kij + Υ kji < 0, i, j, k ∈ Ω r and j > i
This result is directly extended from [START_REF] Tuan | Parameterized linear matrix inequality techniques in fuzzy control system design[END_REF]. The relaxation techniques in [START_REF] Kau | Robust H ∞ fuzzy static output feedback control of T-S fuzzy systems with parametric uncertainties[END_REF][START_REF] Sala | Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya's theorem[END_REF] are also applicable to obtain less conservative results at the price of an increasing computational complexity.

Static Output Control of Constrained T-S Fuzzy Systems

A new LMI-based method to design a SOF controller for the T-S fuzzy system (1) is presented. To obtain less conservative results, a delayed fuzzy Lyapunov function is exploited for theoretical developments.

Theorem 1. Given a T-S fuzzy system (1) with its domain of validity P x described by [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF]. If there exist a scalar α, positive definite matrices P i ∈ R nx×nx , positive diagonal matrices S kij ∈ R nu×nu , and matrices

H kj ∈ R nu×nu , U kij ∈ R nu×nx , K kj ∈ R nu×ny , G 1kj ∈ R nx×nx , G 2kj ∈ R nx×nx , G 3kj ∈ R nu×nx , G 4kj ∈ R nu×nx , for i, j, k ∈ Ω r , such that P k U kji(l) U kji(l) αū 2 l (2S kji(l,l) -α) ≥ 0, i, j, k ∈ Ω r , l ∈ Ω nu (9)
P i M (m) M (m) 1 ≥ 0, i ∈ Ω r , m ∈ Ω q (10)
H kj + H kj > 0, k, j ∈ Ω r (11)    Ψ kii < 0, i, k ∈ Ω r 2 r -1 Ψ kii + Ψ kij + Ψ kji < 0, i, j, k ∈ Ω r with j > i (12)
where the quantity Ψ kji is defined as follows:

Ψ kji = He     Ψ kji(1,1) -G 1kj Ψ kji(3,1) -G 1kj B i Ψ kji(1,2) P j /2 -G 2kj Ψ kji(3,2) -G 2kj B i Ψ kji(1,3) -G 3kj Ψ kji(3,3) S kji -G 3kj B i Ψ kji(1,4) -G 4kj Ψ kji(3,4) -S kji -G 4kj B i     (13)
with

Ψ kji(1,1) = G 1kj A i -P k /2, Ψ kji(1,2) = G 2kj A i + J i K kj C i , Ψ kji(1,3) = G 3kj A i + αK kj C i , Ψ kji(1,4) = -U kji + G 4kj A i , Ψ kji(3,1) = G 1kj B i , Ψ kji(3,2) = G 2kj B i -J i H kj , Ψ kji(3,3) = G 3kj B i -αH kj , Ψ kji(3,4) = G 4kj B i and
J i = B i , B i is column non-full rank B i B i B i -1 , B i is column full rank
Then, the SOF controller (4) solves the control problem defined in Section 2.

Proof. Consider the following delayed fuzzy Lyapunov function:

V(t, x) = x r i=1 η i (t -1)P i x = x P η d x, (14)
and let E x be the set defined as

E x = x ∈ R nx : r i=1 η i x P i x ≤ 1 for some η ∈ Ξ .
Step 1). We show that the SOF control law (4) is well defined. Indeed, for any ξ, η ∈ Ξ, it follows from (11) that r k=1 r j=1 ξ k η j (H kj + H kj ) > 0. This guarantees the nonsingularity of H η d η at any time t.

Step 2). The set E x is contained in the set P x described by [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF]. By convexity and applying Schur complement lemma [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] to [START_REF] Kau | Robust H ∞ fuzzy static output feedback control of T-S fuzzy systems with parametric uncertainties[END_REF], it follows that

P ξ -M (m) M (m) ≥ 0, ∀ξ ∈ Ξ, ∀m ∈ Ω q .
This guarantees the inclusion E x ⊆ P x .

Step 3). Given any t > 0 and any x ∈ R nx . If V(t, x) ≤ 1, then i) x is in P x , and ii) the following inequality holds:

S -1 η d ηη U η d ηη (l) x ≤ ūl , l ∈ Ω nu . (15)
For point i), as x P η d x ≤ 1, then x ∈ E x . Thus, x ∈ P x according to Step 2. For point ii), since (β -α) 2 ≥ 0, it follows that β 2 ≥ α(2β -α) for any α, β ∈ R. Using this inequality with β = S kji(l,l) , it can be deduced from (9) that

P k U kji(l) U kji(l) ū2 l S 2 kji(l,l) ≥ 0, ∀i, j, k ∈ Ω r , ∀l ∈ Ω nu . (16)
By convexity and Schur complement lemma, [START_REF] Lee | Fuzzy static output feedback may be possible in LMI framework[END_REF] implies that

ū2 l P η d -1/S 2 η d ηη(l,l) U η d ηη(l) U η d ηη(l) ≥ 0, ∀l ∈ Ω nu . (17)
Since x ∈ E x , it follows from ( 17) that

ū2 l ≥ ū2 l x P η d x ≥ 1/S 2 η d ηη(l,l) x U η d ηη(l) U η d ηη(l) x, ∀l ∈ Ω nu ,
from which (15) can be easily derived.

Step 4). We prove that, for any given t > 0 and x ∈ R nx such that V(t, x) ≤ 1 and x = 0, the variation ∆V of the Lyapunov function ( 14) along the solution of ( 5) is negative, that is

∆V = V(t + 1, x(t + 1)) -V(t, x)
= x(t + 1) P η x(t + 1) -x P η d x < 0.

To this end, we remark that the matrix Ψ η d ηη based on the matrices Ψ kji defined in (13) can be rewritten in the form

Ψ η d ηη = Q η d ηη + F η d ηη R(η d , η) + R(η d , η) F η d ηη ,
where

Q η d ηη =     -P η d 0 0 -U η d ηη 0 P η 0 0 0 0 0 S η d ηη -U η d ηη 0 S η d ηη -2S η d ηη     , F η d ηη =     G 1η d η 0 G 2η d η J η H η d η G 3η d η αH η d η G 4η d η 0     , R(η d , η) = A η -I B η -B η H -1 η d η K η d η C η 0 -I 0 .
From [START_REF] Nguyen | Simultaneous design of parallel distributed output feedback and anti-windup compensators for constrained Takagi-Sugeno fuzzy systems[END_REF] and Lemma 3, it follows that

Ψ η d ηη < 0. ( 18
)
Denoting

ξ(t) = x x(t + 1) u ψ(u)
, it is easy to show that

∆V + He ψ(u) S η d ηη u -ψ(u) -S -1 η d ηη U η d ηη x = ξ(t) Q η d ηη ξ(t).( 19
)
Moreover, ξ(t) satisfies the equality R(η d , η)ξ(t) = 0, which is simply a reformulation of the closed-loop system equations ( 4) and ( 5). Applying Lemma 2, we can prove that [START_REF] Dong | Robust static output feedback control for linear discretetime systems with time-varying uncertainties[END_REF] implies

ξ(t) Q η d ηη ξ(t) < 0. ( 20
)
Finally, since x satisfies [START_REF] Chadli | LMI solution for robust static output feedback control of discrete Takagi-Sugeno fuzzy models[END_REF] as shown in Step 3, applying Lemma 1 leads to

ψ(u) S η d ηη u -ψ(u) -S -1 η d ηη U η d ηη x ≥ 0. ( 21
)
The conclusion ∆V < 0 is then deduced from ( 19), ( 20) and ( 21).

Step 5). Conclusion. Let E 0 be the set defined by

E 0 = {x ∈ R nx : V(0, x) ≤ 1}. (22)
Note that E 0 ⊂ E x ⊂ P x . It follows from previous steps that the solution x(t) of ( 5) for any initial condition x 0 ∈ E 0 is such that V(t, x(t)) ≤ 1, and so, x(t) ∈ P x for any t ≥ 0. Thus, Property 1 holds. Since it has been proved in Step 4 that V(t, x) is a Lyapunov function for the closed-loop system (5) on the set E x , then Property 2 holds also.

Remark 2. It has been shown that the set E 0 mentioned in Properties 1 and 2 can be defined as in [START_REF] Gershon | Static H 2 and H ∞ output-feedback of discrete-time LTI systems with state multiplicative noise[END_REF]. Since it may be difficult to characterize E 0 for the general case, we consider here the following set of initial conditions (instead of E 0 ):

E V = {x ∈ R nx : x P i x ≤ 1, ∀i ∈ Ω r }. (23)
Obviously, Properties 1 and 2 still hold in this case since E V ⊂ E 0 .

Remark 3. The use of the delayed fuzzy Lyapunov function ( 14) can lead to less conservative results without increasing the number of LMIs compared to the classical one V (t, x) = x P η x intensively studied in the literature [START_REF] Guerra | An efficient Lyapunov function for discrete T-S models: Observer design[END_REF]. This is due to the possibility of this new type of FLFs to introduce some slack variables, namely

K η d η , H η d η , G iη d η , i ∈ Ω 4
, and especially S η d ηη and U η d ηη (for the SOF control context of input-saturated T-S fuzzy systems) instead of K η , H η , G iη , i ∈ Ω 4 , S η and U η . Note also that the proposed method can be extended by using more complex FLF forms such as double-sum Lyapunov function [START_REF] Ding | Further studies on LMI-based relaxed stabilization conditions for nonlinear systems in Takagi-Sugeno's form[END_REF], FLFs incorporating the MFs over several past samples [START_REF] Lee | Approaches to extended non-quadratic stability and stabilization conditions for discrete-time T-S fuzzy systems[END_REF]; or the control approaches based on κ-sample variation of FLFs [START_REF] Guerra | An efficient Lyapunov function for discrete T-S models: Observer design[END_REF] to obtain less conservative results at the expense of high computational cost.

Remark 4. By Finsler's lemma, all complex couplings between Lyapunov matrices, feedback gain matrices and system matrices are avoided. This key point allows an LMI formulation for SOF control without requiring any linear matrix equality constraint on Lyapunov matrices [START_REF] Kau | Robust H ∞ fuzzy static output feedback control of T-S fuzzy systems with parametric uncertainties[END_REF][START_REF] Lee | Fuzzy static output feedback may be possible in LMI framework[END_REF][START_REF] Fang | A new LMI-based approach to relaxed quadratic stabilization of T-S fuzzy control systems[END_REF] and/or rank conditions on system matrices as in most of existing works. Moreover, due to the elimination of these coupling terms, additional performance specifications can be naturally incorporated into the design conditions of Theorem 1, e.g. robustness with respect to norm-bounded uncertainties affecting all matrices (A i , B i , C i ), decay rate, finite L 2 -induced gain. Note that the control approaches based on the cone complementarity linearization algorithm [START_REF] Ghaoui | A cone complementarity linearization algorithm for static output-feedback and related problems[END_REF] cannot have these interesting features [START_REF] Dong | Robust static output feedback control synthesis for linear continuous systems with polytopic uncertainties[END_REF].

Remark 5. Due to the products of α with other decision variables, the conditions in Theorem 1 are a set of LMIs with a line search over α. Since α is a scalar, the design conditions can be effectively solved with some constructive numerical procedures [START_REF] Dong | Robust static output feedback control synthesis for linear continuous systems with polytopic uncertainties[END_REF]. Here, a line search for α is done with 100 points linearly gridded over a logarithmic scale in 10 -4 , 10 4 . Remark 6. Since Theorem 1 is developed in local control context, it is thus desirable to achieve the largest domain of attraction. Note that the domain of attraction is here non-convex due to its associate FLF [START_REF] Cao | Static output feedback stabilization: An ILMI approach[END_REF]. The following optimization is proposed to achieve the control goal:

min µ (24)
subject to µ > 0, trace (P i ) ≤ µ, i ∈ Ω r , LMIs ( 9)-( 12) of Theorem 1.

The optimization [START_REF] Lo | Robust H ∞ nonlinear control via fuzzy static output feedback[END_REF] aims to minimize the largest trace (P i ), i ∈ Ω r , and can be effectively solved with standard LMI numerical solvers.

Numerical Examples

This section presents three examples to demonstrate the effectiveness of the new control method. Note that existing works used for comparison purposes can be only applied to unconstrained T-S fuzzy systems. All LMI optimizations are conducted in Matlab R2011b with YALMIP toolbox and SeDuMi solver.

Example 1. We revisit the motivating example given in (3), which is rewritten as

x(t + 1) = A(x)x(t) + B sat(u(t)) y(t) = C(x)x(t) (25)
where

x = x 1 x 2 x 3 , u = u 1 u 2 and A(x) =   0 -x 3 -1 1 0 0 0 1 0   , B =   1 0 0 0 0 1   , C(x) = 0 2 + x 3 0 0 0 1 .
Note that this nonlinear system is open-loop unstable and subject to the following constraints:

P x = x ∈ R 3 : |x 3 | ≤ 1 , u max = 0.5 0.5 . (26)
Using the sector nonlinearity approach [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] with the premise variable x 3 , the nonlinear system (25) can be exactly represented by (1) for x ∈ P x with

A 1 =   0 1 -1 1 0 0 0 1 0   , B 1 =   1 0 0 0 0 1   , A 2 =   0 -1 -1 1 0 0 0 1 0   , B 2 =   1 0 0 0 0 1   , C 1 = 0 1 0 0 0 1 , C 2 = 0 3 0 0 0 1 , η 1 (x 3 ) = (1 -x 3 )/2, η 2 (x 3 ) = 1 -η 1 (x 3 ). (27)
The condition |x 3 | ≤ 1 is crucial to guarantee that η = η 1 η 2 ∈ Ξ. However, this state constraint is usually neglected in T-S fuzzy control framework. Note also that for this example, SOF design methods proposed in [START_REF] Huang | Robust H ∞ static output feedback control of fuzzy systems: An ILMI approach[END_REF][START_REF] Wu | An ILMI approach to robust H 2 static output feedback fuzzy control for uncertain discrete-time nonlinear systems[END_REF][START_REF] Lo | Robust H ∞ nonlinear control via fuzzy static output feedback[END_REF] are not applicable since C 1 = C 2 whereas all design conditions in [START_REF] Kau | Robust H ∞ fuzzy static output feedback control of T-S fuzzy systems with parametric uncertainties[END_REF][START_REF] Dong | Robust static output feedback control for linear discretetime systems with time-varying uncertainties[END_REF] are infeasible.

Applying Theorem 1 to (27) leads to the following results: with α = 0.298. Figure 1 shows the closed-loop trajectory corresponding to the initial condition x 0 = 1 -1 1 . Note that the controller given in [START_REF] Chadli | LMI solution for robust static output feedback control of discrete Takagi-Sugeno fuzzy models[END_REF] induces unstable behaviors under the same simulation condition. Remark that the system constraints [START_REF] Fang | A new LMI-based approach to relaxed quadratic stabilization of T-S fuzzy control systems[END_REF] have not been explicitly taken into account in [START_REF] Chadli | LMI solution for robust static output feedback control of discrete Takagi-Sugeno fuzzy models[END_REF]. Hence, the resulting design conditions do not provide any guarantee on the domain of attraction of [START_REF] Zhang | T-S fuzzy-model-based piecewise H ∞ output feedback controller design for networked nonlinear systems with medium access constraint[END_REF] in the presence of these constraints. To highlight this, by numerical simulations, we have computed the closed-loop solutions for initial conditions taken on a 20 × 20 × 20 hyper-rectangle grid in the initial condition set [-5, 5] × [-5, 5] × [-1, 1]. Note that this latter set is included inside the validity domain P x defined in [START_REF] Fang | A new LMI-based approach to relaxed quadratic stabilization of T-S fuzzy control systems[END_REF]. It can be verified that, for the proposed method, among the total of 8000 tested trajectories, there are 7267 closed-loop trajectories behave correctly. This is, these trajectories do not leave the validity domain P x and converge towards the origin. For the method proposed in [START_REF] Chadli | LMI solution for robust static output feedback control of discrete Takagi-Sugeno fuzzy models[END_REF], only 2587 solutions induce a correct behavior. This fact strongly confirms the interest of considering the system constraints into the control design procedure.

P 1 =  

Example 2. Consider a family of open-loop unstable T-S fuzzy systems (1) with

following data:

A 1 = 1 -δ -1 -0.1 , B 1 = 5 + δ 2δ , A 2 = 1 δ -1 -0.1 , B 2 = 5 -δ -2δ C 1 = C 2 = 1 0 , η 1 (x 1 ) = (x 1 + δ)/(2δ), η 2 (x 1 ) = 1 -η 1 (x 1 ) (28)
where δ > 0. Note that the validity of both MFs η 1 (•) and η 2 (•) in ( 28) is only guaranteed if |x 1 | ≤ δ. For simulation purposes, the following state and control input limitations are considered:

P x = x ∈ R 2 : |x 1 | ≤ δ, |x 2 | ≤ 1 , ū = 2. ( 29
)
This example is used as a benchmark to study the design conservatism. Note that Theorem 1 and the methods in [START_REF] Kau | Robust H ∞ fuzzy static output feedback control of T-S fuzzy systems with parametric uncertainties[END_REF][START_REF] Chadli | LMI solution for robust static output feedback control of discrete Takagi-Sugeno fuzzy models[END_REF][START_REF] Dong | Robust static output feedback control for linear discretetime systems with time-varying uncertainties[END_REF][START_REF] Fang | A new LMI-based approach to relaxed quadratic stabilization of T-S fuzzy control systems[END_REF] can be applied to [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]. Table 1 provides the maximal value of δ, denoted by δ * , for which a stabilizing SOF controller can be computed with each method. Observe that the new method provides less conservative results even if none of the existing works has dealt with the system constraints [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].

Conditions

Value of δ * Proposed method 0.79 Method in [START_REF] Dong | Robust static output feedback control for linear discretetime systems with time-varying uncertainties[END_REF]: Conditions (18a) and (38a) 0.74 Method in [START_REF] Dong | Robust static output feedback control for linear discretetime systems with time-varying uncertainties[END_REF]: Condition (18b) 0.78 Method in [START_REF] Dong | Robust static output feedback control for linear discretetime systems with time-varying uncertainties[END_REF]: Condition (38b) 0.71 Methods in [START_REF] Kau | Robust H ∞ fuzzy static output feedback control of T-S fuzzy systems with parametric uncertainties[END_REF][START_REF] Chadli | LMI solution for robust static output feedback control of discrete Takagi-Sugeno fuzzy models[END_REF][START_REF] Fang | A new LMI-based approach to relaxed quadratic stabilization of T-S fuzzy control systems[END_REF] Infeasible

Solving the optimization problem [START_REF] Lo | Robust H ∞ nonlinear control via fuzzy static output feedback[END_REF] for system [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF] with δ = 0.

Figure 2 shows the guaranteed region of attraction E V of the closed-loop system which is maximized inside P x . By means of simulations, it can be checked that the corresponding SOF controller provides stable closed-loop behaviors for ∀x 0 ∈ P x .

-0.8 -0.6 -0.4 -0. Remark 7. Note that the size of E V defined in ( 23) is maximized if the intersection size of all the r ellipsoids E(P i ) = {x : x P i x ≤ 1} is maximized. Hence, when solving (24), the Lyapunov matrices P i , i ∈ Ω r , tend to be close, i.e. we approach the quadratic Lyapunov function, see [START_REF] Zou | Stabilization via extended nonquadratic boundedness for constrained nonlinear systems in Takagi-Sugeno's form[END_REF] and Figure 2. However, the fuzzy Lyapunov function [START_REF] Cao | Static output feedback stabilization: An ILMI approach[END_REF] offers more flexibility to reduce the design conservatism since the quadratic one is only a special case of ( 14) by imposing P i = P , ∀i ∈ Ω r .

Example 3. Consider the following T-S fuzzy system (1) borrowed from [START_REF] Dong | Robust static output feedback control for linear discretetime systems with time-varying uncertainties[END_REF]:

This example aims to study not only the design conservatism but also the applicability of some notable methods dealing with SOF control. Since B 1 = B 2 and C 1 = C 2 , the methods in [START_REF] Huang | Robust H ∞ static output feedback control of fuzzy systems: An ILMI approach[END_REF][START_REF] Wu | An ILMI approach to robust H 2 static output feedback fuzzy control for uncertain discrete-time nonlinear systems[END_REF][START_REF] Gershon | Static H 2 and H ∞ output-feedback of discrete-time LTI systems with state multiplicative noise[END_REF][START_REF] Oliveira | Extended H 2 and H ∞ characterizations and controller parametrizations for discrete-time systems[END_REF][START_REF] Lo | Robust H ∞ nonlinear control via fuzzy static output feedback[END_REF] cannot be applied to [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. However, Theorem 1 as well as the methods in [START_REF] Kau | Robust H ∞ fuzzy static output feedback control of T-S fuzzy systems with parametric uncertainties[END_REF][START_REF] Chadli | LMI solution for robust static output feedback control of discrete Takagi-Sugeno fuzzy models[END_REF][START_REF] Lee | Fuzzy static output feedback may be possible in LMI framework[END_REF][START_REF] Dong | Robust static output feedback control for linear discretetime systems with time-varying uncertainties[END_REF][START_REF] Fang | A new LMI-based approach to relaxed quadratic stabilization of T-S fuzzy control systems[END_REF] can be used for this example to design SOF controllers. The stabilization interval of δ obtained with each control method is given in Table 2. Note that differently from all above methods, the following constraints are explicitly considered for (31) when applying Theorem 1:

P x = x ∈ R 3 : |x i | ≤ 5, ∀i ∈ Ω 3 and ū = 3. ( 32
)
The results in Table 2 reconfirm that despite the consideration of (32), the proposed method provides less conservative results than existing ones. Note also that if the state/input limitations are increased, then the stabilization interval of δ given by the new method can be still enlarged.

Conditions

Stabilization intervals of δ Theorem 1

[-3.18, 0.92] Condition (18a) in [START_REF] Dong | Robust static output feedback control for linear discretetime systems with time-varying uncertainties[END_REF] [-2.08, 0.55] Condition (18b) in [START_REF] Dong | Robust static output feedback control for linear discretetime systems with time-varying uncertainties[END_REF] [-2.15, 0.87] Condition (38a) in [START_REF] Dong | Robust static output feedback control for linear discretetime systems with time-varying uncertainties[END_REF] [-2.08, 0.52] Condition (38b) in [START_REF] Dong | Robust static output feedback control for linear discretetime systems with time-varying uncertainties[END_REF] [-2.04, 0.48] Methods in [START_REF] Kau | Robust H ∞ fuzzy static output feedback control of T-S fuzzy systems with parametric uncertainties[END_REF][START_REF] Chadli | LMI solution for robust static output feedback control of discrete Takagi-Sugeno fuzzy models[END_REF][START_REF] Lee | Fuzzy static output feedback may be possible in LMI framework[END_REF][START_REF] Fang | A new LMI-based approach to relaxed quadratic stabilization of T-S fuzzy control systems[END_REF] Infeasible Methods in [START_REF] Huang | Robust H ∞ static output feedback control of fuzzy systems: An ILMI approach[END_REF][START_REF] Wu | An ILMI approach to robust H 2 static output feedback fuzzy control for uncertain discrete-time nonlinear systems[END_REF][START_REF] Gershon | Static H 2 and H ∞ output-feedback of discrete-time LTI systems with state multiplicative noise[END_REF][START_REF] Oliveira | Extended H 2 and H ∞ characterizations and controller parametrizations for discrete-time systems[END_REF][START_REF] Lo | Robust H ∞ nonlinear control via fuzzy static output feedback[END_REF] Non-applicable

Concluding Remarks

This paper presents a new SOF control method for T-S fuzzy systems subject to both state and control input constraints. New design conditions are expressed in terms of LMIs with a line search over a scalar variable. In contrast to most of existing results, these conditions do not require any restriction on the system statespace matrices. In addition, matrix equality constraints between system matrices and Lyapunov matrices are also eliminated via Finsler's lemma. It has been shown that the proposed method can lead to less conservative results compared to existing results. For the control design, the premise vector is assumed to be measured. Further research to deal with unmeasured premise variables is under study.

Figure 1 :

 1 Figure 1: Closed-loop behaviors obtained with the proposed controller for x 0 = [1 -1 1] in Example 1.

Figure 2 :

 2 Figure 2: Guaranteed region of attraction E V and domain of validity Px in Example 2.

B 2

 2 = 1 0 , C 1 = 1 1.5 , C 2 = 1 1 .

Table 1 :

 1 Computed values of δ * for several methods.

Table 2 :

 2 Stabilization interval of δ obtained for different results

Acknowledgment

We would like to thank the reviewers for their valuable suggestions which were useful to improve the manuscript. This work was supported in part by the Agence de l'environnement et de la maîtrise de l'énergie (ADEME) through the "Vehicle for the Future" program, the European Community (through the FEDER European Funds for Regional Development), the region Hauts-de-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).