Anh-Tu Nguyen

Michel Dambrine

Jimmy Lauber

Simultaneous Design of Parallel Distributed Output Feedback and Anti-windup Compensators for Constrained Takagi-Sugeno Fuzzy Systems

Keywords: Takagi-Sugeno fuzzy systems, dynamic output feedback controller, anti-windup compensator, control input saturation, state constraints, domain of attraction

This paper is devoted to develop a novel approach dealing with constrained continuous-time nonlinear systems in the form of Takagi-Sugeno fuzzy models. Here, the disturbed systems are subject to both input and state constraints. The one-step design method is used to simultaneously synthesize the dynamic output feedback controller and its anti-windup strategy. A parameterdependent version of the generalized sector condition is used together with Lyapunov stability theory to derive LMI design conditions. Based on this result and for different design specifications, the synthesis of an anti-windup based dynamic output feedback controller is expressed on the form of convex optimization problems. A physically motivated example is given to illustrate the effectiveness of the proposed method.

I. INTRODUCTION

Computationally intelligent techniques have been demonstrated as powerful tools to cope with real-world complex systems [START_REF] Ruano | Computational intelligence in control[END_REF]- [START_REF] Suykens | Support vector machines: A nonlinear modelling and control perspective[END_REF]. Especially, control methods based on neural networks and fuzzy logic have received more and more attention [START_REF] Ruano | Computational intelligence in control[END_REF]. Due to their outstanding features such as adaptation, learning and universal approximation, neural network control has been successfully applied to many practical systems [START_REF] Ruano | Computational intelligence in control[END_REF], [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF], [START_REF] Bu | High-order tracking differentiator based adaptive neural control of a flexible air-breathing hypersonic vehicle subject to actuators constraints[END_REF]. However, there still remains a number of drawbacks for classical neural network approach, for instance: the existence of multiple local minima solutions, the heavy computational cost [START_REF] Suykens | Support vector machines: A nonlinear modelling and control perspective[END_REF], [START_REF] Wang | Online SVM regression algorithm-based adaptive inverse control[END_REF]. Support vector machines (SVMs) [START_REF] Vapnik | The Nature of Statistical Learning Theory[END_REF] has appeared as an alternative to neural networks. The main advantages of SVM compared to neural network consists in its capacity to provide global optimal solutions [START_REF] Suykens | Support vector machines: A nonlinear modelling and control perspective[END_REF], [START_REF] Vapnik | The Nature of Statistical Learning Theory[END_REF]. Additionally, SVM algorithms offer a higher generalization ability and they more suitable for machine learning in small sample condition [START_REF] Wang | Online SVM regression algorithm-based adaptive inverse control[END_REF]. Therefore, SVMs have become now an efficient methodology in many areas, e.g. classification, function regression, time series prediction [START_REF] Vapnik | The Nature of Statistical Learning Theory[END_REF]. It should be stressed that despite some important advances in the field of control systems [START_REF] Suykens | Support vector machines: A nonlinear modelling and control perspective[END_REF], [START_REF] Wang | Online SVM regression algorithm-based adaptive inverse control[END_REF], systematic framework for rigourous stability analysis of both neural network and SVM control systems is not yet available in the literature [START_REF] Ruano | Computational intelligence in control[END_REF].

Over the past two decades, model-based fuzzy control has been received great interest in the research community [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF], [START_REF] Huang | Application of two-loop robust control to air-conditioning systems[END_REF]- [START_REF] Jafarzadeh | On the stability and control of continuous-time TSK fuzzy systems[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] becomes nowadays one of the most popular and promising approach for complex nonlinear dynamics [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF]. Indeed, T-S fuzzy models have been widely used to represent complex nonlinear systems as they are a universal approximator. Furthermore, using the sector nonlinear approach [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF], an exact T-S fuzzy model can be deduced from a sufficiently smooth nonlinear system. With a T-S representation, a model-based control may be designed to guarantee the stability and achieve some performance requirements for nonlinear systems. Thanks to its polytopic structure, the main interest of T-S control approach is to extend some linear concepts to the case of nonlinear systems [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. Based on Lyapunov stability tools, this control technique provides a general and systematic framework to cope with complex nonlinear systems [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF], [START_REF] Lam | Stable and robust fuzzy control for uncertain nonlinear systems[END_REF]- [START_REF] Jafarzadeh | On the stability and control of continuous-time TSK fuzzy systems[END_REF], [START_REF] Campos | Revisiting the TP model transformation: Interpolation and rule reduction[END_REF].

Among all nonlinear phenomena, actuator saturation is unavoidable in almost all real applications. It can severely degrade closed-loop system performance and in some cases may lead to the system instability [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]. Motivated by this practical control aspect, a great deal of effort has been recently focused on saturated systems, see [START_REF] Bu | High-order tracking differentiator based adaptive neural control of a flexible air-breathing hypersonic vehicle subject to actuators constraints[END_REF], [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]- [START_REF] Du | Fuzzy control for nonlinear uncertain electrohydraulic active suspensions with input constraint[END_REF] and references therein. In [START_REF] Ting | A robust fuzzy control approach to stabilization of nonlinear time-delay systems with saturating inputs[END_REF], the saturation effect is explicitly considered into the design procedure by using a polytopic representation. Norm-bounded approach is proposed in [START_REF] Du | Fuzzy control for nonlinear uncertain electrohydraulic active suspensions with input constraint[END_REF] to deal with T-S systems subject to control input constraints. These approaches often lead to conservative design conditions [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]. A novel adaptive neural controller based on auxiliary error compensation scheme has been recently proposed to deal with flexible air-breathing hypersonic vehicle with magnitude constraints on actuators [START_REF] Bu | High-order tracking differentiator based adaptive neural control of a flexible air-breathing hypersonic vehicle subject to actuators constraints[END_REF]. It is worth mentioning that among all available control approaches, anti-windup (AW) based control scheme [START_REF] Kothare | A unified framework for the study of anti-windup designs[END_REF] is known as the most popular one to handle the input nonlinearity. There are two categories in this case: one-step or two-step design methods. For the two-step method, first a controller is designed without regarding control input nonlinearity and then an additional anti-windup compensator is introduced in order to minimize the undesirable degradation of closed-loop performance caused by input saturation [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF], [START_REF] Kothare | A unified framework for the study of anti-windup designs[END_REF], [START_REF] Ting | Robust anti-windup controller design of time-delay fuzzy systems with actuator saturations[END_REF]. Two-step design methods generally prove to be satisfactory and have an abundant literature, A.-T. Nguyen, M. Dambrine and J. Lauber are with the department of Automatic Control of LAMIH-UMR CNRS 8201, University of Valenciennes, France. see e.g. [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF] for a quick overview. However, the main disadvantages of such a method are that the effect of the nominal controller on the closed-loop performance under saturation is completely ignored [START_REF] Mulder | Simultaneous linear and anti-windup controller synthesis using multiobjective convex optimization[END_REF] and the inherent sub-optimality since the nominal controller and the AW compensator are separately designed [START_REF] Sawada | Generalized sector synthesis of output feedback control with anti-windup structure[END_REF]. The one-step design methods [START_REF] Mulder | Simultaneous linear and anti-windup controller synthesis using multiobjective convex optimization[END_REF] can be used to overcome these drawbacks. For such methods, the dynamic output feedback controller (DOFC) and its associated AW compensator are simultaneously designed. Up to now, this method still remains an open research topic [START_REF] Sawada | Generalized sector synthesis of output feedback control with anti-windup structure[END_REF]. Among some few recent results existing in linear control framework, the following works can be notably cited. In [START_REF] Mulder | Simultaneous linear and anti-windup controller synthesis using multiobjective convex optimization[END_REF], a convex synthesis method guaranteeing global stability and performance was developed for only open-loop stable systems. The authors in [START_REF] Dai | Output feedback design for saturated linear plants using deadzone loops[END_REF] proposed a local synthesis result which is based on parameter elimination approach. Only sub-optimal solution can be achieved with this method since two separated LMI problems must be solved successively to obtain the DOFC and its associated AW compensator, see Procedure 1 in [START_REF] Dai | Output feedback design for saturated linear plants using deadzone loops[END_REF]. A change-of-variable approach for local context was also proposed in [START_REF] Sawada | Generalized sector synthesis of output feedback control with anti-windup structure[END_REF] for undisturbed linear systems. A great advantage of this method is that it can be naturally extended to multi-objective control problem which is not the case in [START_REF] Dai | Output feedback design for saturated linear plants using deadzone loops[END_REF]. However, the Lyapunov matrix has to be imposed with a special structure. Furthermore, it should be stressed that most of works on AW design are available for linear systems; there is a serious lack of literature on the study for nonlinear cases, especially when exogenous disturbance signals are involved in the system dynamics. Recently, a one-step method based on piecewise fuzzy AW dynamic output feedback controller (DOFC) for discrete-time T-S fuzzy systems has been proposed in [START_REF] Zhang | Piecewise fuzzy anti-windup dynamic output feedback control of nonlinear processes with amplitude and rate actuator saturations[END_REF]. Note that those results seem to be valid only for systems that are stable in open-loop since the estimate of the closed-loop domain of attraction is not explicitly defined even with the presence of system disturbances. Another interesting one-step method 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] which can be applied to open-loop unstable T-S systems subject to persistent disturbances. However, this method can be only considered as a "mixed" between one-step and two-step methods since the local L 2 -gain performance cannot be guaranteed whenever the control input is saturated. Besides control input saturation, the models are usually valid on a given subset of the state space. This is not only true for any system in practice due to physical/safety reasons, but also fundamental when using the nonlinear sector decomposition approach to obtain T-S fuzzy models [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. This validity domain of T-S models can be represented by some constraints on the state variables. It is particularly important to consider explicitly these state constraints in the control design to guarantee a desirable closed-loop behavior in response of disturbances [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]. However, this control issue is usually neglected in T-S control framework [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] Ding | Quadratic boundedness via dynamic output feedback for constrained nonlinear systems in Takagi-Sugeno's form[END_REF].

In T-S control framework, state-feedback control based on the concept of parallel distributed compensation (PDC) is usually applied to derive the design conditions [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. However, all system state variables are not always available in many practical cases. Therefore, output feedback control has been intensively investigated in the literature, see for instance [START_REF] Kau | Robust H∞ fuzzy static output feedback control of T-S fuzzy systems with parametric uncertainties[END_REF]- [START_REF] Zhao | Static output feedback control for interval type-2 T-S fuzzy systems based on fuzzy Lyapunov functions[END_REF]. Most of works concern observer-based controller design [START_REF] Liu | New approaches to H∞ controller designs based on fuzzy observers for T-S fuzzy systems via LMI[END_REF], [START_REF] Lin | Improvement on observer-based H∞ control for T-S fuzzy systems[END_REF]. However, observer-based control approaches become much more complicated when dealing with T-S fuzzy systems subject to control input and system state constraints. This control issue has not been well addressed in the literature [START_REF] Ding | Quadratic boundedness via dynamic output feedback for constrained nonlinear systems in Takagi-Sugeno's form[END_REF]. Motivated by the above control issues, we propose in this paper a novel LMI-based method to simultaneously design a DOFC and an AW compensator for a given disturbed T-S fuzzy system subject to control input saturation and also system state constraints. It will be shown that the obtained dynamic controller can satisfy several regional closed-loop requirements, often conflicting. Anti-windup based control design in the presence of energy-bounded disturbances and state constraints is quite novel in T-S fuzzy control framework. The proposed method provides a systematic tool to deal with a very large class of constrained nonlinear systems, which is our main contribution.

The paper is organized as follows. Section II describes the design problem and recalls some preliminaries results needed for the control synthesis. The main result is stated in Section III. In Section IV, different convex optimization problems are formulated to deal with the trade-off between several conflicting control objectives. The effectiveness of the proposed method is clearly demonstrated via an illustrative example in Section V. Finally, Section VI provides some concluding remarks.

Notation. For an integer number r, Ω r denotes the set {1, 2, . . . , r}. I denotes the identity matrix of appropriate dimensions. For any square matrix X, He (X) = X + X T . X > 0 means that the matrix X is symmetric and positive definite. The i-th element of a vector u is denoted u (i) and X (i) denotes the i-th 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 T P x ≤ ρ , and for brevity E (P ) ≡ E (P, 1). The scalar real functions η 1 , . . . , η r are said to verify the convex sum property on a set D, if ∀θ ∈ D, ∀i ∈ Ω r , η i (θ) ≥ 0 and r i=1 η i (θ) = 1. For such functions, we will denote:

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 ,
where the matrices Y i and Z ij are of appropriate dimensions.

II. PROBLEM DEFINITION AND PRELIMINARIES RESULTS

A. Problem Formulation

Consider the following continuous-time T-S fuzzy system described by [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 u + B w i w) y = r i=1 η i (θ) (C y i x + D yw i w) (1)
where x ∈ R nx , u ∈ R nu , w ∈ R nw and y ∈ R ny are respectively the state, the control input, the disturbance and the measured output vectors of the system. In this paper, it is assumed that all components of the premise variable vector θ ∈ R k are measured. For i ∈ Ω r , the matrices

A i ∈ R nx×nx , B u i ∈ R nx×nu , B w i ∈ R nx×nw , C y i ∈ R ny×nx , D yw i ∈ R ny×nw represent
the set of r local linear subsystems and the nonlinear scalar functions η i (θ) satisfy the convex sum property. For system (1), we consider the following assumptions.

Assumption 1. The input vector u is subject to symmetric magnitude limitations

-u max(l) ≤ u (l) ≤ u max(l) , u max(l) > 0, i ∈ Ω nu .
(

) 2
Assumption 2. The disturbance w is a bounded energy signal, i.e. it belongs to the following set of functions

W δ = w : R + → R nw , ∞ 0 w T wdt ≤ δ (3)
where the bound δ > 0 is given.

Consider now the unconstrained dynamic output feedback controller (DOFC) of the form

           ẋc = r i=1 r j=1 η i (θ) η j (θ) A c ij x c + r i=1 η i (θ) B c i y + v u c = r i=1 η i (θ) C c i x c + D c y (4)
where x c ∈ R nx , u c ∈ R nu are respectively the state and output vectors of the controller. The additional input signal v is used to compensate the windup effect. Controller (4) has to be designed to guarantee the stability and some performance requirements for the closed-loop system, see Figure 1.

Because of control input limitations, the actual control signal injected into the system is subject to the saturation effect, i.e. u = sat (u c ), where each component of the saturation function sat (•) is given by

sat u c(l) sign u c(l) min u c(l) , u max(l) , l ∈ Ω nu . (5)
Remark 1. Without loss of generality, we consider symmetric saturation functions in this paper. Asymmetric saturations can be easily handled with the proposed method with some level of conservativeness by taking u max(l) as the minimum absolute value of the negative and positive saturations levels. We note that the symmetrization strategy presented in [START_REF] Boada | Multi-saturation anti-windup structure for satellite control[END_REF] can be also used to deal with asymmetric saturation functions. Then, the proposed design procedure is straightforwardly applied.

The interactions between system (1) and the constrained dynamic controller (4) are given as

u = sat (u c ) , v = r i=1 η i (θ) E c ij (sat (u c ) -u c ). (6)
From ( 4) and ( 6), the DOFC combined with the anti-windup strategy can be expressed as

           ẋc = r i=1 r j=1 η i (θ) η j (θ) A c ij x c + r i=1 η i (θ) B c i y -E c ij ψ (u c ) u c = r i=1 η i (θ) C c i x c + D c y (7)
where E c ij are the anti-windup gains to be designed and ψ (u c ) u c -sat (u c ). The l-th component of the decentralized dead-zone nonlinearity ψ (u c ) is defined as Define x T cl = x T x T c , the constrained closed-loop system can be obtained from ( 1) and ( 7)

ψ u c(l) = 0, if u c(l) ≤ u max(l) u c(l) -sign u c(l) u max(l) , if u c(l) > u max(l) (8)
           ẋcl = r i=1 r j=1 η i (θ) η j (θ) A ij x cl + B w ij w -B ψ i + R ψ E c ij ψ (u c ) y = r i=1 η i (θ) (C i x cl + D i w) (9)
where

A ij = A i + B u i D c C y j B u i C c j B c i C y j A c ij , B w ij = B w i + B u i D c D yw j B c i D yw j , B ψ i = B u i 0 , C i = C y i 0 , D i = [D yw i ] , R ψ = 0 I .
and the controller output is given by

u c = r i=1 η i (θ) (K i x cl + K w i w), (10)
where

K i = D c C y i C c i and K w i = D c D yw i .
Using notations in Section I, the closed-loop system ( 9) is rewritten as

ẋcl = A θθ x cl + B w θθ w -B ψ θ + R ψ E c θθ ψ (u c ) y = C θ x cl + D θ w (11)
and the DOFC output [START_REF] Precup | Fuzzy logic control system stability analysis based on Lyapunov's direct method[END_REF] is rewritten as

u c = K θ x cl + K w θ w. (12)
In this paper, we aim at proposing a systematic method to design a dynamic output feedback controller together with its anti-windup strategy of the form [START_REF] Huang | Application of two-loop robust control to air-conditioning systems[END_REF] such that the closed-loop system satisfies the following properties.

• Property 1 [State constraints]: For any admissible initial state, the states of the closed-loop system (9) are required to remain inside the polyhedral region described by linear inequalities

P x = x cl ∈ R 2nx : h T k x cl ≤ 1, ∀k ∈ Ω q , (13)
where h T k = h T 1k 0 1×nx and the vectors h 1k are given. In this paper, the set P x represents the validity of (1). • Property 2 [Regional quadratic α-stability]: When w = 0, there exist a positive-definite function V (x cl ) = x T cl P x cl , P > 0, and a scalar α > 0, such that V (x cl ) < -2αV (x cl ) along the closed-loop trajectories for ∀x cl (0) ∈ E (P ). This fact implies that these trajectories will converge exponentially to the origin with a decay rate α.

• Property 3 [Regional L 2 -performance]: For a given positive scalar δ, there exist positive real numbers γ, ρ such that, for any energy-bounded signal w ∈ W δ and for all initial conditions in E (P, ρ), the corresponding trajectories of the closed-loop system will never escape the ellipsoid E (P ) ⊃ E (P, ρ). Moreover, the L 2 -norm of the output signal y is bounded by

∞ 0 y T ydt < γ ∞ 0 w T wdt + ρ. (14)

B. Preliminary Results

In what follows, some useful lemmas needed for design problem in Section III are presented.

Lemma 1. Given matrices K i ∈ R nu×2nx and G i = G i 1 G i 2 ∈ R nu×2nx , with G i 1 ∈ R nu×nx , G i 2 ∈ R nu×nx
, for i ∈ Ω r , we define the polyhedral set P u as follows

P u = r i=1 Ξ (K i -G i ), (15)
where

Ξ (K i -G i ) x cl ∈ R 2nx : K i(l) -G i(l) x cl ≤ u max(l) , ∀l ∈ Ω nu . ( 16
)
Consider the function ψ (u c ) 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] with u c defined in [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF]. If x cl ∈ P u , then the following condition is verified

ψ T (u c ) r i=1 η i S i -1 ψ (u c ) - r i=1 η i G i r i=1 η i K w i x cl w ≤ 0, (17)
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.

Proof. Assume that x cl ∈ P u , it implies

x cl ∈ Ξ r i=1 η i (K i -G i )
since scalar functions η i , ∀i ∈ Ω r , satisfy the convex sum propriety. Hence, it follows that

-u max(l) ≤ r i=1 η i K i(l) -G i(l) x cl ≤ u max(l) , ∀ (i, l) ∈ Ω r × Ω nu . (18)
Let S i(l,l) > 0, (i, l) ∈ Ω r × Ω nu , is the element of the l-th row and l-th column of diagonal matrix S i , we will show that the inequality

ψ u c(l) r i=1 η i S i(l,l) -1 ψ u c(l) - r i=1 η i G i(l) r i=1 η i K w i(l) x cl w ≤ 0 (19)
holds, which implies obviously [START_REF] Du | Fuzzy control for nonlinear uncertain electrohydraulic active suspensions with input constraint[END_REF]. For that, there exist only three possible cases according to the value of u c(l) .

• Case 1: u c(l) ≤ u max(l) . It follows that ψ u c(l) = 0 and so, inequality [START_REF] Ting | Robust anti-windup controller design of time-delay fuzzy systems with actuator saturations[END_REF] holds trivially.

• Case 2: u c(l) > u max(l) . Then,

ψ u c(l) = u c(l) -u max(l) = r i=1 η i K i(l) x cl + K w i(l) w -u max(l) > 0. (20)
From [START_REF] Kothare | A unified framework for the study of anti-windup designs[END_REF], it follows that r i=1

η i K i(l) -G i(l) x cl ≤ u max(l) . Hence, ψ u c(l) - r i=1 η i G i(l) r i=1 η i K w i(l) x cl w = r i=1 η i K i(l) -G i(l) x cl -u max(l) ≤ 0. (21)
Since in this case ψ u c(l) > 0, inequality (19) holds.

• Case 3: u (l) < -u max(l) . It follows that

ψ u c(l) = u c(l) + u max(l) = r i=1 η i K i(l) x cl + K w i(l) w + u max(l) < 0. (22)
Then, from [START_REF] Kothare | A unified framework for the study of anti-windup designs[END_REF] it follows that r i=1

η i K i(l) -G i(l) x cl ≥ -u max(l)
, and then

ψ u c(l) - r i=1 η i G i(l) r i=1 η i K w i(l) x cl w = r i=1 η i K i(l) -G i(l) x cl + u max(l) ≥ 0. (23)
Since in this case ψ u c(l) < 0, inequality (19) holds again.

From the results of these three cases, the proof of Lemma 1 can be concluded.

Notice that Lemma 1 presents an extended version of the generalized sector bound condition for input saturation proposed in [START_REF] Da | Anti-windup design with guaranteed regions of stability for discrete-time linear systems[END_REF]. In this version, all involved matrices are allowed to be parameter-dependent which contributes to reduce the conservatism of the results. Moreover, it accounts also the effect of system disturbance signal w into the sector condition.

Lemma 2. The ellipsoid E (P ) is included in the polyhedral set P u defined in [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF] if and only if [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]

K i(l) -G i(l) P -1 K i(l) -G i(l) T ≤ u 2 max(l) , ∀ (i, l) ∈ Ω r × Ω nu . (24)
Lemma 3. The ellipsoid E (P ) is included in the polyhedral set P x defined in [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] if and only if [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]

P h k h T k 1 ≥ 0, ∀k ∈ Ω p . (25)
Lemma 4. [START_REF] Liu | New approaches to H∞ controller designs based on fuzzy observers for T-S fuzzy systems via LMI[END_REF] Let Υ ij , i, j ∈ Ω r be symmetric matrices of appropriate dimensions and η 1 , . . . , η r be a family of functions satisfying the convex sum property. If there exist matrices

Q ij with Q ij = Q T ji , for i, j ∈ Ω r such that                      Υ ii + Q ii < 0, ∀i ∈ Ω r , Υ ij + Υ ji + Q ij + Q ji < 0, ∀i, j ∈ Ω r , i < j,       Q 11 Q 12 • • • Q 1r Q 21 Q 22 . . . . . . . . . Q (r-1)r Q r1 • • • Q r(r-1) Q rr       > 0. (26)
Then, it follows that

r i=1 r j=1 η i (θ) η j (θ) Υ ij < 0.
The relaxation result in Lemma 4 has been further improved 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] at the expense of higher computational cost.

III. MAIN RESULTS

The theorem below provides LMI conditions to design the DOFC together with its anti-windup strategy of the form (7) which solves the control problem defined in Section II.

Theorem 1. Given T-S fuzzy system (1) and positive scalars α, δ. Assume there exist positive definite matrices

P 11 ∈ R nx×nx , X 11 ∈ R nx×nx , positive diagonal matrices S i ∈ R nu×nu , matrices Q ij ∈ R (2nx+nu+ny+nw)×(2nx+nu+ny+nw) , W ij ∈ R nx×nu , U i ∈ R nu×nx , V i ∈ R nu×nx , Âij ∈ R nx×nx , Bi ∈ R nx×ny , Ĉi ∈ R nu×nx
and D ∈ R nu×ny , for i, j ∈ Ω r , and positive scalars γ, ρ such that

X 11 I I P 11 > 0, (27) 
 X 11 I X 11 h 1k * P 11 h 1k * * 1   ≥ 0, ∀k ∈ Ω q , (28)
   X 11 * * I P 11 * C i(l) -U i(l) DC i y (l) -V i(l) u 2 max(l)    ≥ 0, ∀ (i, l) ∈ Ω r × Ω nu , (29)
1 -ρ -γδ ≥ 0, (30)
and conditions (26) hold with

Υ ij        He (H ij ) J ij U T i -B u i S j L ij X 11 (C y i ) T * He (M ij ) V T i -W ij N ij (C y i ) T * * -2S j DD yw i 0 * * * (D yw i ) T (D yw i ) -γI 0 * * * * -I        , (31)
and

H ij A i X 11 + B u i Ĉj + αX 11 , J ij ÂT ij + A i + B u i DC y j + 2αI, M ij P 11 A i + Bi C j y + αP 11 , L ij B u i DD yw j + B w i + X 11 (C y i ) T D yw i , N ij P 11 B w i + Bi D yw j + (C y i )
T D yw i . Let P 12 and X 12 be two matrices satisfying the condition

P 11 X 11 + P 12 X T 12 = I. ( 32
)
Then, the DOFC together with its anti-windup compensation [START_REF] Huang | Application of two-loop robust control to air-conditioning systems[END_REF] given by

E c ij = P -1 12 W ij S -1 j -P -1 12 P 11 B u i , D c = D, C c i = Ĉi -D c C y i X 11 X -T 12 , B c i = P -1 12 Bi -P 11 B u i D c , A c ij = P -1 12 Âij -P 11 A i + B u i D c C y j X 11 -P 12 B c i C y j X 11 -P 11 B u i C c j X T 12 X -T 12 , ∀i, j ∈ Ω r . ( 33
) θ ψ (u c ) -G θ K w θ x cl w < 0. ( 38
)
Since E (P ) ⊆ P u , by Lemma 1, the satisfaction of condition [START_REF] Yang | Robust model predictive control for discrete-time Takagi-Sugeno fuzzy systems with structured uncertainties and persistent disturbances[END_REF] implies that

V (x cl ) + 2αV (x cl ) + y T y -γw T w < 0. ( 39
)
We distinguish the two following cases.

• Case 1. If w = 0, ∀t ≥ 0, it follows from (39) that: V (x cl ) < -2αV (x cl ). This inequality implies that the set E (P ) is positively invariant with respect to the closed-loop system [START_REF] Lam | Stable and robust fuzzy control for uncertain nonlinear systems[END_REF], and that, for any initial state in this set, the corresponding trajectory converges exponentially to the origin with a decay rate α. This fact proves Property 2.

• Case 2. If w = 0, ∀t ≥ 0 and w ∈ W δ , integrating both sides of (39) from 0 to T f , with T f > 0, it follows that

V (x cl (T f )) -V (x cl (0)) + 2α T f 0 V (x cl ) dt -γ T f 0 w T wdt + T f 0 y T ydt < 0. ( 40
)
From ( 30) and ( 40), it can be deduced that, for any initial state in E (P, ρ), we have V (x cl (T f )) < ρ + γδ ≤ 1. This means that the corresponding closed-loop trajectories are confined in the set E (P ) ⊆ P x . Furthermore, considering the limit case

T f → ∞ in (40), then we obtain ∞ 0 y T ydt < γ ∞ 0 w T wdt + V (x (0)
), this means that the L 2 -norm of the output signal y is upper bounded by y 2 2 < γ w 2 2 + ρ. These facts proves Properties 1 and 3 of the closed-loop system. From the results of the two above cases, the proof of the theorem can be concluded.

Remark 2. When w ∈ W δ \{0}, ∀t ≥ 0, and x cl (0) = 0, there is a trade-off between the size of the set of initial conditions and the maximal level energy of disturbances characterized by δ. Indeed, the lower is the admissible δ (i.e. the lower is the admissible energy of the disturbance signal), the larger is the estimate of the set of admissible initial states E (P, ρ), and by extension, the ellipsoid E (P ) [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].

Remark 3. Using quadratic Lyapunov function may make the results conservative. To overcome this drawback, multiple Lyapunov function approach [START_REF] Tanaka | A multiple Lyapunov function approach to stabilization of fuzzy control systems[END_REF] could be applied. Although this latter has recently received increasing attention, it still remains an open research topic for continuous-time T-S systems. This fact is due to the presence of the time-derivatives of the membership functions involved in the theory development (which is not the case of discrete-time T-S systems [START_REF] Guerra | LMI-based relaxed non-quadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form[END_REF]). It should be stressed that these non-quadratic Lyapunov based approaches generally lead to extremely complex design conditions with heavy computation cost even with a static state-feedback controller [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF]. Moreover, in terms of control law implementation, the control structure is also very complex requiring online parameter-dependent matrix inversion. Therefore, for practical uses this work keeps using quadratic Lyapunov function regardless of its inducing conservativeness.

IV. ANTI-WINDUP BASED DYNAMIC OUTPUT FEEDBACK CONTROL DESIGN

Theorem 1 provides conditions to check easily the feasibility of the control problem defined in Section II. Since the results are developed for regional stability and performance, there is a trade-off between different requirements. According to the design purposes, two following optimization problems can be formulated. Optimization Problem 1. Find a DOFC and its AW strategy [START_REF] Huang | Application of two-loop robust control to air-conditioning systems[END_REF] such that the closed-loop system satisfies the properties defined in Section II-A and the disturbance rejection effect is maximized for a given bound δ on the L 2 -norm of admissible disturbances. This problem can be straightforwardly solved with the following LMI optimization

min P11,X11,Si,Ui,Vi,Wij , Âij , Bi, Ĉi, D,γ,ρ γ, (41)
subject to: γ > 0 and LMI conditions in 1. Optimization Problem 2. Find a DOFC and its AW strategy [START_REF] Huang | Application of two-loop robust control to air-conditioning systems[END_REF] such that the closed-loop system satisfies the properties defined in Section II-A and both ellipsoids E (P, ρ) and E (P ) can be maximized. In order to avoid obtaining a flat estimate of the domain of attraction, we can minimize the following optimization function [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF], [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF], [START_REF] Yang | Robust model predictive control for discrete-time Takagi-Sugeno fuzzy systems with structured uncertainties and persistent disturbances[END_REF]

α R = λ trace (R) + (λ -1) ρ, (42)
where 0 ≤ λ ≤ 1 is a tuning parameter and the matrix R R 11 R 12 * R 22 satisfies R > P . Using successively Schur complement lemma and congruence transformation with diag I, Π T 2 , it can be proved that

R > P ⇔     R 11 R 12 I P 11 * R 22 0 P T 12 * * X 11 I * * * P 11     > 0. (43)
From (43), it follows that trace (R) > trace (P ). Then, the minimization of trace (R) implies the minimization of trace (P ).

Then, Problem 2 can be solved through the convex optimization problem

min P11,X11,P12,Si,Ui,Vi,Wij , Âij , Bi, Ĉi, D,γ,ρ,R α R , (44)
subject to: R > 0, LMI conditions in Theorem 1 and LMI (43). Several remarks on the obtained results can be reported as follows.

Remark 4. The equation ( 32) has an infinite number of solutions P 12 and X 12 parameterizing an infinite number of controllers with the same closed-loop performance. Indeed, the choice of matrices P 12 , X 12 is irrelevant for control design since it corresponds to a change of coordinates of the controller states [START_REF] Ghaoui | Control of rational systems using linear-fractional representations and linear matrix inequalities[END_REF]. Up to now this degree of freedom has not been yet exploited for the selection of the most desirable controller [START_REF] Dai | Output feedback design for saturated linear plants using deadzone loops[END_REF]. However, with the proposed method it should be stressed from (43) that P 12 can be considered as a slack variable of the optimization problem (44) which aims at selecting, among all possible solutions of (32), the most desirable one maximizing the domain of attraction with respect to a predefined optimization criterion. The corresponding "optimal" DOFC with its AW compensation [START_REF] Huang | Application of two-loop robust control to air-conditioning systems[END_REF] can be now computed by ( 44) and [START_REF] Sala | Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya's theorem[END_REF].

Remark 5. It follows from [START_REF] Liu | New approaches to H∞ controller designs based on fuzzy observers for T-S fuzzy systems via LMI[END_REF] that P 11 > X -1 11 . Consequently, the set E (P 11 , ρ) is always contained in E X -1 11 , ρ . Assume that x c (0) = 0, then x cl (0) ∈ E (P, ρ) if x (0) ∈ E (P 11 , ρ). In the absence of disturbances, the set E (P ) is positively invariant, and so x cl (t) ∈ E (P ) for ∀t ≥ 0. Therefore, x (t) will never escape the projection of E (P ) onto the plane defined by x c = 0. This projection is nothing but the ellipsoid E X -1 11 .

V. ILLUSTRATIVE EXAMPLE

In this section, the effectiveness of the proposed method is demonstrated through a physically motivated nonlinear system. All LMI problems are solved with the YALMIP toolbox in Matlab R2013b. Let us consider the following nonlinear massspring-damper mechanical system

     ẋ1 = -x 1 -1 + 0.05x 2 2 x 2 + 0.5 + 0.075x 3 2 u + 0.1w ẋ2 = x 1 + 0.1w y = x 2 (45)
where x 1 , x 2 and w are respectively the velocity, the position and the system disturbance. The following assumptions are considered for this example. First, the state vector x T [x 1 x 2 ] always remains in the validity domain described by P x = x ∈ R 2 : |x 1 | ≤ 1.5, |x 2 | ≤ 1.5 . Second, the control saturation limit is: u max = 0.5. Third, only the position is measured as system output. Finally, the system disturbance w is energy-bounded and of the following form

w (t) = 0.1 cos (t) , if 0 ≤ t ≤ 15 0, otherwise (46)
where the corresponding energy bound is given δ ≈ 0.07. Since x ∈ P x , two nonlinearities f (x 2 ) = 1 + 0.05x 2 2 and g (x 2 ) = 0.5 + 0.075x 3 2 of system (45) are then bounded f min ≤ f (x 2 ) ≤ f max and g min ≤ g (x 2 ) ≤ g max . Using sector nonlinearity approach [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF], these nonlinearities can be represented as follows

f (x 2 ) = ω 1 f min + ω 2 f max , g (x 2 ) = ω 3 g min + ω 4 g max , (47)
where

ω 1 (x 2 ) = f max -f (x 2 ) f max -f min , ω 2 (x 2 ) = 1 -ω 1 (x 2 ) , ω 3 (x 2 ) = g max -g (x 2 ) g max -g min , ω 4 (x 2 ) = 1 -ω 3 (x 2 ) .
The normalized nonlinear membership functions of T-S model can be now obtained by

η 1 (x 2 ) = ω 1 (x 2 ) ω 3 (x 2 ) , η 2 (x 2 ) = ω 1 (x 2 ) ω 4 (x 2 ) , η 3 (x 2 ) = ω 2 (x 2 ) ω 3 (x 2 ) , η 4 (x 2 ) = ω 2 (x 2 ) ω 4 (x 2 ) . (48)
Then, the nonlinear system (45) can be exactly represented by T-S model (1) in the polyhedral set P x , where the subsystem matrices are given by

A 1 = -1 -f min 1 0 , B u 1 = g min 0 , A 2 = -1 -f min 1 0 , B u 2 = g max 0 , B w i = 0.1 0.1 , A 3 = -1 -f max 1 0 , B u 3 = g min 0 , A 4 = -1 -f max 1 0 , B u 4 = g max 0 , ∀i ∈ Ω 4 .
For numerical experiments, it is noticed that the decay rate α in Property 2 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]. A large value of this tuning parameter leads to a fast convergence time; however the corresponding controller could induce some aggressive closed-loop behaviors. Especially, this situation can get worst if the disturbance is directly involved in the system dynamics. In this work, we take α = 0.01 to illustrate the results. Moreover, without loss of generality the effectiveness of the proposed method will be demonstrated through the solution of the optimization problem (44). To this end, let us take the weighting factor λ = 0.

In what follows, we will show that the corresponding DOFC satisfies the three predefined properties stated in Section II.

Regional quadratic α-stability and set inclusion properties. In the absence of disturbances, i.e. w (t) = 0, it can be seen in Figure 2 that the projection of E (P ) onto the plane defined by x c = 0, that is S 1 E X -1 11 , is an invariant set (corresponding to system states) of the closed-loop system, i.e. all system trajectories initialized in this set will never escape it. From (49), one has X -1 11 ≈ P 11 . Then, we note that both ellipsoids E X -1

11 and E (P 11 , ρ) are (almost) superimposed in Figure 2 and the set E X -1 11 includes, of course, E (P 11 , ρ) since ρ < 1. Furthermore, as can be also observed, the ellipsoid S 1 is maximized along the polyhedral set S 2 P u P x which is, in turn, contained in the S 3 Ξ r i=1 η i (K i -G i ) . Finally, all closed-loop trajectories converge to the origin. Closed-loop finite L 2 -gain performance. It is assumed now that the disturbance w defined in (46) is directly involved in the dynamics of system (45), Figure 3 (up) shows the state evolution of the closed-loop system (45) obtained with initial condition

x T 0 = [1 -0.5].
As can be observed, the energy-bounded disturbance w is well attenuated and the ratio

T f 0 y T ydt γ T f 0 w T wdt+ρ , with
T f > 0, is always bounded by 1, see Figure 3 (bottom). This latter guarantees the closed-loop L 2 -gain performance defined in [START_REF] Campos | Revisiting the TP model transformation: Interpolation and rule reduction[END_REF]. The corresponding control signal u, the controller output u c and the controller state x c are depicted in Figure 4. We can see that despite an important level of control input saturation at the beginning, the proposed AW-based dynamic controller induces stable closed-loop behaviors. It is noticed that in this case, the classical DOFC (without anti-windup structure) proposed in [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] is no longer effective as shown in Figure 5. Now, let us examine a limit case where the initial condition x T 0 = [1.5 1.5] is a vertex of the validity domain P x . It can be clearly observed in Figure 6 that the proposed controller can provide stable behaviors despite a huge level of saturation of input signal u and also system state x 2 . Notice that the classical DOFC in [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] is not able to guarantee the closed-loop stability in such a situation.

VI. CONCLUDING REMARKS

A novel approach to design a dynamic output feedback controller together with anti-windup strategy for continuous-time nonlinear systems in T-S form has been proposed. In this approach, the disturbed systems are subject to control input and system state constraints. Based on Lyapunov stability theory, a constructive procedure is given to design simultaneously the dynamic output feedback controller and its anti-windup compensator. The control design is reformulated as convex optimization problems such that several regional closed-loop specifications can be achieved. The interests of the proposed method are clearly demonstrated by means of an physically motivated example. Moreover, the proposed design conditions are relatively simple which can be applied to a wide class of nonlinear disturbed systems. Notice that if properly extended, the proposed control method also provides possible application prospects for T-S fuzzy systems subject to both input magnitude and rate saturations, or more generally nested saturations [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF], which will be pursued in future works.

ACKNOWLEDGMENT

We are grateful for the valuable suggestions from the reviewers, which are constructive to improve the paper quality.

Fig. 1 .

 1 Fig. 1. Closed-loop system with anti-windup strategy.

Fig. 2 .

 2 Fig. 2. Projection of E (P ) onto the plane defined by xc = 0 and system trajectories.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Closed-loop system states (up) and finite L 2 -gain performance (bottom) with x T 0 = [1 -0.5].

Fig. 5 .

 5 Fig.5. Closed-loop state responses (top); evolution of the control input and controller output (bottom) obtained with classical DOFC[START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF].

Fig. 6 .

 6 Fig. 6. Closed-loop behaviors obtained with the proposed AW-based dynamic controller for x T 0 = [1.5 1.5].

solves the control problem stated in Section II.

Proof. We use the linearizing approach of [START_REF] Scherer | Multiobjective output-feedback control via lmi optimization[END_REF]: properties [START_REF] Liu | New approaches to H∞ controller designs based on fuzzy observers for T-S fuzzy systems via LMI[END_REF] and [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] imply the existence of two matrices P 22 and X 22 such that the block matrices P and X given by

are positive definite. Note also that the same properties imply that the matrices P 12 and X 12 are regular. Let us now introduce the matrices

By congruence transformation with diag (Π 1 , I), inequality ( 25) is shown to be equivalent to [START_REF] Lin | Improvement on observer-based H∞ control for T-S fuzzy systems[END_REF]. This implies that the ellipsoid E (P ) is included in the polyhedral set P x defined in [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. Similarly, by Schur complement lemma and congruence transformation with diag (Π 1 , I), inequality ( 29) is shown to be equivalent to [START_REF] Nguyen | Anti-windup based dynamic output feedback controller design with performance consideration for constrained Takagi-Sugeno systems[END_REF], so that the ellipsoid E (P ) is included in the polyhedral set P u defined in [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF] with

By Lemma 4, inequalities [START_REF] Kau | Robust H∞ fuzzy static output feedback control of T-S fuzzy systems with parametric uncertainties[END_REF] imply clearly that Υ θθ r i=1 r j=1 η i (θ) η j (θ) Υ ij < 0. After a congruence transformation with diag Π 1 , S -1 θ , I, I , this inequality is proved to be equivalent to    


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

Antoine Dequidt

Michel Dambrine

Anti-Windup Based Dynamic Output Feedback Controller Design with Performance Consideration for Constrained Takagi-Sugeno Systems

Keywords: Nonlinear systems, Takagi-Sugeno fuzzy model, dynamic output feedback controller, anti-windup compensator, ℒ2 performance, domain of attraction

This paper proposes an LMI-based method to guarantee the closed-loop regional stability and performance for Takagi-Sugeno systems that are subject to input saturation, state constraints and also amplitude-bounded disturbance.

Based on Lyapunov stability tool, the proposed method provides conditions to simultaneously design the dynamic output feedback controller and its anti-windup compensator. By solving a convex optimization problem, these conditions are derived such that a tradeoff between the upper bound on the nominal ℒ2 gain for exogenous disturbance and the minimal size of the domain of attraction can be found. This method is simple and systematic, allowing dealing with a very large class of constrained nonlinear systems. The effectiveness of the proposed method is illustrated with numerical examples.

Introduction

Over the past two decades, Takagi-Sugeno (T-S) fuzzy models [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF] have been intensively studied in the control community [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. It is motivated by the fact that these models have general approximation capability for complex dynamical system [START_REF] Cao | Analysis and design for a class of complex control systems, Parts I and II[END_REF]. Moreover, under weak conditions, a nonlinear model can be, globally or more often regionally, rewritten on a fuzzy T-S form. Stability analysis or controller synthesis is then facilitated due to their polytopic structure [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. As a consequence, this approach has become now a very attractive research topic in control theory [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF]. Stability analysis of a given T-S system is investigated in most cases via the direct Lyapunov method through the use of a quadratic Lyapunov function; the derived stability conditions being expressed as linear matrix inequalities (LMIs) [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] for which efficient solvers are available. For the controller design, the choice of a parallel distributed compensation (PDC) control law is usually done for T-S models [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]). An abundant literature is available on this nonlinear state feedback control law; see e.g. [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF] for a quick overview.

For technological or economic reasons, the state variables are not all measured in most of real-world applications. In order to deal with this practical problem, output feedback control must be used. In general, observer-based control scheme is proposed for unconstrained T-S systems [START_REF] Liu | New approaches to controller designs based on fuzzy observers for T-S fuzzy systems via LMI[END_REF][START_REF] Guerra | Conditions of Output Stabilization for Nonlinear Models in the Takagi-Sugeno's Form[END_REF]. However, the design problem becomes much more complicated when state and/or input constraints have to be explicitly considered. As highlighted in [START_REF] Ding | Quadratic boundedness via dynamic output feedback for constrained nonlinear systems in Takagi-Sugeno's form[END_REF], this control issue is not well addressed in the literature.

Due to physical/technical limitations and/or safety constraints, actuator saturation is unavoidable in almost all real applications. This phenomenon can severely degrade the closed-loop system performance and, in some cases, may lead to system instability. Motivated by this practical control aspect, a great deal of effort has been focused on saturated systems [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]. In the literature, several methods now exist to handle saturation effects, but the most popular and effective one remains the anti-windup (AW) approach (see for instance [START_REF] Kothare | A unified framework for the study of anti-windup designs[END_REF]). LMI-based synthesis of anti-windup compensators has been proposed recently to synthesize either static (Gomes da Silva [START_REF] Gomes Da Silva | Anti-windup design with guaranteed regions of stability: an LMI-based approach[END_REF][START_REF] Mulder | Simultaneous linear and anti-windup controller synthesis using multiobjective convex optimization[END_REF] or dynamic anti-windup compensators [START_REF] Cao | An anti-windup approach to enlarging domain of attraction for linear systems subject to actuator saturation[END_REF][START_REF] Grimm | Antiwindup for stable linear systems with input saturation: an LMI-based synthesis[END_REF][START_REF] Hu | Anti-windup synthesis for linear control systems with input saturation: achieving regional, nonlinear performance[END_REF] , an overview of these results can be found in the survey [START_REF] Tarbouriech | Anti-windup design: an overview of some recent advances and open problems[END_REF]. Most of these works deal with a twostep method in which the controller and the AW strategy are designed separately. This method often proves to be satisfactory but it has however some drawbacks. First, only sub-optimal solution can be achieved since the controller and its AW compensator are designed separately [START_REF] Sawada | Generalized Sector Synthesis of Output Feedback Control with Antiwindup Structure[END_REF]. Second, the effect of the nominal controller on the closed-loop performance under saturation is completely ignored [START_REF] Mulder | Simultaneous linear and anti-windup controller synthesis using multiobjective convex optimization[END_REF]. To overcome these drawbacks, an alternative solution called one-step method, which designs simultaneously the controller and the AW compensator, has been proposed. Among few works existing in the literature, we can notably cite [START_REF] Mulder | Simultaneous linear and anti-windup controller synthesis using multiobjective convex optimization[END_REF] in which an LMI solution ensuring global stability and performance is proposed for systems that are stable in open loop; [START_REF] Sawada | Generalized Sector Synthesis of Output Feedback Control with Antiwindup Structure[END_REF]) proposes a regional approach based on a change of variable; whereas [START_REF] Dai | Output feedback design for saturated linear plants using deadzone loops[END_REF] relies on the parameter elimination approach of [START_REF] Skelton | A Unified Algebraic Approach to Linear Control Design[END_REF]. Both latter approaches are based on some preliminary results proposed in [START_REF] Hu | Stability and performance for saturated systems via quadratic and non-quadratic Lyapunov functions[END_REF].

Up to now, a large amount of works on AW-based design are available for linear systems [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF], but very few works deal with nonlinear cases, especially when exogenous disturbance signals are actively present [START_REF] Gomes Da | Dynamic output feedback stabilization for systems with sector-bounded nonlinearities and saturating actuators[END_REF]. In the T-S control framework, there are a couple of works devoted to the analysis or control design of saturated systems, for instance [START_REF] Cao | Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation[END_REF][START_REF] Tseng | H∞ fuzzy control design for nonlinear systems subject to actuator saturation[END_REF][START_REF] Du | Fuzzy control for nonlinear uncertain electrohydraulic active suspensions with input constraint[END_REF][START_REF] Bezzaoucha | Stabilization of nonlinear systems subject to actuator saturation[END_REF][START_REF] Ariño | Guaranteed cost control analysis and iterative design for constrained Takagi-Sugeno systems[END_REF]. However, very few papers are dedicated to AW synthesis for T-S systems: in [START_REF] Ting | Robust anti-windup controller design of time-delay fuzzy systems with actuator saturations[END_REF], the authors addressed a two-step approach to deal with a continuous time-delay T-S systems; in [START_REF] Zhang | Piecewise fuzzy anti-windup dynamic output feedback control of nonlinear processes with amplitude and rate actuator saturations[END_REF], an interesting one-step approach based on piecewise fuzzy AW dynamic output feedback controller (DOFC) for discrete-time T-S systems has been proposed, note that these results seems to be valid only for systems that are stable in open-loop since no admissible sets of initial conditions are defined; and, at last, [START_REF] Song | Robust stabilization of state delayed TS fuzzy systems with input saturation via dynamic antiwindup fuzzy design[END_REF] which extends the approach proposed in (Gomes da [START_REF] Gomes Da Silva | Anti-windup design with guaranteed regions of stability: an LMI-based approach[END_REF] to the case of T-S systems. It is noteworthy that an important point is neglected in all these results: besides control input saturation, the T-S model is only valid on a given subset of the state space. This is of course true for any model of real-world systems, but is fundamental in the writing of a T-S model using the nonlinear sector decomposition approach [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. This validity domain can be represented by some constraints on the state variables (see Example 2 for illustration of this fact). It is particularly important to consider explicitly these state constraints in the control design to ensure a good behavior of the closed-loop system in response of disturbances. This fact has been very recently emphasized in [START_REF] Nguyen | Advanced control design tools for automotive applications[END_REF][START_REF] Nguyen | Lyapunov-based robust control design for a class of switching nonlinear systems subject to input saturation: application to engine control[END_REF][START_REF] Klug | Fuzzy dynamic output feedback control through nonlinear Takagi-Sugeno models[END_REF]. The two first cited references concern stabilization of nonlinear switching systems under saturation.

The latter one deals with the design of a dynamic output feedback control for nonlinear systems represented in Takagi-Sugeno form. Apart from the fact that we consider the effect of bounded disturbances on the design, a fundamental difference with the work in [START_REF] Klug | Fuzzy dynamic output feedback control through nonlinear Takagi-Sugeno models[END_REF] is that they considered discrete-time systems, whereas this paper concerns continuous-time systems. This allows them to use elegantly non-quadratic Lyapunov functions for the control law design.

Multiple or non-quadratic Lyapunov functions are also developed in the continuous-time case allowing to obtain less conservative conditions for stability analysis [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]. Concerning synthesis of control laws, the situation becomes much more complex and remains open due to the presence of the time-derivative of the membership functions in the stability conditions. Results so far developed lead to very complex design conditions or to controllers requiring the inversion in real time of a parameter-dependent matrix (see for instances [START_REF] Guelton | Robust dynamic otput feedback fuzzy Lyapunov stabilization of Takagi-Sugeno systems -A descriptor redundancy approach[END_REF][START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF][START_REF] Bouarar | Robust non-quadratic static output feedback controller design for Takagi-Sugeno systems using descriptor redundancy[END_REF]. The goal of this paper is to obtain a controller that may be easily implemented in practice. For this reason, the obtained results rely on the quadratic stabilization approach. However, as will be shown in Section 5 through numerical example, in the context of input-saturated T-S systems, the proposed results may be competitive in some sense with respect to those presented in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF] which actually provides the less conservative results among all non-quadratic approaches found in the open literature.

In this paper, we address a novel one-step method to design simultaneously a DOFC and an AW compensator for a disturbed T-S system subject to control input and system state constraints. The proposed approach has some special features deserving particular attention: quadratic boundedness of the trajectories (see [START_REF] Ding | Quadratic boundedness via dynamic output feedback for constrained nonlinear systems in Takagi-Sugeno's form[END_REF] and references therein) is ensured for any admissible initial condition and disturbance signal, as well as a maximal 2 gain for the unsaturated system. Note that this 2 performance may be ensured regionally for the saturated case as in [START_REF] Dai | Output feedback design for saturated linear plants using deadzone loops[END_REF], however this decoupling allows reducing the conservatism of the results. Moreover, it will be shown that the control design can be formulated as a multi-objective LMI optimization problem. In such a way, the obtained controller can solve the tradeoff between some predefined closed-loop requirements.

The paper is organized as follows. Section 2 describes the design problem and recalls some preliminaries results. The main result is stated in Section 3. In Section 4, a constructive control design is presented as a multi-objective LMI optimization problem. The results of the paper are effectively illustrated through an example in Section 5. Finally, Section 6 gives some concluding remarks.

The notations and terminology used in this paper are standard. For an integer number r, r  denotes the set

 

1, 2, , r .

 

0,   is the set of non-negative real numbers.

 

2 n e denotes the extended 2 -space composed of measurable functions

  0 :, n ft  such that   0 2 0 ,; T t f t dt T t      and     0 1/ 2 2 2, . T T t f f t dt     i x
     and 1 1 r i i    
. The following notations are occasionally used:

        1 1 1 1 1 1 ;; r r r r i i i i i j ij i i i j Y Y Y Y Z Z                           (

Problem Definition and Preliminaries Results

Control Problem Definition

Closed-loop system description

Consider the following fuzzy T-S model described by [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] valid on a polyhedral domain x : Assumption 2. The input vector u is subject to symmetric magnitude limitations:

         1 1 1 r uw i i i i i r z zu zw i i i i i r y yw i i i i x A x B u B w z C x D u D w y C x D w                          (2)
        max max max ; 0; . u n i i i i u u u u i        (3)
Assumption 3. The disturbance signal w is assumed to belong to the set W  defined by:

        2 0
,, :

w T n e W w t Rw t t t w        (4)
where the matrix 0 R  and the bound 0   are known.

Assumption 4. The validity domain x of (2) is a polyhedral set defined by:

  1 : 1; , x n xq Tk xh xk      (5)
where the q vectors 1

x n k h  are given.

Consider now the unconstrained dynamic output feedback controller (DOFC) in the form of [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]:

          1 1 1 1 ; 0 0 r r r ij i c i j c c i c c i j i r i c i c c c i x A x B y v x u C x D y                           (6)
where

x n c x  , u n c u 
are respectively the state and the output vectors of the controller. The additional term v is introduced to minimize the undesirable degradation of closed-loop performance caused by input saturation. The controller (6) has to be designed to guarantee the local stability and some performance requirements for the closed-loop system.

Because of the input limitation, the actual control signal injected into the system is subject to the saturation effect:

 , c u sat u  (7)
where each component of the saturation function

 

sat  is given by:

              max min , ;
.

u cn c i c i i i sat u sign u u u i  (8)
Then, the interactions between the system (2) and the constrained controller are given as:

        11 ;
.

rr c c i ij c c ij u sat u v E sat u u       (9)
From ( 6) and ( 9), the DOFC combined with the anti-windup strategy can be expressed as:

            1 1 1 1 r r r c c c c i j ij c ij c i i i j i r cc c i i c i x A x E u B y u C x D y                           (
                    max max max 0 if if ii i i i i i i uu u u sign u u u u          (11) Define T T T cl c
x x x   , the constrained closed-loop system is obtained from (2) and ( 10):

                  11 11 rr c cl i j cl i w ij ij w ij ij ij c ij rr i j cl i c ij x x w R E u z x w u                                (12) where   0 ; ; ; 0 ;; u c y u c w u c yw u i i j i j i i j i i c y c c yw i j ij i j z zu c y zu c zw zu c yw zu i i j i j i i j i ij ij w ij iw i j A B D C B C B B D D B R B C A B D I C D D C D C D D D D D                                       (13)
and the controller output is given as:

   1 , r w c i i cl i i u x K w     (14)
where

c y c i i i D C C    and w c yw ii K D D  .
Using notations (1), the closed-loop system (12) is rewritten as:

      c cl cl c w cl c w x x w R E u z x w u                         (15)
and the DOFC output ( 14) is rewritten as:

.

w c cl u x K w   (16)

Control problem definition

Local stability and performance specifications of the closed-loop system (12) will be presented in terms of Lyapunov analysis tools. Our goal is to propose a systematic method to design a DOFC together with its AW compensator of the form (10) such that the closed-loop system satisfies the following properties:

 Property 1 [Regional Quadratic Stability]. For 0,

w  there exist a quadratic function   T cl cl cl V x x Px  with 0 P  and a real number 1 0   such that     1 cl cl xx VV 
 along the trajectory of the closed-loop system for any initial augmented state in the ellipsoid   P . This fact means that, for any   0 ()

cl xt P 
, the corresponding trajectory of the closed-loop system will converge exponentially to the origin with a rate less than 1 /2  .

 Property 2 [State Constraints].

For any initial augmented state in   P and any disturbance signal in W  , the trajectory of the closed-loop system (12) remains in the polyhedral region described by linear inequalities: f Tt 

  2 1; : , c x l x l c n c l q T k h x x k      (

Preliminaries

In this section, some important preliminaries results needed for design problem in Section 3 will be presented.

Generalized sector bound condition for input saturation

Given the matrices

2 ux nn i   and 2 12 ux nn ii i GG     , with 1 ux nn i G   , 2 ux nn i G  
, for r i  , we define the polyhedral set u as follows:

  1 , i u r i i   ( 18
)
where

            2 max ; . : x u n i i cl cl n i l i l l x x u l         (19)
The following lemma is an extended version of Lemma 1 of (Gomes da [START_REF] Gomes Da Silva | Anti-windup design with guaranteed regions of stability: an LMI-based approach[END_REF]. x  , then the following condition is verified:

    1 1 1 1 0, r r r cl Tw c i i c i i i i i i i x u S u K w                              (20)
where

uu nn i S  
are any positive diagonal matrix and 1 ,, r  are scalar functions satisfying the convex sum propriety.

Proof. The proof of this result can be done in the same fashion as the one given in [START_REF] Nguyen | Advanced control design tools for automotive applications[END_REF].

Other preliminary results

Some other results that will be useful in the proof of our main result are recalled below.

Lemma 2. [START_REF] Tuan | Parameterized linear matrix inequality techniques in fuzzy control system design[END_REF]

  0, 2 0, an 1 , d ii ii ij ij r rr i i r j j i                       (21)
Remark 1. Note that more performing relaxation result exists in [START_REF] Liu | New approaches to controller designs based on fuzzy observers for T-S fuzzy systems via LMI[END_REF][START_REF] Ariño | Relaxed LMI conditions for closed-loop fuzzy systems with tensor-product structure[END_REF]. However, Lemma 2 constitutes a good tradeoff between complexity and conservatism since it does not require the introduction of auxiliary variables. Lemma 3. [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] The ellipsoid   P is included in the polyhedral set u defined in (18) if and only if:

                12 max ; , , u T rn i l i l i l i l l P u i l          (22)
where   il ,   il are respectively the th l rows of the matrices i and i .

Lemma 4. [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] The ellipsoid   P is included in the polyhedral set

Main Results

The theorem below provides LMI conditions to design the DOFC together with its anti-windup strategy of the form (10) which solves the control problem defined in Subsection 2.1.2. ;. 00

* 0; 1 k k q X I X h P h k          (25)               11 11 2 max ** * 0; , u rn i il y i l i l l l X I P i l U C V u               (26) 12 0     (27)   0, 2 0, an 1 , d ii ii ij ij r rr i i r j j i                       (28)   , 0, 2 0, and
1 ii ii ij ij r rr i ij ij r                       (
u y j ij i i ij i i j ij i y u u y j ij i T j ij i T j i ij ij i i i j ij i y A X B X A B C P A sym I sym P C A X B A B C P A mm C sy sy              (
TT X I I P P XP                 (
  1 2 0. T cl cl T cl x Px V x w R w     (39)
From ( 27) and ( 39), it follows that:

      2 1 1 0. T cl cl T cl V x w w x Px R      ( 40
)
Satisfaction of ( 40) ensures the ellipsoid   P is a robustly positively invariant set [START_REF] Blanchini | Set invariance in control[END_REF] with respect to the closed-loop system (15). In additional, since   , it can be found that (29) implies inequality (42). This fact shows that Property 3 holds which completes the proof.

Anti-Windup Based Dynamic Output Feedback Controller Synthesis

The feasibility problem stated in Theorem 1 provides the conditions to design the controller satisfying some predefined requirements in Subsection 2.1.2. In this section, the design method will be formulated as a multi-objective optimization problem. Two objectives are considered: maximization of the disturbance rejection and of the estimated domain of attraction.

In order to obtain a sufficiently large estimated domain of attraction, we can consider, for example, the reference ellipsoid set

2 x n R X  containing the origin:   2 0 : 1 , x n T R cl cl cl X x x P x    (43)
where 0 0. P  Then, finding the largest ellipsoid   P can be formulated as the following optimization problem:

  ,0 max subject to: .

R P XP     (44)
It is worth noting that R X defines the directions in which we want to maximize  

P . Condition   R XP   is equivalent to 2 0 PP  
. By Schur complement lemma, this condition is equivalent to, with  

1 2 1 11 , , , ˆˆˆ, , , ,, , , , , min 1 ,
i i i ij i i P X S U V W        (47)
such that the following LMI conditions hold: 0   , 0   ; LMIs (24)-(29); LMI (46).

The positive weighting factor 01   is chosen according to desired trade-off between the closed-loop performance (characterized by  ) and the size of the domain of attraction   P of the closed-loop system (characterized by  ).

Remark 2. It is important to stress that equation ( 33) has an infinite number of solutions and the choice of matrices 12 P , 12 X is totally irrelevant for closed-loop performance [START_REF] El Ghaoui | Control of rational systems using linear-fractional representations and linear matrix inequalities[END_REF]. As remarked in [START_REF] Dai | Output feedback design for saturated linear plants using deadzone loops[END_REF], up to now this degree of freedom is not yet exploited for the selection of the most desirable controller. However, we have clearly pointed out in (46) that the matrix 12 P can be considered as a slack variable contributing to enlarge the domain of attraction. For matrix inversion property, the condition 12 12 0 T PP  should be added to guarantee that 12 P is nonsingular. Then 12

X is deduced from condition (33). Now, the DOFC with its AW strategy (10) can be computed from (34).

Remark 3. Note that the condition 0 X  is equivalent to x  . This projection is nothing but the ellipsoid   1 11 X  .

Illustrative Examples

In this section, numerical examples are given for illustrating the effectiveness of the proposed method. The implementations are done using the SeDuMi solver and the YALMIP toolbox [START_REF] Lofberg | YALMIP: a toolbox for modeling and optimization in MATLAB[END_REF] with option sdpsettings ('solver','sedumi','shift',1e-4) for the resolution of all LMI problems.

Example 1. Consider the T-S system (2) adapted from Example 1 in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF] with,

2 i    :             1 1 2 2 1 2 1 1 1 2
i i i i ab A B A B B B C C D D D x x x x                                                  (48)
The feasibility of ( 48) is checked using the proposed quadratic anti-windup based approach (QAWA) and the nonquadratic approach (NQA) (Theorem 3 in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF]) for all the 806 points of a 31 by 26 rectangle grid on the

parameter space   , ab ,   20, 25 a  ,   0, 25 b 
. It is noteworthy that in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF] static state-feedback controller is considered for undisturbed T-S systems and closed-loop performance was not taken into account for the control design. Thus, it is not objective to compare these two approaches since their control contexts are different.

However, checking the feasibility of their conditions provides us an idea about the conservatism of each approach. For this study, the following data are considered: Note also that all considered T-S systems are open-loop unstable. As can be observed in Figure 1, the feasibility space obtained with QAWA is larger than that of NQA for the considered grid. This interesting fact can be explained by the over-conservativeness of the unsaturated control approach in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF] which is not the case when applying the parameter-dependent sector condition proposed in Lemma 1. Moreover, the degree of relaxation of the NQA depends strongly on the control bound of the system. These serious drawbacks of the NQA are inherently due to the handling of the time-derivative of membership functions involved in the derivation of the design conditions as stated in Section 1.

Note that if closed-loop performance is considered for NQA, the obtained results are expected to be much more conservative [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF] x , it is impossible to obtain a T-S model that is globally equivalent to (49). One solution consists in restricting the range of evolution of variable 2

x , for instance 2 1.5.

x  We also consider an additional constraint on the first state variable, leading to the validity domain

  2 12 : 1.5; 1.5 x x x x    
. The control saturation limit will be taken as: max 1 u  .

On

x , functions f and g are such that:

        1m 34 in 2 max min max min max min max ; f f f f g g g ff g g g                      (
                (
T u u w y z i i i u u yw zw ii f g f g A B A B B C C f g f g A B A B D D                                                                    1. zu i D  (53)
In what follows, it will be shown that the proposed DOFC satisfies all predefined properties stated in Subsection 2.1.2. To that end, let us take 1 0.1   , the reference shape Figure 3 shows that the closed-loop system is stabilized and the disturbance is well attenuated. By means of simulation it can be also checked that the obtained controller provides stable behaviors for all closed-loop trajectories initialized in x and these trajectories remain inside this validity domain. Figure 4 shows the evolution of the performance variable for two designs of the controller. The first design (Case 1) corresponds to the minimization of 2e -gain performance whose solution is given in (55), while in Case 2 this performance is not taken into account in the control design: controller parameters are obtained solving the feasibility LMIs of Theorem 1 without condition (29). From this figure, the interest of the consideration of 2e -gain performance in the control design is clearly confirmed.

Concluding Remarks

In this paper, a novel LMI-based approach has been proposed to design simultaneously a dynamic output feedback controller together with an anti-windup compensator for nonlinear systems under input and state constraints and subject to bounded persistent disturbances. Based on Lyapunov stability theory, conditions for the stability in closed-loop are given, which allows formulating the design of the control as a multi-objective convex optimization problem. In this way, the controller and anti-windup compensator gains can be efficiently computed with semidefinite programming solvers.

By means of examples, the paper has pointed out a very interesting fact, namely, the combination of a generalized sector condition with quadratic Lyapunov approach for input-saturated T-S systems leads to simple and efficient conditions which may be competitive with respect to recent non-quadratic approaches. This is particularly true when the control bounds are not excessively large. It has been also shown that the resulting controller satisfies several predefined closedloop properties. The proposed approach can be applied to a large class of constrained nonlinear systems and the obtained controllers are without real difficulties in terms of implementation for real-time applications.

i

   are functions sharing the convex sum property.

D

  state, the control input, the disturbance, the measured output and the scheduling variable vectors of the system. The regulated output vector represent the set of r local linear subsystems and the nonlinear scalar functions   i  satisfy the convex sum property. For system (2), we consider the following assumptions: Assumption 1. The scheduling variable vector k   is assumed to be a function of all measured signals of interest (states, external disturbances …) with the exception of the control input value . u

  implying that the system state x remains in the validity domain x of the model (2)).  Property 3 [Finite 2e -Gain Performance].

Theorem 1 .

 1 Given positive real numbers  and 1  , assume there exists positive definite matrices 11

  1, the condition (38) implies that the following condition is verified along the trajectories of the closed-loop system (12):

  the well-known Bounded Real Lemma[START_REF] Scherer | Multiobjective output feedback control via LMI optimization[END_REF], the 2e -gain from the admissible disturbance w to the regulated output z will be less than  if:

   amounts to minimize  . Pre-and post-multiplying condition (45) by on the result of Theorem 1, the following LMI optimization problem solves the control design problem stated in Subsection 2.1.2 while maximizing the disturbance rejection level and the estimated domain of attraction of the system (12):

.

  Since the set   P is robustly positively invariant with respect to closed-loop system, it follows that

Figure 2 .Figure 2 .

 22 Figure2. It is noteworthy that this set is not invariant. However, since the ellipsoid   P is robustly invariant for all

Figure 3 .

 3 Figure 3. Closed-loop system trajectories and disturbance attenuation (left); corresponding control signal response and evolution of the controller state of the DOFC (right)

Figure 4 .

 4 Figure 4. Performance variable of Case 1 (solid line) and Case 2 (dashed line)

  Let ij

	 ,   , ij    be symmetric matrices of appropriate dimensions and 1 ,, r rr 
		rr
	be a family of numbers satisfying the convex sum property. Condition	11 ij    i j ij 	0	is satisfied if:

  .

				Figure1. Feasibility spaces obtained with QAWA (+) and NQA (o)
	Example 2. Consider the following nonlinear open-loop unstable system:
	 0.1 0.1 0.12 0.1 w z x u 1 1 21 2 x x x x w             	  23 2 2 2 1.48 0.16 x x x u   		0.1 w	(49)
	2 y x   	0.2	w
	Since the two nonlinearities   22 0.1 0.12 2 f xx 	and   22 1.48 0.16 3 x g x 	are unbounded functions of 2

Acknowledgements

This work was developed in the framework of the CERVIFER project and of the International Campus on Safety and Intermodality in Transportation. This project is supported in part by the European Community (through the FEDER European Funds for Regional Development) and the Nord Pas-de-Calais Region. The authors gratefully acknowledge the support of these institutions.