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

Chouki Sentouh

Jean-Christophe Popieul

Fuzzy Steering Control for Autonomous Vehicles under Actuator Saturation: Design and Experiments

Keywords: Autonomous vehicles, lane keeping control, Takagi-Sugeno fuzzy systems, actuator saturation, fuzzy Lyapunov functions, linear matrix inequality

This paper presents a new control method for autonomous vehicles. The design goal is to perform the automatic lane keeping under multiple system constraints, namely actuator saturation of the steering system, roads with unknown curvature and uncertain lateral wind force. Such system constraints are explicitly taken into account in the control design procedure. To achieve this goal, we propose a new constrained Takagi-Sugeno fuzzy model-based control method using fuzzy Lyapunov control framework. The resulting non-parallel distributed compensation controller is able to handle not only various system constraints but also a large variation range of vehicle speed. In particular, Taylor's approximation method is exploited to reduce not only the numerical complexity for real-time implementation but also the conservatism of the results. The design conditions are strictly expressed in terms of linear matrix inequalities which can be efficiently solved with available numerical solvers. The effectiveness of the proposed control method is demonstrated through both simulation and hardware experiments with various driving scenarios.

Introduction

Nowadays, automobiles have become essential in our society since they provide individuals a great freedom for traveling. At the same time, road accident still remains one of the main mortality causes of our daily life despite huge prevention efforts from governments and automotive industry. As a consequence, the field To be published in the Journal of The Franklin Institute of intelligent vehicles, including the issue of autonomous vehicles, has attracted a growing attention from both academic and industrial settings with the aim of improving safety, comfort, and efficiency [START_REF] Baskar | Traffic control and intelligent vehicle highway systems: A survey[END_REF][START_REF] Rajamani | Vehicle Dynamics and Control[END_REF][START_REF] Li | Development of a new integrated local trajectory planning and tracking control framework for autonomous ground vehicles[END_REF][START_REF] Li | A three-dimensional dynamics control framework of vehicle lateral stability and rollover prevention via active braking with mpc[END_REF][START_REF] Zhang | Vehicle lateral dynamics control through AFS/DYC and robust gain-scheduling approach[END_REF]. Intelligent vehicles make use of sensing and intelligent algorithms to understand the vehicle's immediate environment, for either assisting the driver or fully controlling the vehicle [START_REF] Bishop | Intelligent vehicle applications worldwide[END_REF][START_REF] Nguyen | Driver-automation cooperative approach for shared steering control under multiple system constraints: Design and experiments[END_REF][START_REF] Wang | A gain-scheduling driver assistance trajectory-following algorithm considering different driver steering characteristics[END_REF][START_REF] Wang | Output-feedback robust control for vehicle path tracking considering different human drivers' characteristics[END_REF]. In this context, our research is concerned with the automatic control of the steering system (also known as lateral control) for autonomous vehicles.

Up to now, several lateral controllers have been developed for the lane keeping control problem in the literature [START_REF] Li | Development of a new integrated local trajectory planning and tracking control framework for autonomous ground vehicles[END_REF][START_REF] Li | A three-dimensional dynamics control framework of vehicle lateral stability and rollover prevention via active braking with mpc[END_REF][START_REF] Nguyen | Driver-automation cooperative approach for shared steering control under multiple system constraints: Design and experiments[END_REF][START_REF] Enache | Driver steering assistance for lane-departure avoidance based on hybrid automata and composite Lyapunov function[END_REF][START_REF] Tagne | Design and comparison of robust nonlinear controllers for the lateral dynamics of intelligent vehicles[END_REF]. The authors in [START_REF] Naranjo | Power-steering control architecture for automatic driving[END_REF] have presented an automatic-steering control architecture based on a combination of fuzzy logic and PID control. In that work, the driver actions have been considered as a system disturbance which is systematically rejected by the control system. A switching control scheme based on Lyapunov stability theorem and LMI (linear matrix inequality) optimization has been proposed in [START_REF] Enache | Driver steering assistance for lane-departure avoidance based on hybrid automata and composite Lyapunov function[END_REF] to avoid lane departures when the driver has a lapse of attention. In [START_REF] Cerone | Combined automatic lane-keeping and driver's steering through a 2-DOF control strategy[END_REF], an automatic lane-keeping control is combined with driver's steering for obstacle avoidance and lane-change maneuvers without using switching strategies between these both control actions. A nested PID steering control strategy has been proposed and experimentally validated for an autonomous vehicle in [START_REF] Marino | Nested PID steering control for lane keeping in autonomous vehicles[END_REF] in the case of roads with unknown curvature. A real-vehicle application being able to manage autonomous-steering and perform human-like tracking has been also developed in [START_REF] Sotelo | Virtuous: Vision-based road transportation for unmanned operation on urban-like scenarios[END_REF]. Robust dynamic output feedback controllers based on a driver-vehicle model have been proposed in [START_REF] Wang | A gain-scheduling driver assistance trajectory-following algorithm considering different driver steering characteristics[END_REF][START_REF] Wang | Output-feedback robust control for vehicle path tracking considering different human drivers' characteristics[END_REF] to assist the driver for tracking the reference trajectory. Note that in most of the available works, the longitudinal speed has been considered as a constant to ease the control design. Moreover, existing works have not explicitly taken into account the saturation effects of the steering system in the control design procedure. This can lead to serious degradation of control performance, in many cases, the stability may be lost [START_REF] Nguyen | Static output feedback design for a class of constrained Takagi-Sugeno fuzzy systems[END_REF][START_REF] Nguyen | Anti-windup based dynamic output feedback controller design with performance consideration for constrained Takagi-Sugeno systems[END_REF][START_REF] Zou | Stabilization via extended nonquadratic boundedness for constrained nonlinear systems in Takagi-Sugeno's form[END_REF].

In recent years, stability analysis and 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] have become the most popular research platform in fuzzy model-based control [START_REF] Nguyen | Static output feedback design for a class of constrained Takagi-Sugeno fuzzy systems[END_REF][START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF][START_REF] Xu | Fuzzy logic based fault-tolerant attitude control for nonlinear flexible spacecraft with sampled-data input[END_REF][START_REF] Sadeghi | LMI-based stability analysis and robust controller design for a class of nonlinear chaotic power systems[END_REF][START_REF] Sala | Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya's theorem[END_REF]. This fact is due to many outstanding features of T-S fuzzy models for control purposes [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF]. First, they can be used as a universal approximator for any smooth nonlinear system. In particular, the sector nonlinearity approach provides an exact T-S representation of a given nonlinear model in a compact set. Second, thanks to their polytopic structure with linear systems in the consequent parts, T-S fuzzy models allow to extend some linear control concepts to nonlinear systems. Moreover, T-S fuzzy-model-based control techniques have been successfully applied to various engineering applications [START_REF] Nguyen | Driver-automation cooperative approach for shared steering control under multiple system constraints: Design and experiments[END_REF][START_REF] Sadeghi | LMI-based stability analysis and robust controller design for a class of nonlinear chaotic power systems[END_REF][START_REF] Du | Fuzzy control for nonlinear uncertain electrohydraulic active suspensions with input constraint[END_REF][START_REF] Dahmani | Observer-based state feedback control for vehicle chassis stability in critical situations[END_REF].

In T-S fuzzy control framework, a norm-bounded approach has been used in [START_REF] Du | Fuzzy control for nonlinear uncertain electrohydraulic active suspensions with input constraint[END_REF] to handle the control input limitations. The resulting low-gain non-saturated con-trollers are generally conservative and offer poor control performance [START_REF] Nguyen | Anti-windup based dynamic output feedback controller design with performance consideration for constrained Takagi-Sugeno systems[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]. Polytopic representation of the saturation nonlinearity has been employed 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] Kim | H ∞ state-feedback control for fuzzy systems with input saturation via fuzzy weighting-dependent Lyapunov functions[END_REF][START_REF] Saifia | Fuzzy control for electric power steering system with assist motor current input constraints[END_REF]. Based on the technique of extended non-quadratic boundedness, the authors in [START_REF] Zou | Stabilization via extended nonquadratic boundedness for constrained nonlinear systems in Takagi-Sugeno's form[END_REF] have proposed non-parallel distributed compensation controllers for T-S fuzzy systems subject to input and state constraints and bounded noise. An equivalent augmentation form of the closed-loop system has been exploited together with a generalized saturation sector condition in [START_REF] Nguyen | An augmented system approach for LMI-based control design of constrained Takagi-Sugeno fuzzy systems[END_REF] for the control design of a class of input-constrained Takagi-Sugeno fuzzy systems. In [START_REF] Benzaouia | Stabilization of positive constrained T-S fuzzy systems: Application to a Buck converter[END_REF], the control input limitations have been taken into account in the design procedure using an LP (linear programming) approach.

Motivated by the above control issues, this paper aims at developing a new robust control method for automatic lane keeping of autonomous vehicles subject to multiple system constraints, i.e. actuator saturation, roads with unknown curvature and uncertain lateral wind force. The contributions are summarized as follows.

• Using T-S fuzzy modeling to represent the vehicle dynamics, the proposed automatic lane keeping method can handle a large variation range of vehicle speed. Moreover, Taylor's approximation method is used to reduce significantly the numerical complexity of the vehicle T-S fuzzy model. This eases the real-time control implementation and also reduces the design conservatism.

• The actuator saturation of the steering system is explicitly taken into account in the control design via a generalized sector condition. In particular, a fuzzy Lyapunov function is used for theoretical developments to reduce further the conservatism. The design conditions are expressed in terms of LMIs which can be easily solved with numerical solvers.

• The practical performance of the proposed lane keeping control method is successfully validated through both simulations and hardware experiments.

The paper is organized as follows. Section 2 presents the key elements of vehicle modeling. The transformation from a continuous vehicle model to its corresponding discrete version via Euler's approximation is also given. In Section 3, we first formulate the control problem, then the design conditions are derived in fuzzy Lyapunov control framework. Section 4 highlights the application of the proposed method to the studied autonomous vehicle. Both simulation and hardware experiments to demonstrate the lane keeping performance are presented in Section 5.

Finally, concluding remarks are reported in Section 6.

Notation. For an integer number r, Ω r denotes the set {1, 2, . . . , r}. I denotes the identity matrix of appropriate dimension. For a square matrix X, X > 0 means that X is positive definite. The ith element of a vector u is denoted u (i) and X (i)

denotes the ith row of matrix X. The symbol stands for matrix blocks that can be deduced by symmetry. For a positive definite function V(x) defined on R nx , we denote E V = {x ∈ R nx : V(x) ≤ 1}. The scalar functions η 1 , . . . , η r are said to verify the convex sum property if

η i ≥ 0, r i=1 η i = 1 (1)
For such scalar functions with any argument θ, we denote

Y θ = r i=1 η i (θ)Y i , Y -1 θ = r i=1 η i (θ)Y i -1 (2)
where the matrices Y i are of appropriate dimensions. Throughout this paper, the time argument will be dropped when convenient.

Vehicle Modeling

This section details the modeling of the studied autonomous vehicle.

Road-Vehicle Model

In order to investigate the vehicle motions and to evaluate the control performance, the vehicle handling dynamics in the horizontal plane are represented by the non-linear single track vehicle model [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF][START_REF] Marino | Nested PID steering control for lane keeping in autonomous vehicles[END_REF], see Figure 1. Then, the vehicle dynamics is given by

M ( vx -rv y ) = F xf cos δ -F yf sin δ + F xr M ( vy + rv x ) = F xf sin δ + F yf cos δ + F yr + f w I z ṙ = l f (F xf sin δ + F yf cos δ) -l r F yr + l w f w (3)
For brevity, the vehicle nomenclature used in this work are given in Table 1. The front and rear longitudinal forces F xi with i = f, r and the front and rear lateral forces F yi with i = f, r are modeled according to the Pacejka's tire model [START_REF] Pacejka | Tire and Vehicle Dynamics[END_REF] (known also as magic formula):

F ki (α i ) = D i sin {C i arctan [(1 -E i )B i α i + E i arctan (B i α i )]} (4)
where k = x, y and i = f, r. The Pacejka parameters B i , C i , D i and E i in (4) depend on the characteristics of the tyre, road and the vehicle operating conditions. For lateral control purposes, the nonlinear vehicle model (3) will be simplified. To this end, we consider normal driving situations and small angle approximation [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF]. As a result, the lateral speed and the lateral tire forces can be computed with the following expressions:

v y = v x β F yf = 2C f α f = 2C f v y + l f r v x -δ F yr = 2C r α r = 2C r v y -l r r v x (5)
where C f and C r are respectively the front and rear cornering stiffness coefficients. This work focuses on the lateral motion of the vehicle. Then, the longitudinal dynamics is decoupled from the lateral dynamics and the longitudinal speed is considered as a time-varying parameter for control purposes. Combining this fact with (3)-( 5), the vehicle lateral dynamics is given by

β ṙ = a 11 a 12 a 21 a 22 β r + b 1 b 2 δ + e 1 e 2 f w (6)
where the matrix elements in (6) are defined as follows:

a 11 = - 2(C r + C f ) M v x , a 12 = 2(l r C r -l f C f ) M v 2 x -1, b 1 = 2C f M v x , e 1 = 1 M v x a 21 = 2(l r C r -l f C f ) I z , a 22 = -2(l 2 r C r + l 2 f C f ) I z v x , b 2 = 2l f C f I z , e 2 = l w I z .

Road-Vehicle Positioning

The lane-keeping dynamics can be represented via two supplementary measurements provided by the vision system [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF], namely the lateral deviation error y L from the centerline of the lane projected forward a lookahead distance l s , and the heading error ψ L between the tangent to the road and the vehicle orientation, see Figure 1. Then, the dynamics representing the vehicle positioning on the road is given by ẏL

= v x β + l s r + v x ψ L ψL = r -v x ρ r (7)
where ρ r denotes the unknown road curvature.

Vehicle Control-Based Model

From the bicycle vehicle model ( 6), the dynamics for lane tracking [START_REF] Nguyen | Driver-automation cooperative approach for shared steering control under multiple system constraints: Design and experiments[END_REF], the road-vehicle model used for control purposes can be expressed as follows:

ẋ(t) = A v x(t) + B vu u(t) + B vw w(t) (8)
where x = β r ψ L y L is the vehicle state vector, w = f w ρ r is the disturbance vector, and the control input is the steering angle u = δ. The system matrices of the control-based model ( 8) are given by

A v =     a 11 a 12 0 0 a 21 a 22 0 0 0 1 0 0 v x l s v x 0     , B vu =     b 1 b 2 0 0     , B vw =     e 1 0 e 2 0 0 -v x 0 0     .
For simulation and experiment studies, the following system parameters are considered in this paper:

M = 2025 [kg], l f = 1.3 [m], l r = 1.6 [m], l w = 0.4 [m], l s = 5 [m], η t = 0.13 [m], I z = 2800 [kgm 2 ], C f = 57000 [N/rad], C r = 59000 [N/rad].
To ease the real-time implementation on the vehicle Electronic Control Unit (ECU), the numerical state-feedback controller is directly synthesized in the discrete-time domain. To this end, the well-known Euler's approximation

ẋ(t) ≈ x(κ + 1) -x(κ) T e
is used to obtain the following discrete-time version of the continuous-time vehicle system (8):

Σ v (v x ) : x(κ + 1) = Ax(κ) + B u u(κ) + B w w(κ) (9)
where z (κ) denotes the value of the signal z taken at the κ-instant, T e = 0.01s is the sampling time of the ECU, and the system matrices are given by

A = I + T e A v , B u = T e B vu , B w = T e B vw (10)
The discrete-time vehicle system Σ v in ( 9) is exploited later for control purposes.

Control Design for Input-Saturated Takagi-Sugeno Fuzzy Systems

This section presents the theoretical development of a new control method for disturbed T-S fuzzy systems subject to actuator saturation. The design conservatism of the proposed method compared to existing literature is also studied.

Problem Definition

Consider the discrete-time T-S fuzzy system subject to control input saturation of the following form:

           x(κ + 1) = r i=1 η i (θ) (A i x(κ) + B u i sat(u(κ)) + B w i w(κ)) z(κ) = r i=1 η i (θ)C i x(κ) (11)
where x ∈ R nx is the system state, u ∈ R nu is the control input, w ∈ R nw is the disturbance, z ∈ R nz is the performance output, and θ ∈ R p is the vector of premise variables. The constant matrices A i , B u i , B w i , C i , i ∈ Ω r , are of appropriate dimensions and r is the number of model rules. It is worth noting that the normalized membership functions η i (θ), i ∈ Ω r , satisfy the convex sum property defined in [START_REF] Baskar | Traffic control and intelligent vehicle highway systems: A survey[END_REF]. The standard input saturation is defined by

sat(u (l) ) = sign(u (l) ) min u (l) , u max(l) , l ∈ Ω nu
where the control bounds u max(l) > 0 are given. The disturbance signal w in ( 11) is bounded in amplitude, i.e. it belongs to the following class of function:

W ∞ φ = w : R + → R nw , w(κ) w(κ) ≤ φ, κ ≥ 0
for some φ > 0.

For the control design of the T-S fuzzy system [START_REF] Tagne | Design and comparison of robust nonlinear controllers for the lateral dynamics of intelligent vehicles[END_REF], let us consider the nonparallel distributed compensation (non-PDC) control law of the form

u(κ) = r i=1 η i (θ)G i r i=1 η i (θ)H i -1 x(κ) (12)
where H i , i ∈ Ω r , are nonsingular matrices. Using the notations defined in (2), the closed-loop system can be rewritten from ( 11) and ( 12) as follows:

x(κ + 1) = A θ + B u θ G θ H -1 θ x(κ) -B u θ Ψ(u(κ)) + B w θ w(κ) z(κ) = C θ x(κ) (13)
where

Ψ(u) = u -sat(u).
The control goal is to propose a constructive LMI-based method to design a non-PDC controller of the form [START_REF] Naranjo | Power-steering control architecture for automatic driving[END_REF] such that the closed-loop system (13) satisfies the following properties.

Property 1 (Local internal stability). There exists a positive definite function

V(x(κ)) = x(κ) n i=1
η i (θ)P i x(κ) and its associate set E V such that all closed-loop trajectories starting from E V converge exponentially to the origin in the absence of disturbances, i.e. w = 0.

Property 2 (Input-to-state stability and disturbance attenuation). If w = 0 and w ∈ W ∞ φ for some φ > 0, then all closed-loop trajectories of ( 13) initialized inside E V will be confined in this set. Moreover, there exists a positive scalar γ such that the L ∞ -norm of the performance output signal z is bounded by

z(κ) 2 ∞ ≤ γ, x(0) = 0, ∀κ ≥ 0.

Preliminaries

In the sequel, some useful preliminaries for the control design are presented.

Fact 1. Given positive definite matrix Φ, the following matrix inequality holds for any matrix M of appropriate dimension:

M ΦM ≥ M + M -Φ -1 (14)
Proof. Since Φ > 0, it is clear that

(M -Φ -1 ) Φ(M -Φ -1 ) ≥ 0 (15)
for any matrix M of appropriate dimension. Then, the inequality ( 14) follows directly by developing [START_REF] Sotelo | Virtuous: Vision-based road transportation for unmanned operation on urban-like scenarios[END_REF].

Lemma 1. Consider matrices G i ∈ R nu×nx , H i ∈ R nx×nx and W i ∈ R nu×nx , for i ∈ Ω r
, we define the following set:

P u = x ∈ R nx : G θ H -1 θ -W θ H -1 θ (l) x ≤ u max(l) , l ∈ Ω nu (16)
If x ∈ P u , then the following inequality on the dead-zone nonlinearity Ψ(u), where u is defined in [START_REF] Naranjo | Power-steering control architecture for automatic driving[END_REF]:

Ψ(u) S -1 θ Ψ(u) -W θ H -1 θ x ≤ 0 (17)
holds for any positive diagonal matrices S i ∈ R nu×nu , and for any scalar functions η i (θ), i ∈ Ω r , satisfying the convex sum property.

Proof. If x ∈ P u , then it follows that

-u max(l) ≤ (G θ H -1 θ -W θ H -1 θ ) (l) x ≤ u max(l) , l ∈ Ω nu (18)
In order to prove Lemma 1, we have to show that

Ψ(u (l) ) r i=1 η i S i(l,l) -1 Ψ(u) -W θ H -1 θ x (l) ≤ 0, l ∈ Ω nu ( 19
)
where S i(l,l) , i ∈ Ω r , l ∈ Ω nu , denotes the diagonal element at the lth row and lth column of the matrix S i . To this end, three possible cases are distinguished according to the value of u (l) .

• Case 1:

If -u max(l) ≤ u (l) ≤ u max(l)
, it follows that Ψ(u (l) ) = 0. Therefore, the relation ( 17) holds trivially.

• Case 2: If u (l) > u max(l) , then Ψ(u (l) ) = u (l) -u max(l) = G θ H -1 θ (l) x -u max(l) > 0 (20)
It follows from [START_REF] Zou | Stabilization via extended nonquadratic boundedness for constrained nonlinear systems in Takagi-Sugeno's form[END_REF] that

G θ H -1 θ -W θ H -1 θ (l) x ≤ u max(l) . Hence Ψ(u (l) ) -W θ H -1 θ (l) x = G θ H -1 θ -W θ H -1 θ (l) x -u max(l) ≤ 0 (21)
Since Ψ(u (l) ) > 0 in this case, then the inequality (19) holds.

• Case 3: If u (l) < -u max(l) , then

Ψ(u (l) ) = u (l) + u max(l) = G θ H -1 θ (l) x + u max(l) < 0 (22)
From ( 18), we have that

G θ H -1 θ -W θ H -1 θ (l) x ≥ -u max(l) . Hence Ψ(u (l) ) -W θ H -1 θ (l) x = G θ H -1 θ -W θ H -1 θ (l) x + u max(l) ≥ 0 (23)
Combining the fact that Ψ(u (l) ) < 0 in this case with the inequality [START_REF] Sala | Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya's theorem[END_REF], it follows that (19) holds.

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

Lemma 2. Let Υ k ij , i, j, k ∈ Ω r ,
ν k ω i ω j Υ k ij < 0 holds if    Υ k ii < 0, i, k ∈ Ω r 2 r -1 Υ k ii + Υ k ij + Υ k ji < 0, i, j, k ∈ Ω r , and i = j ( 24
)
Lemma 2 is directly extended from the relaxation result in [32, Theorem 2.2]. Other more efficient relaxation techniques can be found in [START_REF] Sala | Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya's theorem[END_REF] at the expense of high computational costs due to additional slack variables.

LMI-based Controller Computation for Constrained T-S Fuzzy Systems

The following theorem provides conditions to design a non-PDC controller [START_REF] Naranjo | Power-steering control architecture for automatic driving[END_REF] for the input-saturated T-S system [START_REF] Tagne | Design and comparison of robust nonlinear controllers for the lateral dynamics of intelligent vehicles[END_REF].

Theorem 1. Given the T-S fuzzy system [START_REF] Tagne | Design and comparison of robust nonlinear controllers for the lateral dynamics of intelligent vehicles[END_REF] where w ∈ W ∞ φ for some φ > 0 and a positive scalar τ 1 < 1. If there exist positive definite matrices

X i ∈ R nx×nx , positive diagonal matrices S i ∈ R nu×nu , matrices H i ∈ R nx×nx , G i ∈ R nu×nx , W i ∈ R nu×nx and positive scalars γ, τ 2 such that H i + H i -X i G i(l) -W i(l) u 2 max(l) > 0, i ∈ Ω r , l ∈ Ω nu ( 25
)
τ 1 -τ 2 φ > 0 (26)
H i + H i -X i C j H i γI ≥ 0, i, j ∈ Ω r (27)
Φ k ii < 0, i, k ∈ Ω r (28) 2 r -1 Φ k ii + Φ k ij + Φ k ji < 0, i, j, k ∈ Ω r , and i = j (29)
where

Φ k ij =     (τ 1 -1)(H i + H i -X i ) W i -2S i 0 0 -τ 2 I A j H i + B u j G i -B u j S i B w j -X k     (30)
Then, the non-PDC controller [START_REF] Naranjo | Power-steering control architecture for automatic driving[END_REF] solves the control problem in Section 3.1.

Proof. Note that if [START_REF] Dahmani | Observer-based state feedback control for vehicle chassis stability in critical situations[END_REF] is verified, then it follows that

H θ + H θ -X θ > 0.
Then, the weighting matrix H θ is nonsingular since X θ > 0. This guarantees the existence of the inverse matrix H -1 θ . By the relaxation result in Lemma 2 with

ν k = η k (θ(κ + 1)), ω i = η i (θ(κ)), i, k ∈ Ω r ,
we can deduce from ( 28)- [START_REF] Nguyen | An augmented system approach for LMI-based control design of constrained Takagi-Sugeno fuzzy systems[END_REF] with

Φ k ij defined in (30) that     (τ 1 -1)(H θ + H θ -X θ ) W θ -2S θ 0 0 -τ 2 I A θ H θ + B u θ G θ -B u θ S θ B w θ -X θ+     < 0 (31)
Using the matrix property ( 14) of Fact 1 with M = H θ and Φ = X -1 θ = P θ , the inequality [START_REF] Pacejka | Tire and Vehicle Dynamics[END_REF] implies clearly that

    (τ 1 -1)H θ P θ H θ W θ -2S θ 0 0 -τ 2 I A θ H θ + B u θ G θ -B u θ S θ B w θ -X θ+     < 0 (32)
Pre-and post-multiplying [START_REF] Tuan | Parameterized linear matrix inequality techniques in fuzzy control system design[END_REF] with diag H -1 θ , S -1 θ , I, I yields    and P θ+ = X -1 θ+ . Pre-and postmultiplying (34) by the vector x(κ) Ψ(u(κ)) w(κ) and its transpose leads to the following inequality after some simple manipulations:

V(x(κ + 1)) + (τ 1 -1)V(x(κ)) -τ 2 w(κ) w(κ) -2Ψ(u) S -1 θ Ψ(u) -W θ H -1 θ x(κ) < 0 (35)
where the positive definite function V(x(κ)) is defined as follows:

V(x(κ)) = x(κ) n i=1 η i (θ)P i x(κ) = x(κ) P θ x(κ) (36)
Using Schur complement lemma and the matrix property in ( 14), it can be deduced from ( 25) that

H θ P θ H θ - G θ(l) -W θ(l) G θ(l) -W θ(l) u 2 max(l) ≥ 0 (37)
Pre-and post-multiplying (37) with H - θ , then it is easily proved that (37) implies E V ⊆ P u . Since E V ⊆ P u , by Lemma 1 it follows from (35) that

V(x(κ + 1)) + (τ 1 -1)V(x(κ)) -τ 2 w(κ) w(κ) < 0, ∀x ∈ E V (38)
• If w = 0, it can be deduced from (38) that

∆V < -τ 1 V(x(κ)), ∀x ∈ E V (39)
which means that all closed-loop trajectories stating from the set E V converge asymptotically to the origin with a decay rate less than τ 1 /2.

• If w = 0 and w ∈ W ∞ φ , the satisfaction of ( 26) and (38) implies that

∆V + τ 1 (V(x(κ)) -1) + τ 2 φ -w(κ) w(κ) < 0, ∀x ∈ E V (40)
The condition (40), in turn, guarantees that the set E V is robustly positively invariant [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF] with respect to the closed-loop system [START_REF] Cerone | Combined automatic lane-keeping and driver's steering through a 2-DOF control strategy[END_REF]. Moreover, it follows from the condition ( 27) that

P θ C θ C θ γI ≥ 0 (41)
Applying Schur complement lemma to (41), we can prove that

z z = x C θ C θ x ≤ γx P θ x ≤ γ, ∀x ∈ E V (42)
which means that the L ∞ -norm of the output signal z is bounded:

z 2 ∞ ≤ γ.
The proof of Theorem 1 can be now concluded.

Remark 1. The design conditions in Theorem 1 are based on the choice of the fuzzy Lyapunov function [START_REF] Guerra | LMI-based relaxed non-quadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form[END_REF]. This type of Lyapunov functions allows reducing effectively the design conservatism compared to the common quadratic Lyapunov function V(x(κ)) = x(κ) P x(κ). Indeed, the latter is simply a special case of (36) by imposing P i = P , i ∈ Ω r . Note that the proposed method can be easily generalized by using more complex fuzzy Lyapunov functions [START_REF] Nguyen | Static output feedback design for a class of constrained Takagi-Sugeno fuzzy systems[END_REF] to reduce further the conservatism at the expense of computational cost.

Remark 2. The design conditions presented in Theorem 1 are strictly expressed in terms of linear matrix inequalities which can be effectively solved with available numerical solvers [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. In this work, the feedback gains G i , H i , i ∈ Ω r in ( 12) are computed with SeDuMi solver and YALMIP toolbox [START_REF] Löfberg | YALMIP: A toolbox for modeling and optimization in MATLAB[END_REF].

Remark 3. The decay rate τ 1 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 signals are directly involved in the system dynamics, for example the lateral wind force and the road curvature in the case of the road-vehicle system (9).

Comparison Study between Different Control Methods

By means of an academic example, we study here the design conservatism and the numerical complexity of the proposed method compared to the work in [START_REF] Zou | Stabilization via extended nonquadratic boundedness for constrained nonlinear systems in Takagi-Sugeno's form[END_REF] dealing with a similar control problem, i.e. constrained T-S fuzzy systems subject to actuator saturation and L ∞ disturbances.

Example 1. Let us consider the constrained T-S fuzzy system [START_REF] Tagne | Design and comparison of robust nonlinear controllers for the lateral dynamics of intelligent vehicles[END_REF] with the following system data [START_REF] Zou | Stabilization via extended nonquadratic boundedness for constrained nonlinear systems in Takagi-Sugeno's form[END_REF][START_REF] Guerra | LMI-based relaxed non-quadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form[END_REF]:

A 1 = 1 -β -1 -0.5 , B u 1 = 5 + β 2β , B w 1 = β/2 0 , C 1 = 1 0 A 2 = 1 β -1 -0.5 , B u 2 = 5 -β -2β , B w 2 = -β/2 0 , C 2 = 1 0 (43)
where β > 0 and u max = 1. It is also assumed that the T-S fuzzy system (43) is subject to the amplitude-bounded disturbance w(κ) = 0.5 sin(κ).

Note that the computational complexity of an LMI optimization problem can be estimated as being proportional to N 3 var N row , where N var is the total number of scalar decision variables and N row the total row size of the LMIs. These numbers for two different control methods are given as follows.

• For the design conditions of Theorem 1:

N row = 2 + rn u (n x + 1) + r 3 (2n x + n u + n w + n z ) N var = 2 + rn x (n x + 2n u ) + n x (n x + 1)/2 + n u (n u + 1)/2
(44)

• For the design conditions of Theorem 2 in [START_REF] Zou | Stabilization via extended nonquadratic boundedness for constrained nonlinear systems in Takagi-Sugeno's form[END_REF]:

N row =r 2 (2n x + n w )(2 + 3r(r -1)/2 + (r(r -1)/2) 2 ) + 2r(3n x + n u + n w ) + r 2 (r -1)(3n x + n u + n w )/2 N var =1 + n x (n x + 1)/2 + n u (n u + 1)/2 + rn x (n x + n u ) + r 2 (r 2 + (n x + n u ) 2 + n x (n x + 1)/2 + (2n x + n w ) 2 ) + r 4 (2n x + n w ) 2 (45)
Table 2 shows the maximal β, denoted by β * , for which a stabilizing controller can be designed from two different methods and also the numbers characterizing the complexity of these methods. Observe that the new method provides not only less conservative results but also much lower computational cost. Compared to (44), it can be seen from ( 45) that the computational complexity of the method in [START_REF] Zou | Stabilization via extended nonquadratic boundedness for constrained nonlinear systems in Takagi-Sugeno's form[END_REF] becomes excessively important with high dimensional T-S fuzzy systems and/or T-S fuzzy systems with important number of linear subsystems. This represents a major advantage of the proposed control method for real-world applications.

Automatic Steering Control of Autonomous Vehicle

This section first presents the application of Theorem 1 to the lane keeping control of the autonomous vehicle described in Section 2. Then, simulation results are given to show the effectiveness of the new control method.

T-S Fuzzy Modeling for Vehicle System

For the control design, let us define the performance output of the vehicle system [START_REF] Wang | Output-feedback robust control for vehicle path tracking considering different human drivers' characteristics[END_REF]. This variable should represent both lane keeping and driving comfort

z = a y ψ L y L (46)
Note that the lane keeping performance is represented by the heading error ψ L and the lateral deviation error y L . The driving comfort is represented by the lateral acceleration a y ∼ = v x r. Note also that all components of z can be expressed by those of the state x in (9) as follows:

z = Cx =   0 v x 0 0 0 0 1 0 0 0 0 1   x (47)
We note also that the matrices A, B w in ( 9)- [START_REF] Enache | Driver steering assistance for lane-departure avoidance based on hybrid automata and composite Lyapunov function[END_REF] and C in (47) depend nonlinearly on the vehicle speed which is measured and bounded

v x , 1/v x , 1/v 2 x , v min ≤ v x ≤ v max (
* = v x 1/v x 1/v 2 x ∈ R 3 .
Using the sector nonlinearity approach [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF], this choice leads to an exact T-S fuzzy model of (9) with 2 3 = 8 linear subsystems. However, this vehicle T-S model would be too expensive in terms of numerical computation for control design, and especially for real-time implementation. In this work, we make use of the well-known Taylor's approximation (first order) to exploit the strong relationship between v x , 1/v x and 1/v 2

x . Therefore, the numerical complexity of the proposed control method can be significantly reduced. Concretely, one has

1 v x = 1 v 0 + 1 v 1 ∆ x , v x ∼ = v 0 1 - v 0 v 1 ∆ x , 1 v 2 x ∼ = 1 v 2 0 1 + 2 v 0 v 1 ∆ x ∆ min ≤ ∆ x ≤ ∆ max , ∆ min = -1, ∆ max = 1 (49)
where the measured parameter ∆ x is used to describe the variation of v x between its lower and upper bounds. The two constants v 0 and v 1 in (49) are given by

v 0 = 2v min v max v min + v max , v 1 = 2v min v max v min -v max .
Replacing (49) into ( 9), we obtain a vehicle model Σ v (∆ x ) where the matrices A and B w in ( 10) depend now exclusively on the time-varying parameter ∆ x . Choosing θ = ∆ x ∈ R as new premise variable and using the sector nonlinearity approach, the T-S fuzzy model ( 11) of this vehicle model has only 2 linear subsystems whose matrices are given by

Σ v1 : A (∆ min ) , B u (∆ min ) , B w (∆ min ) Σ v2 : A (∆ max ) , B u (∆ max ) , B w (∆ max ) (50)
where the expressions of A, B u and B w are given in [START_REF] Enache | Driver steering assistance for lane-departure avoidance based on hybrid automata and composite Lyapunov function[END_REF]. The two corresponding membership functions of this T-S fuzzy model are defined as follows:

η 1 (∆ x ) = 1 -∆ x 2 , η 2 (∆ x ) = 1 -η 1 (∆ x ) .
As stated in Section 1, the actuator saturation of the steering system should be explicitly taken into account in the design procedure to prevent the loss of closedloop stability during some specific driving scenarios. Here, the constraint on the control input is u max = 10 [deg]. This input constraint is the limitation of the steering angle imposed to the studied autonomous vehicle. We can now design the vehicle steering control actions by solving the LMI conditions of Theorem 1.

Some obtained numerical results are given as follows:

G 1 = -0.025 -0.387 -0.027 -0.080 , G 2 = -0.023 -0.491 -0.022 -0.054 , Note that two Lyapunov matrices are significantly different which justifies a posteriori the interest of the non-quadratic Lyapunov function [START_REF] Guerra | LMI-based relaxed non-quadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form[END_REF] for the proposed T-S fuzzy control method.

H 1 =     0.

Simulation Results

For the following numerical simulations, the designed non-PDC controller is tested with the road-vehicle model presented in Section 2.

Scenario 1: Control input saturation

We assume that the vehicle system is not well initialized where its initial state x 0 = 0 0 0.25 0.5 does not correspond to the system origin, i.e. the lane centreline. The vehicle responses in this case are indicated in Figure 2. It can be clearly observed that despite an important level of actuator saturation at the beginning of the simulation, all vehicle state variables converge to the origin.

Note that the non-PDC controller proposed in [START_REF] Guerra | LMI-based relaxed non-quadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form[END_REF] cannot provide a stable closed-loop behaviors for the studied autonomous vehicle under the same simulation conditions as shown in Figure 3. Indeed, since the input constraint was not explicitly considered in the control procedure with the design conditions in [START_REF] Guerra | LMI-based relaxed non-quadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form[END_REF], therefore the obtained fuzzy controller cannot provide any guarantee on the stability of the vehicle in case of important actuator saturation. For the following driving test scenarios, the vehicle will be initialized at the lane centreline.

Scenario 2: Lane keeping performance

This scenario aims to show the lane-keeping performance of the proposed controller in the case where the road is composed by several sections with different levels of curvature. To this end, we use the digital database of the Satory test track near Paris, France [START_REF] Nguyen | Sensor reduction for driverautomation shared steering control via an adaptive authority allocation strategy[END_REF] for simulation purposes, see Figure 4 (left). The corresponding road curvature and vehicle speed are respectively depicted in Figure 4 (right). As can be observed, the vehicle speed for this test strongly varies within its range v x ∈ [START_REF] Wang | A gain-scheduling driver assistance trajectory-following algorithm considering different driver steering characteristics[END_REF][START_REF] Benzaouia | Stabilization of positive constrained T-S fuzzy systems: Application to a Buck converter[END_REF]. This clearly justifies the interest of the proposed T-S fuzzy model-based control method. Observe in Figure 4 (left) that the proposed T-S fuzzy controller is able to guarantee a good lane keeping performance for the studied autonomous vehicle. This is also confirmed by the fact that the variables representing lane keeping errors are relatively small and strictly remain within their practical domain of variation during the whole simulation test as indicated in Figure 5.

Experimental Results

To further examine the practical performance of the designed controller, a series of experiments are implemented on the advanced SHERPA dynamic simulator, see Figure 6. This simulator is in the form of a Peugeot 206 vehicle fixed on a Stewart platform, the whole is positioned in front of five flat panel displays providing a visual field of 240 • . Based on a distributed computing architecture, this complex simulator is structured around a SCANeR network connecting fifteen PC-type workstations. The whole software of the SHERPA simulator is developed

Scenario 3: Rejection of wind disturbance

For this scenario, the vehicle speed is fixed at v x = 15 [m/s] and the vehicle dynamics is affected by an important lateral wind force for a duration of 5s as shown in Figure 7. The vehicle variables are therefore perturbed. However, the proposed T-S fuzzy controller is able to reject effectively the disturbance effect and all states converge to the origin at the end of the test scenario. Moreover, we can see also from Figure 7 that state variables representing the vehicle performance, namely sideslip angle β, vehicle yaw rate r, heading error ψ L , and lateral deviation error y L are very small under the effect of the strong wind force. This also illustrates the performance of the proposed controller.

Scenario 4: Vehicle stability performance with ISO chicane test

The normalized chicane test is widely employed to verify the stability performance of autonomous vehicles in an extreme driving situation. This corresponds to an obstacle avoidance scenario as indicated in Figure 8. For this experiment, the vehicle speed is fixed as v x = 10 [m/s]. The lateral trajectory performed with the proposed non-PDC controller and the steering control signal corresponds to this scenario are respectively shown in Figures 9 (a 11 that the proposed non-PDC controller guarantees a good control performance for the whole test with small lane keeping errors. In particular, for the first four curves although the vehicle speed is different between Scenarios 2 and 5, the vehicle responses (steering angle and vehicle variables) obtained for both cases are rather similar. This fact demonstrates not only the advantage of considering the speed variation into the control design procedure but also the strong usefulness of numerical simulations.

Concluding Remarks

A new LMI-based control method for the automatic lane keeping of autonomous vehicles subject to actuator saturation has been proposed. The vehicle system is approximated by means of T-S fuzzy modeling to deal with a large variation range of vehicle speed. This method relies on the use of a fuzzy Lyapunov function to reduce the conservatism of the results. Moreover, two specific realizations have been proposed to ease the real-time control implementation: (1) Taylor's approximation method is used to reduce the model complexity, (2) the control design of an input-saturated non-PDC controller has been directly synthesized in discrete-time domain. Extensive validation with both numerical simulations and experimental tests has been carried out to verify the practical performance of the proposed control method. Future works focus on exploiting the proposed design method for the control issues of semi-autonomous vehicles. To this end, the human driver behaviors should be integrated into the control design procedure [START_REF] Nguyen | Sensor reduction for driverautomation shared steering control via an adaptive authority allocation strategy[END_REF].

Figure 1 :

 1 Figure 1: Single track vehicle model.

  be symmetric matrices of appropriate dimensions and {ν k } k∈Ωr , {ω i } i∈Ωr , be any family of scalar functions satisfying the property of convex sum. The condition

  48) where v min = 8 [m/s] and v max = 30 [m/s]. Hence, a natural choice of premise variables to derive the T-S fuzzy representation (11) of vehicle model Σ v would be θ

Figure 2 :

 2 Figure 2: Vehicle responses in the presence of actuator saturation with the proposed method.

Figure 3 :

 3 Figure 3: Vehicle responses obtained with the control method in [36] for the test in Scenario 1.

Figure 4 :

 4 Figure 4: Vehicle trajectory performed by the automatic lane-keeping controller (left). Digital database corresponding to the test track of Scenario 2 (right).

Figure 5 :

 5 Figure 5: Simulation results of Scenario 2: steering control actions and vehicle responses.

with

  RTMaps environment composed by several modules which are in charge of different tasks: perception, planning, human-machine interface.

Figure 6 :

 6 Figure 6: SHERPA interactive dynamic driving simulator (left). Data acquisition system (right).

Figure 7 :

 7 Figure 7: Control performance in terms of lateral wind force attenuation.

  ) and (d). As indicated in Figures9 (b), (c), (e) and (f), the studied autonomous vehicle is perfectly able to perform the

Figure 8 :

 8 Figure 8: ISO chicane test for vehicle stability evaluation.

Figure 9 :

 9 Figure 9: Lateral trajectory performed with the proposed non-PDC controller (a); steering control angle (d); response of the autonomous vehicle (b), (c), (e) and (f).

5. 3 .

 3 Scenario 5: Lane keeping with the complete Satory test track This experiment aims to show the lane keeping performance of the proposed T-S fuzzy controller during the whole Satory test track, see Figure 10 (a). Figures 10 (b), (c), and (d) show respectively the road curvature of Satory test track, the vehicle speed for this test, and the designed steering control signal. Despite a large variation of vehicle speed, it can be observed from the vehicle variables representing the lane keeping performance in Figure

Figure 10 :

 10 Figure 10: Satory track and lane keeping test condition.

Figure 11 :

 11 Figure 11: Automatic lane keeping performance during the whole Satory test track.

Table 1 :

 1 Vehicle model parameters

	Symbol Description
	v x,y	longitudinal/lateral speed
	β	sideslip angle at the center of gravity (CG)
	r	vehicle yaw rate
	f w	lateral wind force
	ρ r	road curvature
	δ	steering angle
	M	total mass of the vehicle
	l f	distance from CG to the front axle
	l r	distance from CG to the rear axle
	l w	distance from CG to the impact center of the wind force
	l s	look-ahead distance
	η t	tire length contact
	I z	vehicle yaw moment of inertia
	C f	front cornering stiffness
	C r	rear cornering stiffness

Table 2 :

 2 Comparison between different design methods

	Design conditions Theorem 1 Theorem 2 in [18]
	β *	1.68	1.55
	N row	56	168
	N var	25	581

Acknowledgement

This work has been done in the framework of the CoCoVeA research program (ANR-13-TDMO-0005), funded by the National Research Agency. This work was also sponsored by the International Campus on Safety and Intermodality in Transportation, the Nord-Pas-de-Calais Region, the European Community, the Regional Delegation for Research and Technology, the Ministry of Higher Education and Research, and the French National Center for Scientific Research.


Anh-Tu Nguyen

Michio Sugeno

Victor Campos

Michel Dambrine

LMI-based Stability Analysis for Piecewise Multi-Affine Systems

Keywords: Piecewise multi-affine systems, fuzzy systems, singleton consequents, piecewise Lyapunov functions, stability analysis, linear matrix inequalities (LMIs)

This paper provides a computational method to study the asymptotic stability of piecewise multi-affine systems. Such systems stem from a class of fuzzy systems with singleton consequents and can be used to approximate any smooth nonlinear system with arbitrary accuracy. Based on the choice of piecewise Lyapunov functions, stability conditions are expressed as a feasibility test of a convex optimization with linear matrix inequality constraints. The basic idea behind these conditions is to exploit the parametric expressions of piecewise multi-affine systems by means of Finsler's lemma. Numerical examples are given to point out the effectiveness of the proposed method.

I. INTRODUCTION

Fuzzy logic is a powerful tool for the control of complex systems [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF], [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF]. The success of fuzzy control systems comes from their linguistic knowledge representation and their powerful ability to cope with very general nonlinearities [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF]. Moreover, experimental results have illustrated their remarkable robustness properties [START_REF] Braae | Theoretical and linguistic aspects of the fuzzy logic controller[END_REF]. Since Mamdani's pioneering works [START_REF] Mamdani | An experiment in linguistic synthesis with a fuzzy logic controller[END_REF], various fuzzy systems have been proposed in the literature which are categorized into three following types according to the consequent parts of IF-THEN rules [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF].

• Mamdani fuzzy systems are with fuzzy variables in both premises and consequences so that they can be linguistically understandable [START_REF] Mamdani | An experiment in linguistic synthesis with a fuzzy logic controller[END_REF]. • Mamdani-like fuzzy systems [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF] are with singletons in consequent parts. Here, these fuzzy systems are called piecewise multi-affine (PMA) systems as explained later. • For Takagi-Sugeno (T-S) fuzzy systems, the consequences are functions (linear/affine in most cases) of premise variables. T-S fuzzy modeling has been well discussed in [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF], [START_REF] Taniguchi | Model construction, rule reduction, and robust compensation for generalized form of Takagi-Sugeno fuzzy systems[END_REF]. Although Mamdani-type systems are the first fuzzy models used for control purposes and the most meaningful to human beings, they have still received serious criticisms from the control community. This fact is mainly due to the lack of a systematic and rigorous method for studying their stability issues [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF]. Indeed, establishing such a method is expected to be very hard because of the novelty of the fuzzy mathematics involved, and the fuzzy language [START_REF] Braae | Theoretical and linguistic aspects of the fuzzy logic controller[END_REF]. The situation has been A.-T. Nguyen and M. Dambrine are with the CNRS laboratory LAMIH UMR 8201, Valenciennes, France. M. Sugeno is with the Tokyo Institute of Technology, Japan. Victor Campos is with the Department of Electrical Engineering, Federal University of Ouro Preto, João Monlevade, Brazil.

E-mail addresses: {nguyen.trananhtu; michio.sugeno}@gmail.com, victor@cpdee.ufmg.br, michel.dambrine@univ-valenciennes.fr. greatly improved with T-S fuzzy systems whose key feature consists in their polytopic structure with linear systems [START_REF] Taniguchi | Model construction, rule reduction, and robust compensation for generalized form of Takagi-Sugeno fuzzy systems[END_REF], [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] or affine systems [START_REF] Johansson | Piecewise quadratic stability of fuzzy systems[END_REF], [START_REF] Wang | H∞ controller design for affine fuzzy systems based on piecewise Lyapunov functions in finite-frequency domain[END_REF] in consequent parts. This feature allows to extend some linear control concepts to deal with nonlinear systems [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. Indeed, tremendous investigations have been devoted to the studies of T-S systems such that T-S model-based stability analysis and control design have become the most popular research platform in model-based fuzzy control [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF]. However, T-S fuzzy systems are not meaningful to human beings and do not allow for an incorporation of the qualitative expertise of human operators into the control design procedure [START_REF] Kim | A new computational approach to stability analysis and synthesis of linguistic fuzzy control system[END_REF]. Notice that until now most of engineering applications are still based on Mamdani fuzzy models. In [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF], [START_REF] Sugeno | On improvement of stability conditions for continuous Mamdani-like fuzzy systems[END_REF], several outstanding features of PMA systems have been highlighted. First, PMA models can be straightforwardly obtained from mathematical expression of a given nonlinear system. They are also easily obtained from input/output data using the identification method developed in [START_REF] Sugeno | A fuzzy-logic-based approach to qualitative modeling[END_REF]. Second, these systems can be naturally implemented with look-up tables which are intensively employed in industry as a powerful tool for dynamic model approximation and controller implementation. Third, PMA systems have universal approximation capability for any smooth nonlinear function. Recently, this outstanding feature has been effectively exploited in [START_REF] Tong | Fuzzy adaptive output feedback control of MIMO nonlinear systems with partial tracking errors constrained[END_REF] (respectively in [START_REF] Li | Observer-based adaptive fuzzy tracking control of MIMO stochastic nonlinear systems with unknown control directions and unknown dead zones[END_REF]) to identify unknown nonlinear functions for the design of an adaptive fuzzy backstepping outputfeedback tracking controller of nonlinear systems (respectively stochastic nonlinear systems). Finally, both state rate vector and state vector itself of PMA systems can be expressed in a specific form, called parametric expressions [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF]. This representation of PMA systems allows a simple and efficient mathematical analysis. Up to now, the simplicity and the efficiency of analysis are less obvious; however this paper aims to demonstrate this fact. These advantages make PMA systems particularly interesting when dealing with nonlinear systems.

Despite these great advantages, only very few theoretical works regarding PMA systems are available in the literature. Using the knowledge on the membership functions (MFs), quadratic approaches for stability analysis of PMA systems were first presented in [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF] with only sufficient conditions and then improved in [START_REF] Sugeno | On improvement of stability conditions for continuous Mamdani-like fuzzy systems[END_REF] with necessary and sufficient conditions. Sufficient stability conditions regardless of the knowledge on the MFs were suggested in [START_REF] Kim | A new computational approach to stability analysis and synthesis of linguistic fuzzy control system[END_REF] (respectively [START_REF] Kim | A new approach to numerical stability analysis of fuzzy control systems[END_REF]) with quadratic (respectively piecewise quadratic) Lyapunov functions. Although these works offer some guidelines to study rigorously the stability of PMA systems, it should be stressed that they suffer two serious drawbacks. First, the proposed stability conditions are expressed in terms of bilinear matrix inequalities which are hardly tractable with available numerical solvers. Second and more importantly, these conditions are only applicable to second-order systems. We note that statespace representation has been exclusively employed to study the stability of PMA systems in all previous works. However, such a representation is hard to be obtained for higher order PMA systems and not convenient for theoretical developments [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF]. Therefore, extensions to high dimensional systems of existing approaches are not obvious. These drawbacks make the use of PMA modeling for studying nonlinear systems too restrictive and, without a doubt, have prevented PMA systems to be commonly used in control theory. This paper aims at developing a novel stability framework for PMA systems which can overcome the aforementioned drawbacks. The main contributions can be summarized as follows.

1) For the first time in the open literature, the specific parametric expressions of PMA systems [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF] will be judiciously exploited via Finsler's lemma [START_REF] Skelton | A Unified Algebraic Approach to Linear Control Design[END_REF] to derive the stability conditions. We note that differently from the state-space representation, these expressions are straightforwardly obtained for high-order systems. Therefore, the proposed results can be applied to PMA systems of any order which is not the case of all previous works [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF], [START_REF] Kim | A new computational approach to stability analysis and synthesis of linguistic fuzzy control system[END_REF], [START_REF] Sugeno | On improvement of stability conditions for continuous Mamdani-like fuzzy systems[END_REF], [START_REF] Kim | A new approach to numerical stability analysis of fuzzy control systems[END_REF]. 2) The proposed method relies on the use of a piecewise Lyapunov function and the S-procedure [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] to take fully advantage of the information on piecewise statespace partition associated to PMA systems. This allows to effectively reduce the conservatism of the results.

3) The proposed stability conditions are strictly expressed in terms of linear matrix inequalities (LMIs) [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. Then, stability analysis of PMA systems amounts to solving a set of LMIs which can be easily done with available numerical solvers.

We wish that these outstanding contributions of the proposed results will contribute to promote the research on PMA systems to the control community.

Notice that general nonlinear systems can be also approximated by piecewise linear (PL) models [START_REF] Johansson | Piecewise Linear Control Systems: A Computational Approach[END_REF] with any given accuracy. However, there are significant differences between PL and PMA systems. For each piecewise region, PMA models can be equivalently expressed as a polytope of affine systems [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF] whereas PL model is defined by an affine system. Especially, by construction PMA systems are inherently continuous over piecewise regions and they are structurally simpler because all piecewise regions are partitioned in the same fashion, i.e. hyper-rectangle. Therefore, PMA modeling is more attractive for approximating nonlinear systems since rectangular grids are easier to work with, especially for large dimensional systems. Actually, PL modeling has been mostly used as a powerful tool for a class of hybrid systems [START_REF] Johansson | Piecewise Linear Control Systems: A Computational Approach[END_REF]. Notice also that piecewise linear differential inclusions have been used to model an important class of affine Takagi-Sugeno fuzzy systems in [START_REF] Johansson | Piecewise quadratic stability of fuzzy systems[END_REF], [START_REF] Wang | H∞ controller design for affine fuzzy systems based on piecewise Lyapunov functions in finite-frequency domain[END_REF], [START_REF] Johansson | Piecewise Linear Control Systems: A Computational Approach[END_REF].

The paper is organized as follows. The description of PMA systems and useful preliminaries are presented in Section II. Section III is devoted to the theoretical developments of the new stability conditions for PMA systems. Numerical examples are given in Section IV to illustrate the effectiveness of the proposed method. Finally, Section V provides some concluding remarks. Notation. For an integer number n, Ω n represents the subset {1, 2, . . . , n}. For a vector x ∈ R n and i ∈ Ω n , x (i) denotes the ith entry of x. For a square matrix X, X denotes its transpose, and He (X) = X + X . For a symmetric matrix X, X > 0 means it is positive definite, and X 0 denotes the matrix with all nonnegative entries. We denote I n as the identity matrix of dimension n and 0 n×m as the n × m zero matrix. Throughout the paper, the explicit time-dependence of the variables is omitted when convenient.

II. SYSTEM DESCRIPTION AND PRELIMINARIES

In this section, the procedure to derive the parametric expressions of a PMA model from a given dynamical system is reviewed, see [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF] for more details. Then, some useful preliminaries for the theoretical developments are also presented.

A. Description of Piecewise Multi-Affine Systems

Let us consider a system described by the following statespace representation

ẋ = f (x) (1)
where x ∈ R n is the state vector and the vector field f (•) is sufficiently smooth. As common in practice, the state variables of (1) are assumed to be limited in amplitude for physical and/or safety reasons, i.e. x i,min ≤ x (i) ≤ x i,max , i ∈ Ω n , where x i,min and x i,max denote respectively the minimal and maximal bounds of the ith entry of vector x.

As a consequence, the state vector x belongs to the set

R = [x 1,min , x 1,max ] × . . . × [x n,min , x n,max ].
In order to approximate system (1) by means of PMA modeling, its state space is partitioned as follows

x j,min = χ j,1 < χ j,2 < . . . < χ j,Nj +1 = x j,max (2)
for j ∈ Ω n . Multi-index notations are used to represent both elementary regions and vertices of each partition. Let

K v = Ω N1+1 × . . . × Ω Nn+1
be the set of multi-indexes corresponding to all the vertices induced by the partition (2) and K r = Ω N1 × . . . × Ω Nn the set of multi-indexes corresponding to the regions.

For i = (i 1 , . . . , i n ) ∈ K r , the region [χ 1,i1 , χ 1,i1+1 ] × . . . × χ n,in , χ n,in+1 is denoted by R i and K i = {i 1 , i 1 + 1} × . . . × {i n , i n + 1} is the set of multi-indexes corresponding to all vertices of R i . For k ∈ K i , χ k is the vertex of R i whose jth component is defined by χ k(j) = χ j,kj for j ∈ Ω n . Finally, we denote f k = f (χ k ).
Using these notations, the PMA model derived from system (1) valid on R can be expressed as follows

       ẋ = k∈Kv η k (x) f k x = k∈Kv η k (x) χ k (3)
where η k (x) = n j=1 η j,kj x (j) and the membership functions

η j,kj x (j) , j ∈ Ω n , k j ∈ Ω Nj +1 , are defined by η j,kj x (j) =          x (j) -χ j,k j -1 χ j,k j -χ j,k j -1 , if x (j) ∈ χ j,k j -1 , χ j,k j and j ≥ 2 χ j,k j +1 -x (j) χ j,k j +1 -χ j,k j , if x (j) ∈ χ j,k j , χ j,k j +1 and j ≤ Nj 0, otherwise (4)
On a region R i , the expressions in (3) can be simplified to

       ẋ = k∈Ki η k (x) f k x = k∈Ki η k (x) χ k (5)
and the membership functions (4) satisfy the following properties [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF]:

       η j,kj x (j) ≥ 0, ij +1 kj =ij η j,kj x (j) = 1, j ∈ Ω n k∈Ki η k (x) = 1 (6)
Note that system (1) can be approximated by PMA model (3) with arbitrary accuracy on R by increasing the number of partition points in [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF]. In this paper, we exclusively focus on the stability analysis of PMA system (3) for a given state-space partition. For engineering applications, such a partition is often carried out with experiments, for example in automotive and aerospace industries [START_REF] Bohn | An optimization-based approach for the calibration of lookup tables in electronic engine control[END_REF].

Remark 1. When n = 2, PMA model (3) with membership functions (4) corresponds to the parametric expressions presented in [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF]. This model is associated to a class of fuzzy systems whose rules are defined by singleton consequents.

Here, the term piecewise multi-affine system is originated from the fact that both x and ẋ in each region R i can be expressed on the multi-affine form (5).

Remark 2. The choice of triangular membership functions (4) for PMA systems seems restrictive since several other types of MFs are available in the literature. However, this type of MFs allows conveniently deriving canonical forms, namely parametric expression and also state-space representation, of PMA systems [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF]. In addition, this is the most employed in fuzzy control systems and also in practice. The advantages of triangular MFs are discussed in detail in [START_REF] Pedrycz | Why triangular membership functions?[END_REF].

The following notations are introduced for convenience:

x = x 1 , ẋ = ẋ 0 , fk = f k 0 , χk = χ k 1 X i = k∈Ki η k (x) χ k , Xi = k∈Ki η k (x) χk F i = k∈Ki η k (x) f k , Fi = k∈Ki η k (x) fk (7)
With the notations defined in [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF], system (3) on R i can be rewritten on the following compact form:

       ẋ = k∈Ki η k (x) fk = Fi x = k∈Ki η k (x) χk = Xi (8)
For stability analysis, without loss of generality we assume that the origin x = 0 is the equilibrium of system (3) which corresponds to the vertex χ k0 of the state-space partition, for a given k 0 ∈ K v . Let K Z be the set of multi-indexes for regions containing the origin which will be called zero-regions, and K N Z = K r \K Z (called non-zero regions). In each zeroregion, every point can be computed by interpolating its 2 n -1 vertices (except for the origin). Denoting K * i = K i \ {k 0 }, the interpolation procedure for zero-regions can be expressed by

         ẋ = k∈K * i η k (x) f k = F 0 i x = k∈K * i η k (x) χ k = X 0 i (9)

B. Preliminaries

Since the state space of PMA systems is partitioned into hyper-rectangle regions, it is natural to consider piecewise quadratic Lyapunov functions [START_REF] Johansson | Piecewise quadratic stability of fuzzy systems[END_REF] and the S-procedure [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] for their stability analysis. In this way, the information on the region structure of PMA systems can be fully exploited to reduce the conservatism of the results. To this end, let us construct the following matrices:

Li = L i l i , Mi = M i m i (10) such that Li x ≥ 0, x ∈ R i , i ∈ K r (11) Mi x = Mj x, x ∈ R i ∩ R j , i, j ∈ K r (12)
where l i = 0, and m i = 0, for i ∈ K Z .

Remark 3. Systematic methods to compute Li and Mi are available in [START_REF] Johansson | Piecewise quadratic stability of fuzzy systems[END_REF], [START_REF] Johansson | Piecewise Linear Control Systems: A Computational Approach[END_REF]. It is worth remarking that the statespace partition (2) of PMA systems is structurally simple with n-rectangular regions, therefore in contrast to [START_REF] Johansson | Piecewise Linear Control Systems: A Computational Approach[END_REF], the dimensions of all constraint matrices in ( 11)-( 12) are exactly known, i.e.

L i ∈ R 2n×n , Li ∈ R 2n×(n+1) , M i ∈ R N ×n and Mi ∈ R N ×(n+1) with N = n i=1 N i .
We will consider a piecewise quadratic Lyapunov function parameterized as follows

V (x) = x P i x, for x ∈ R i , i ∈ K Z x Pi x, for x ∈ R i , i ∈ K N Z (13)
where

P i = M i T M i , for i ∈ K Z Pi = M i T Mi , for i ∈ K N Z (14)
Remark 4. All free parameters of the Lyapunov function V (x) are collected in the symmetric matrix T and the expression of P i is linear in T . This allows an LMI formulation for the stability analysis of PMA systems. Moreover, since the piecewise quadratic Lyapunov function [START_REF] Tong | Fuzzy adaptive output feedback control of MIMO nonlinear systems with partial tracking errors constrained[END_REF] combines the power of quadratic functions near an equilibrium point with the flexibility of piecewise functions in the large, it can lead to less conservative results compared to those based on a common quadratic Lyapunov function. Indeed, the latter can be regarded as a special case of piecewise quadratic Lyapunov function ( 13) by taking M i = I and L i = 0.

The following lemma will be useful for the theoretical developments in next section.

Lemma 1 (Finsler's Lemma [START_REF] Skelton | A Unified Algebraic Approach to Linear Control Design[END_REF]). For a vector ξ ∈ R n and two matrices

Q = Q ∈ R n×n , R ∈ R m×n such that rank (R) < n, the following statements are equivalent i) ξ Qξ < 0, ∀ξ ∈ {ξ ∈ R n : ξ = 0, Rξ = 0 m×1 } ii) ∃F ∈ R n×m such that Q + FR + R F < 0
Lemma 1 can be used to convert checking the sign of a quadratic form over a subspace into solving an LMI problem. By this lemma, the parametric expressions in (5) will be effectively exploited for the stability study.

III. STABILITY ANALYSIS OF PIECEWISE MULTI-AFFINE SYSTEMS

Hereafter, a novel LMI-based method for studying the stability of PMA systems is presented. This method relies on the choice of the piecewise quadratic Lyapunov function [START_REF] Tong | Fuzzy adaptive output feedback control of MIMO nonlinear systems with partial tracking errors constrained[END_REF]. Moreover, the S-procedure [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] is also introduced to take advantage of the piecewise-region feature of PMA systems via constraint matrices [START_REF] Sugeno | On improvement of stability conditions for continuous Mamdani-like fuzzy systems[END_REF]. These facts contribute to reduce the conservatism inherent to this type of Lyapunov-based method.

Theorem 1 (Piecewise Quadratic Stability for PMA Systems). Given PMA system (3), assume there exist a symmetric matrix T ∈ R N ×N , symmetric matrices with nonnegative entries

U i ∈ R 2n×2n and W i ∈ R 2n×2n (for i ∈ K r ), matrices Y 1i ∈ R n×n , Y 2i ∈ R 1×n (for i ∈ K Z ), Ȳ1i ∈ R (n+1)×(n+1)
and Ȳ2i ∈ R 1×(n+1) (for i ∈ K N Z ) satisfying the following linear matrix inequalities:

P i -L i U i L i > 0, i ∈ K Z ( 15
)
He Y 1i + L i W i L i /2 -Y 1i χ k + P i f k Y 2i -Y 2i χ k < 0, i ∈ K Z , k ∈ K * i (16) Pi -L i U i Li > 0, i ∈ K N Z (17)
He Ȳ1i + L i W i Li /2 -Ȳ1i χk + Pi fk Ȳ2i -Ȳ2i χk < 0, i ∈ K N Z , k ∈ K i (18)
where P i and Pi are defined in [START_REF] Li | Observer-based adaptive fuzzy tracking control of MIMO stochastic nonlinear systems with unknown control directions and unknown dead zones[END_REF]. Then, the equilibrium x = 0 of PMA system (3) is asymptotically stable. In addition, the piecewise-quadratic function ( 13) is a Lyapunov function of this system.

Proof. Consider the piecewise function [START_REF] Tong | Fuzzy adaptive output feedback control of MIMO nonlinear systems with partial tracking errors constrained[END_REF] which can be rewritten in a more compact form

V (x) = x Pi x, x ∈ R i , i ∈ K r ( 19
)
The construction of the constraint matrices Li in [START_REF] Sugeno | On improvement of stability conditions for continuous Mamdani-like fuzzy systems[END_REF] and the satisfaction of conditions ( 15) and [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] imply that there exist positive real numbers α and β such that [8]

α|x| 2 ≤ V (x) ≤ β|x| 2 , x ∈ R (20)
Now, we will first consider the non-zero regions, i.e. x ∈ R i , for i ∈ K N Z . To derive the stability conditions, the timederivative of V (x) along the solution of system ( 8) is required to be negative definite, i.e.

V (x) = ẋ Pi x + x Pi ẋ < 0.

From the expression of ẋ in [START_REF] Johansson | Piecewise quadratic stability of fuzzy systems[END_REF], this inequality can be rewritten as follows

V (x) = x 1 0 Pi Fi * 0 x 1 < 0 (21) Let Qi = L i W i Li Pi Fi * 0 and Fi = I n+1 -Xi . Since
W i 0 and by the construction of Li in ( 11), the condition

x 1 Qi x 1 < 0, x ∈ R i , i ∈ K N Z ( 22
)
implies obviously [START_REF] Rhee | A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design[END_REF]. Moreover, from ( 8) it follows that

Fi x 1 = 0, x ∈ R i , i ∈ K N Z (23)
By Finsler's lemma, inequality [START_REF] Mozelli | A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems[END_REF] holds under condition [START_REF] Khalil | Nonlinear Systems[END_REF] if there exist matrices Mi such that

Qi + Mi Fi + F i M i < 0, i ∈ K N Z ( 24
)
Let us partition Mi = Ȳ1i Ȳ2i , then condition (24) can be rewritten in the form

He Ȳ1i + L i W i Li /2 -Ȳ1i Xi + Pi Fi Ȳ2i -Ȳ2i Xi < 0, i ∈ K N Z (25)
By the convexity property (6), if condition ( 18) is verified, then (25) holds as well. In other words, condition [START_REF] Johansson | Piecewise Linear Control Systems: A Computational Approach[END_REF] implies that V (x) < 0 for all x in non-zero regions.

The proof for zero-regions will slightly differ to that of non-zero regions since the parametric expressions given in [START_REF] Wang | H∞ controller design for affine fuzzy systems based on piecewise Lyapunov functions in finite-frequency domain[END_REF] do not correspond to convex combinations. This issue can be addressed via some special changes of variables as follows. Let x ∈ R i \{0} for i ∈ K Z , and α (x) =

k∈K * i η k (x). Since x = 0 and α (x) = 1 -η k0 (x), it follows that α (x) > 0. Denoting x * = x/α (x), x * = x * 1 , η * k (x) = η k (x)/α (x), F * i = k∈K * i η * k (x) f k , X * i = k∈K * i η * k (x) χ k .
By the convexity property [START_REF] Taniguchi | Model construction, rule reduction, and robust compensation for generalized form of Takagi-Sugeno fuzzy systems[END_REF], inequality [START_REF] Skelton | A Unified Algebraic Approach to Linear Control Design[END_REF] implies clearly that

He Y 1i + L i W i L i /2 -Y 1i X * i + P i F * i Y 2i -Y 2i X * i < 0, i ∈ K Z (26)
which can be also represented in the following form:

Q i + M i F i + F i M i < 0, i ∈ K Z (27) with Q i = L i W i L i P i F * i F * i P i 0 , F i = I n -X * i , M i = Y 1i Y 2i .
We note that F i x * = 0. From this latter and inequality (27), applying again Finsler's lemma it follows that

x * Q i x * < 0, x ∈ R i \{0}, i ∈ K Z ( 28
)
Remarking that 28)-(29) that V (x) < 0 for all x in zero-regions, x = 0. This completes the proof of Theorem 1.

α 2 (x) x * Q i x * = V (x) + x L i W i L i x (29) for x ∈ R i \{0}, i ∈ K Z . Since W i 0, it is easily deduced from (
The stability conditions in Theorem 1 are expressed in strict LMIs. The terms involving the matrices U i , W i , i ∈ K r , are related to the S-procedure to reduce the conservatism of those inequalities. 3) is strictly equivalent to the membership functions (4). For the results of Theorem 1, we fully exploit this expression of x to derive the stability conditions. This fact means that the knowledge on the MFs of PMA systems is taken into account in the stability analysis to reduce the conservatism of the proposed method. Remark 6. A natural question arises is whether conditions of Theorem 1 imply the stability of the original system (1). In fact, if the approximation errors between the system (1) and PMA model (3) can be characterized, then it could be possible to take these errors into account in the analysis so that the successful analysis of PMA model (3) guarantees the stability of system (1). This error characterization is left for future study. By now, it is worth noting that a fine repartition of a PMA model decreases the approximation error and increases the flexibility of the piecewise Lyapunov function [START_REF] Tong | Fuzzy adaptive output feedback control of MIMO nonlinear systems with partial tracking errors constrained[END_REF], but leads to additional LMI conditions to be solved. Thus, there is a trade-off between computational effort and precision in the stability analysis problem.

Remark 5. It has been proved in [1] that the parametric expression

x = k∈Ki η k (x) χ k in (
Remark 7. It is noteworthy that the computational cost of an LMI optimization problem can be estimated as being proportional to N 3 var N row , where N var is the total number of scalar decision variables and N row the total row size of the LMIs [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. In the case of Theorem 1, one has

N row = 2 n [(2n + 1)(2 n -1) + (2n + 3)(N r -2 n )] N var = N (N + 1)/2 + (5n 2 + 5n + 2)N r -(2n + 2)2 n where N = n i=1 N i and N r = n i=1 N i .

IV. NUMERICAL EXAMPLES

We will demonstrate the effectiveness of the proposed method through three examples. All computations are done in Matlab R2013b with YALMIP toolbox and MOSEK solver.

Example 1. Consider the following nonlinear pendulum system with friction:

ẋ1 = x 2 ẋ2 = - g l sin x 1 -bx 2 (30)
where x 1 is the pendulum angle, x 2 is its angular velocity, g = 9.87 is the gravity acceleration, l = 4 is the length of the pendulum and b = 0.5 is the friction coefficient. Let us construct the PMA model (3) of system (30) from 121 points of a 11 by 11 rectangular grid on the state-space

(x 1 , x 2 ): R = [-3.1, 3.1] × [-3.1, 3.1].
The stability conditions in Theorem 1 are feasible in this case and the corresponding Lyapunov function is depicted in Figure 1 (a). Thus, the local asymptotic stability of the considered PMA system can be proved. Some simulated trajectories of system (30) and the estimated domain of attraction of the equilibrium x = 0 of its PMA model are plotted in Figure 1 (b). All decision matrices obtained from Theorem 1 for this example are not shown here for brevity.

Example 2. Consider the following family of nonlinear systems taken from [START_REF] Rhee | A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design[END_REF]:

ẋ = 2 i=1 2 j=1 η 1i (x) η 2j (x) A ij x (31)
where i = 1, 2 and

η 1i (x i ) =    (1 -sin (x i ))/2, |x i | ≤ π/2 0, x i > π/2 1, x i < -π/2 η 2i (x i ) = 1 -η 1i (x i ) .
The system matrices in (31) are given by

A 11 = -5 -4 -1 a , A 12 = -4 -4 (3b -2)/5 (3a -4)/5 A 21 = -3 -4 (2b -3)/5 (2a -6)/5 , A 22 = -2 -4 b -2
for given constant parameters a and b. This example aims at studying the conservatism of the proposed method. To this end, the stability of system (31) for different parameter pairs (a, b) will be studied via three approaches: 1) PMA approximation approach with the conditions in Theorem 1, 2) Approximate T-S representation approach with the conditions in [START_REF] Johansson | Piecewise quadratic stability of fuzzy systems[END_REF], 3) Exact T-S representation approach with recent contributions in [START_REF] Rhee | A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design[END_REF], [START_REF] Mozelli | A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems[END_REF]. Notice that at the present stage of progress, the feasibility of the conditions in Theorem 1 does not directly imply the stability of (31), see Remark 6. However, the comparison carried out within this example provides an idea on the conservatism of the proposed method. Moreover, the PMA models of (31) will be constructed with very high number of piecewise regions such that they are tightly closed. Now, the PMA approximation of (31) is constructed from 961 points of a 31 by 31 rectangular grid on the statespace (x 1 , x 2 ): R = [-1.5, 1.5] × [-1.5, 1.5] resulting in 900 piecewise regions. Therefore, for comparison purposes the approximate T-S fuzzy model used with the stability conditions in [START_REF] Johansson | Piecewise quadratic stability of fuzzy systems[END_REF] has been constructed with 900 membership functions, corresponding to the number of regions of the PMA approximation.

Using Theorem 1, the asymptotic stability of PMA model is checked for different parameter pairs (a, b). For illustration, in Figure 2 circles correspond to pairs (a, b) for which stability conditions of Theorem 1 are satisfied whereas the grey area corresponds to feasibility space of parameters (a, b) obtained through Theorem 1 in [START_REF] Mozelli | A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems[END_REF] assuming that ḣij (x) ≤ 0.85, with h ij (x) = η 1i (x) η 2j (x), ∀ (i, j) ∈ Ω 2 . We note that the method in [START_REF] Mozelli | A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems[END_REF] is based on multiple Lyapunov functions and less conservative than those of Theorem 3 in [START_REF] Rhee | A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design[END_REF] (which is based on line-integral Lyapunov function). It is observed from Figure 2 that the proposed method allows proving (approximately) the stability of system (31) for a larger parameter domain. Notice that the stability conditions given in [START_REF] Johansson | Piecewise quadratic stability of fuzzy systems[END_REF] provide the same (a, b) parameter stability domain as those of Theorem 1. Therefore, in order to compare the conservativeness of these results, we have searched for the largest values of b, denoted by b * , for which the stability conditions are still feasible for two cases a = 0 and a = -2. The obtained results are presented in Table I.

Remark 8. Notice that one of the drawbacks of Takagi-Sugeno fuzzy modeling is the lack of a systematic procedure for selecting, among the infinite possibilities, a single parametrization for f (x) in (1) (in the form of ẋ = A (θ (t)) x where θ (t) is premise vector) which achieves stability. The use of PMA approximation approach allows overcoming this major drawback. Taking a = 0, b = 0 as a particular case, neither [START_REF] Rhee | A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design[END_REF] nor [START_REF] Mozelli | A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems[END_REF] are capable of proving the stability of system (31). Figure 3 (a) depicts the phase plane of system (31) and the level curves of the Lyapunov function obtained with Theorem 1. The corresponding estimate of the domain of attraction and some simulated trajectories of are depicted in Figure 3 (b). It is important to note that all simulations in Figure 3 are done with the original system (31). Example 3. Consider an electric circuit represented by the following Hopfield artificial neural network model [START_REF] Khalil | Nonlinear Systems[END_REF]:

ẋi = λ C i 1 1 + tan 2 πxi 2Vm   3 j=1 T ij x j - 2V m λπR i tan πx i 2V m   (32)
where the state variables x i , i ∈ Ω 3 , are the voltages at the amplifier outputs. These voltages can only take values in the set S = x ∈ R 3 : -V m < x i < V m . The system data are given as follows C 1 = 3.3µF, C 2 = 1.5µF, C 3 = 5.6µF, R 1 = 1.0kΩ, R 2 = 3.3kΩ, R 3 = 1.2kΩ, λ = 7.0 × 10 -2 , V m = 5V and

T ij = 10 -3 ×   1 2 1 2 2 3 1 3 3   .
The physically motivated system (32) is used to illustrate the outstanding feature of the proposed method in terms of dealing with complex nonlinearities. We note also that existing results in [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF], [START_REF] Kim | A new computational approach to stability analysis and synthesis of linguistic fuzzy control system[END_REF], [START_REF] Sugeno | On improvement of stability conditions for continuous Mamdani-like fuzzy systems[END_REF], [START_REF] Kim | A new approach to numerical stability analysis of fuzzy control systems[END_REF] can not be applied to this threedimensional system. Let us construct the PMA model (3) of system (32) from 1331 points of a 11 × 11 × 11 hyper-rectangular grid on the state-space (x 1 , x 2 , x 3 ): R = [-4.99, 4.99] × [-4.99, 4.99] × [-4.99, 4.99]. The stability conditions in Theorem 1 are feasible in this case and we are able to find a piecewise Lyapunov function proving the local asymptotic stability of the considered PMA model. Some simulated trajectories of system (32) are depicted in Figure 4 (a) and the time evolution of the Lyapunov function for these same trajectories are given in Figure 4 (b). All decision matrices obtained from Theorem 1 for this example are not shown here for brevity.

V. CONCLUDING REMARKS

Based on the use of piecewise quadratic Lyapunov functions, a constructive method for stability analysis of PMA systems has been proposed. By exploiting the parametric expressions of PMA systems, stability conditions are derived and expressed in terms of LMIs efficiently solvable with numerical solvers. Moreover, the proposed method is able to deal with PMA systems of any order. The effectiveness of the method is clearly demonstrated via numerical examples.

Despite many outstanding contributions of the proposed method, there still remain some challenging issues which will be the focus of future works. The first topic consists in considering, in the stability analysis, the modeling error between the original system and its PMA model. To this end, PMA model-based methods can be directly employed to study general nonlinear systems. Second, Lyapunov-based control design for PMA systems is another important research topic for future investigation.

Fig. 1 .

 1 Fig. 1. (a) Lyapunov function obtained with the PMA model of system (30); (b) Estimate of the domain of attraction for equilibrium x = 0 of PMA model and simulated trajectories of nonlinear system (30).

Fig. 2 .

 2 Fig.2. Parameter stability domain obtained with Theorem 1 (circles) and Theorem 1 in[START_REF] Mozelli | A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems[END_REF] (grey area).

Fig. 3 .

 3 Fig. 3. (a) Phase plane of system (31) for a = 0, b = 0 and level curves associated to the obtained Lyapunov function; (b) Estimate of the domain of attraction of PMA model and some simulated trajectories of system (31).

Fig. 4 .

 4 Fig. 4. (a) Simulated trajectories of system (32) with different initial conditions; (b) Time evolution of the Lyapunov function found in Example 3 for the corresponding trajectories in (a).

ACKNOWLEDGMENT

We would like to thank anonymous reviewers for their helpful comments and for bringing to our attention important references.