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

Member, IEEE Jagat Jyoti Rath

Thierry-Marie Guerra

Member, IEEE Reinaldo Martínez Palhares

Senior Member, IEEE Hui Zhang

Robust Set-Invariance Based Fuzzy Output Tracking Control for Vehicle Autonomous Driving Under Uncertain Lateral Forces and Steering Constraints

Keywords: Autonomous driving, path tracking, vehicle dynamics, output feedback control, Takagi-Sugeno fuzzy systems, robust set invariance

This paper is concerned with a new control method for path tracking of autonomous ground vehicles. We exploit the fuzzy model-based control framework to deal with the timevarying feature of the vehicle speed and the highly uncertain behaviors of the tire-road forces involved in the nonlinear vehicle dynamics. To avoid using costly vehicle sensors for feedback control while favoring the simplest control structure for real-time implementation, a new fuzzy static output feedback (SOF) scheme is proposed. In particular, though the robust set-invariance property and Lyapunov-based arguments, the physical constraints on the steering input saturation and the vehicle state can be taken into account in the control design to improve the driving safety and comfort. The theoretical development relies on the use of fuzzy Lyapunov functions and the non-parallel distributed compensation control concept to reduce the design conservatism. Exploiting some specific convexification techniques, the control design is reformulated as an optimization problem under linear matrix inequalities with a single line search, which are efficiently solved via semidefinite programming techniques. The proposed fuzzy path tracking controller is evaluated through various dynamic driving tests conducted with high-fidelity CarSim/Matlab co-simulations. Moreover, to emphasize the advantages of the new fuzzy SOF controller, a performance comparison with the CarSim driver model is also performed.

I. INTRODUCTION

N OWADAYS, autonomous vehicles have become a reality thanks to significant recent advances in computation and communication technologies. It has been demonstrated that autonomous ground vehicles can improve not only the safety, accessibility and comfort of passengers but also the energysaving efficiency [START_REF] Broggi | Autonomous vehicles control in the VisLab intercontinental autonomous challenge[END_REF]. For these reasons, driverless vehicle technology has attracted a great deal of attention from academic researchers, industrial companies and local governments [START_REF] Paden | A survey of motion planning and control techniques for self-driving urban vehicles[END_REF].

As one of the most important parts of vehicle autonomous navigation, path tracking control of autonomous vehicles has been a significant research topic [START_REF] Paden | A survey of motion planning and control techniques for self-driving urban vehicles[END_REF]- [START_REF] Alcala | Autonomous vehicle control using a kinematic Lyapunov-based technique with LQR-LMI tuning[END_REF].

Path tracking control is concerned with designing a steering control law to guide the vehicle to follow a desired trajectory, generated by a vehicle path planner [START_REF] Dixit | Trajectory planning for autonomous high-speed overtaking in structured environments using robust MPC[END_REF], [START_REF] Benloucif | Cooperative trajectory planning for haptic shared control between driver and automation in highway driving[END_REF]. The key control goal is to achieve path following performance with acceptably small tracking errors and smooth steering actions under various driving conditions [START_REF] Paden | A survey of motion planning and control techniques for self-driving urban vehicles[END_REF]. Remarkable contributions on path tracking of autonomous vehicles have been reported in the literature. Generally speaking, the path tracking control strategies can be classified into three main categories [START_REF] Xu | Design, analysis, and experiments of preview path tracking control for autonomous vehicles[END_REF]: model-free control, vehicle geometric-kinematic-based control and dynamic model-based feedback control. First, model-free controllers, such as PID control [START_REF] Zhao | Design of a control system for an autonomous vehicle based on adaptive-PID[END_REF] or fuzzy logic control [START_REF] Naranjo | Powersteering control architecture for automatic driving[END_REF], [START_REF] Wang | Lateral control of autonomous vehicles based on fuzzy logic[END_REF], [START_REF] Wang | Lateral control of autonomous vehicles based on fuzzy logic[END_REF]- [START_REF] Taghavifar | Path-tracking of autonomous vehicles using a novel adaptive robust exponential-like-sliding-mode fuzzy type-2 neural network controller[END_REF], generate the steering actions exclusively based on the tracking errors. Without requiring the vehicle information, these control methods generally require a laborintensive tuning task to achieve an acceptable path tracking performance [START_REF] Amer | Modelling and control strategies in path tracking control for autonomous ground vehicles: a review of state of the art and challenges[END_REF]. Second, the geometric-kinematic-based path tracking controllers are designed using a vehicle kinematics model [START_REF] Alcala | Autonomous vehicle control using a kinematic Lyapunov-based technique with LQR-LMI tuning[END_REF] and/or geometric relations [START_REF] Snider | Automatic steering methods for autonomous automobile path tracking[END_REF]. Despite their simplicity, these control approaches are only suitable for driving situations, in which the vehicle dynamics can be neglected. Third, the dynamic model-based control design is based on the information of vehicle parameters and dynamics. Here, we focus on these model-based feedback control approaches since they can be used to overcome the above-mentioned drawbacks as indicated in recent comparative studies [START_REF] Paden | A survey of motion planning and control techniques for self-driving urban vehicles[END_REF], [START_REF] Amer | Modelling and control strategies in path tracking control for autonomous ground vehicles: a review of state of the art and challenges[END_REF], [START_REF] Lee | Synthesis of robust lane keeping systems: Impact of controller and design parameters on system performance[END_REF].

Numerous works on path tracking feedback control of autonomous vehicles are available in the open literature, which are mostly based on conventional control theory [START_REF] Snider | Automatic steering methods for autonomous automobile path tracking[END_REF], [START_REF] Amer | Modelling and control strategies in path tracking control for autonomous ground vehicles: a review of state of the art and challenges[END_REF], [START_REF] Barbosa | Robust path-following control for articulated heavy-duty vehicles[END_REF]. Sliding mode control technique was applied to vehicle control in [START_REF] Tagne | Design and comparison of robust nonlinear controllers for the lateral dynamics of intelligent vehicles[END_REF], [START_REF] Jeong | Nonlinear hybrid impedance control for steering control of rack-mounted electric power steering in autonomous vehicles[END_REF]. Despite its robustness with respect to parametric uncertainties and external disturbances, dealing with the chattering issue still leads to both theoretical and practical difficulties [START_REF] Amer | Modelling and control strategies in path tracking control for autonomous ground vehicles: a review of state of the art and challenges[END_REF]. A Lyapunov-based hybrid faulttolerant controller was proposed in [START_REF] Yang | Optimal fault-tolerant pathtracking control for 4WS4WD electric vehicles[END_REF] with experimental validations on path tracking of a prototype vehicle. However, the stability and robustness aspects of this control design should be further investigated. An active disturbance rejection controller (ADRC) was developed in [START_REF] Chu | Active disturbance rejection control applied to automated steering for lane keeping in autonomous vehicles[END_REF] for vehicle steering control in the presence of uncertainties and external disturbance. Note that the ADRC requires a fine tuning of several design parameters to ensure a good real-time control performance. Thanks to its intuitive principle and its advantages in dealing with system constraints, model predictive control (MPC) technique has been largely applied to the motion planning [START_REF] Dixit | Trajectory planning for autonomous high-speed overtaking in structured environments using robust MPC[END_REF], [START_REF] Suh | Stochastic model predictive control for lane change decision of automated driving vehicles[END_REF], [START_REF] Wang | Crash mitigation in motion planning for autonomous vehicles[END_REF], path tracking [START_REF] Yang | Path following predictive control for autonomous vehicles subject to uncertain tire-ground adhesion and varied road curvature[END_REF] and obstacle avoidance [START_REF] Funke | Collision avoidance and stabilization for autonomous vehicles in emergency scenarios[END_REF], [START_REF] Guo | Simultaneous trajectory planning and tracking using an MPC method for cyberphysical systems: A case study of obstacle avoidance for an intelligent vehicle[END_REF] of autonomous vehicles. However, MPC technique requires solving an online optimization problem at each control step, which leads too expensive computational burden for real-time implementation, especially for nonlinear MPC schemes. To avoid this major practical issue while taking into account the forward prediction ability, the preview tracking control has been proposed [START_REF] Xu | Design, analysis, and experiments of preview path tracking control for autonomous vehicles[END_REF]. However, handling the model uncertainties and the system constraints is still open for this control technique. Takagi-Sugeno (T-S) fuzzy approaches [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF] have been proposed to deal with the vehicle tracking control issues [START_REF] Zhang | A novel fuzzy observer-based steering control approach for path tracking in autonomous vehicles[END_REF], [START_REF] Dahmani | Observer-based state feedback control for vehicle chassis stability in critical situations[END_REF], [START_REF] Nguyen | Fuzzy static output feedback control for path following of autonomous vehicles with transient performance improvements[END_REF]. In contrast to model-free fuzzy control [START_REF] Naranjo | Powersteering control architecture for automatic driving[END_REF], [START_REF] Boada | Fuzzy-logic applied to yaw moment control for vehicle stability[END_REF], [START_REF] Wu | The heterogeneous systems integration design and implementation for lane keeping on a vehicle[END_REF], rigorous stability analysis and performance proofs can be achieved with T-S fuzzy control approaches [START_REF] Nguyen | Fuzzy control systems: Past, present and future[END_REF]. However, fuzzy modelbased output feedback control for uncertain systems remains challenging [START_REF] Zhang | A novel fuzzy observer-based steering control approach for path tracking in autonomous vehicles[END_REF], [START_REF] Nguyen | Constrained output-feedback control for discrete-time fuzzy systems with local nonlinear models subject to state and input constraints[END_REF].

Despite a significant advance on path tracking of autonomous vehicles, the three following control challenges still require a great deal of research efforts. First, the use of full state feedback design where the effectiveness of the control depends on data acquisition rate is a major constraint [START_REF] Zhang | A novel observer design for simultaneous estimation of vehicle steering angle and sideslip angle[END_REF], [START_REF] Hu | MME-EKF-based path-tracking control of autonomous vehicles considering input saturation[END_REF]. Most of works on the path tracking assume that all the vehicle states can be measured for the real-time control implementation [START_REF] Snider | Automatic steering methods for autonomous automobile path tracking[END_REF], [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF]. However, this assumption is generally not realistic in practice. For instance, the sideslip angle and/or the lateral speed, which is crucial for vehicle control, cannot be reliably measured with low-cost sensors [START_REF] Nguyen | Sensor reduction for driver-automation shared steering control via an adaptive authority allocation strategy[END_REF]- [START_REF] Ting | An output-feedback fuzzy approach to guaranteed cost control of vehicle lateral motion[END_REF]. Therefore, various methods have been proposed to design observers for the estimation of unmeasured vehicle variables, see [START_REF] Zhang | A novel observer design for simultaneous estimation of vehicle steering angle and sideslip angle[END_REF], [START_REF] Nam | Estimation of sideslip and roll angles of electric vehicles using lateral tire force sensors through RLS and Kalman filter approaches[END_REF]- [START_REF] Nguyen | Unknown input observers for simultaneous estimation of vehicle dynamics and driver torque: Theoretical design and hardware experiments[END_REF] and references therein. Then, an observer-based control scheme can be designed for path tracking of autonomous vehicles [START_REF] Zhang | A novel fuzzy observer-based steering control approach for path tracking in autonomous vehicles[END_REF], [START_REF] Dahmani | Observer-based state feedback control for vehicle chassis stability in critical situations[END_REF], [START_REF] Ting | An output-feedback fuzzy approach to guaranteed cost control of vehicle lateral motion[END_REF]. Unfortunately, the design of an effective observer-based controller for vehicle path tracking under modeling uncertainties and system constraints still remains open [START_REF] Hu | MME-EKF-based path-tracking control of autonomous vehicles considering input saturation[END_REF]. The second challenge is the reliance of the control design on vehicle modeling. Describing precisely the vehicle nonlinear dynamics is very hard or even impossible [START_REF] Amer | Modelling and control strategies in path tracking control for autonomous ground vehicles: a review of state of the art and challenges[END_REF]. On the other hand, an accurate vehicle model may (unnecessarily) induce many difficulties in the control design. For example, although several semi-empirical models are available to represent accurately the tire-road friction, such as Pacejka magic formula, LuGre model, Burckhardt model [START_REF] Li | Integrated longitudinal and lateral tire/road friction modeling and monitoring for vehicle motion control[END_REF], these are however hardly exploitable for control purposes due to their strong nonlinearities. Hence, the design of a model-based controller robust to the unmodeled vehicle dynamics, timevarying parameters and external disturbances, is of crucial importance [START_REF] Du | Stabilizing vehicle lateral dynamics with considerations of parameter uncertainties and control saturation through robust yaw control[END_REF]. The third challenge consists in considering the system constraints, e.g., the physical limitations of the steering actuator and the vehicle state constraints, in the control design procedure to improve the driving safety and comfort [START_REF] Yang | Optimal fault-tolerant pathtracking control for 4WS4WD electric vehicles[END_REF], [START_REF] Nguyen | Driver-automation cooperative approach for shared steering control under multiple system constraints: Design and experiments[END_REF]. This important issue has not been well addressed in the existing literature on the path tracking control [START_REF] Nguyen | Fuzzy steering control for autonomous vehicles under actuator saturation: Design and experiments[END_REF], [START_REF] Hu | MME-EKF-based path-tracking control of autonomous vehicles considering input saturation[END_REF], especially with output feedback control schemes.

This paper proposes a new fuzzy static output feedback (SOF) controller to simultaneously address the three above challenges of vehicle path following. As in [START_REF] Zhang | A novel fuzzy observer-based steering control approach for path tracking in autonomous vehicles[END_REF], fuzzy modelbased control is exploited to deal with the nonlinear uncertain vehicle dynamics. Then, the time-varying feature of the vehicle speed and the highly uncertain behaviors of the lateral tireroad forces can be taken into account in the control design to guarantee a satisfactory path tracking performance under different driving conditions. Moreover, our proposed control solution leads to the following new contributions.

• Exploiting the concept of robust invariant set together with Lyapunov-based arguments, the steering input saturation and the vehicle state constraints can be explicitly considered in the control design for safety and confort reasons. Moreover, differently from [START_REF] Nguyen | Sensor reduction for driver-automation shared steering control via an adaptive authority allocation strategy[END_REF]- [START_REF] Ting | An output-feedback fuzzy approach to guaranteed cost control of vehicle lateral motion[END_REF], here a fuzzy Lyapunov function and a non-parallel distributed compensation (non-PDC) scheme [START_REF] Nguyen | Gain-scheduled static output feedback control for saturated LPV systems with bounded parameter variations[END_REF] are used to reduce further the design conservatism.

• No additional fuzzy observers as in [START_REF] Zhang | A novel fuzzy observer-based steering control approach for path tracking in autonomous vehicles[END_REF], [START_REF] Dahmani | Observer-based state feedback control for vehicle chassis stability in critical situations[END_REF], [START_REF] Hu | MME-EKF-based path-tracking control of autonomous vehicles considering input saturation[END_REF], [START_REF] Ting | An output-feedback fuzzy approach to guaranteed cost control of vehicle lateral motion[END_REF] or dynamic control schemes as in [START_REF] Nguyen | Sensor reduction for driver-automation shared steering control via an adaptive authority allocation strategy[END_REF], [START_REF] Wang | Output-feedback robust control for vehicle path tracking considering different human drivers' characteristics[END_REF] are required to deal with the online unavailability of vehicle state. Then, the proposed fuzzy SOF controller is of much simpler structure than that of these dynamic output feedback control schemes. As a result, the new controller not only improves the reliability of the vehicle control system but also is easier to be implemented in practice. • The control design procedure is recast as an optimization problem under linear matrix inequalities (LMIs) which can be easily solved with semidefinite programming techniques [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. Note that the previous robust output feedback control methods in [START_REF] Zhang | A novel fuzzy observer-based steering control approach for path tracking in autonomous vehicles[END_REF] are reformulated as nonconvex problems, inducing numerical difficulties. • The effectiveness of the proposed fuzzy SOF controller clearly is demonstrated with high-fidelity CarSim/Matlab experiments under dynamic driving conditions. Notation. For a vector x, x i denotes its ith entry. For a square matrix X, X denotes its transpose, X 0 means that X is positive definite, X (i) denotes its ith row and He X = X + X . diag(X 1 , X 2 ) denotes a block-diagonal matrix composed of X 1 , X 2 . I denotes the identity matrix of appropriate dimension. The set of nonnegative integers is denoted by Z + . For r ∈ Z + , we denote I r = {1, . . . , r} ⊂ Z + . The symbol ' ' stands for the terms deduced by symmetry in symmetric block matrices. For convenience, arguments are omitted when their meaning is straightforward.

II. VEHICLE MODELING AND PROBLEM FORMULATION

We present hereafter the vehicle modeling for path tracking control, see Fig. 1. The nomenclature is given in Table I.

A. Road-Vehicle Model

A nonlinear single track vehicle model is used to study the vehicle motions, whose dynamics is given as follows [START_REF] Nguyen | Fuzzy steering control for autonomous vehicles under actuator saturation: Design and experiments[END_REF]:

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 , I z ṙ = l f (F xf sin δ + F yf cos δ) -l r F yr . (1)
The front and rear longitudinal forces F xi and lateral forces F yi , for i ∈ {f, r}, are caused by the tire-road interaction. The behaviors of these forces depend on several factors, including slip angles, tire/road characteristics, normal load and so forth [START_REF] Kim | Identification of lateral tyre force dynamics using an extended Kalman filter from experimental road test data[END_REF]. Various nonlinear tire-road friction models such as Pacejka magic formula, Brush model, Burckhardt model, etc., have been developed to represent the tires properties in the extreme handling situations with high lateral acceleration and slip angles [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF], [START_REF] Li | Integrated longitudinal and lateral tire/road friction modeling and monitoring for vehicle motion control[END_REF]. However, the strong nonlinearities involved in these models induce major challenges in the control design. Here, to reduce the design complexity, the following norm-bounded uncertain tire model is used [START_REF] Du | Stabilizing vehicle lateral dynamics with considerations of parameter uncertainties and control saturation through robust yaw control[END_REF]:

F yf = 2C f α f = 2C f δ - v y + l f r v x , F yr = 2C r α r = 2C r l r r -v y v x .
(

) 2
where the cornering stiffness are time-varying to take into account the road friction changes or the nonlinear tire behaviors [START_REF] Zhang | A novel fuzzy observer-based steering control approach for path tracking in autonomous vehicles[END_REF]. These bounded parameters

C f ∈ [C f min , C f max ] and C r ∈ [C r min , C r max ]
, can be represented by

C f = C f + ∆C f ζ f (t), C r = C r + ∆C r ζ r (t), (3)
where |ζ i (t)| ≤ 1, i ∈ {f, r}, are unknown parameters, and

C f = C f max + C f min 2 , C r = C r max + C r min 2 , ∆C f = C f max -C f min 2 , ∆C r = C r max -C r min 2 .
For steering control purposes, we consider the small angles assumption and a slow vehicle speed variation. Then, the vehicle lateral dynamics can be derived from ( 1) and (2) as

β ṙ = - 2(Cr+C f ) M vx 2(lrCr-l f C f ) M v 2 x - 1
2(lrCr-l f C f ) Iz -2(l 2 r Cr+l 2 f C f ) Izvx β r + 2C f M vx 2l f C f Iz δ(4)
where v y = v x β. The vehicle positioning on the road can be represented by the lateral error y L from the lane centerline at a lookahead distance l s , and the heading error ψ L between the tangent to the road and the vehicle orientation [START_REF] Nguyen | Driver-automation cooperative approach for shared steering control under multiple system constraints: Design and experiments[END_REF]. The dynamics of these error variables are given as follows [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF], [START_REF] Nguyen | Fuzzy steering control for autonomous vehicles under actuator saturation: Design and experiments[END_REF]: where ρ r is the road curvature. Note that this amplitudebounded signal can be obtained from the vehicle vision system.

ẏL = v x β + l s r + v x ψ L , ψL = r -v x ρ r , (5)

B. Vehicle Control-Based Model

From the vehicle model ( 4), the dynamics for path tracking ( 5), the road-vehicle model can be represented as follows:

ẋ(t) = Âv (t)x(t) + Bv (t)δ(t) + E v (t)w(t), (6)
where x = β r ψ L y L is the vehicle state, w = ρ r is the system disturbance. The steering control input δ(t) of the vehicle system ( 6) is subject to the actuator saturation as

δ(t) = sat(u(t)) = sign(u(t)) min (|u(t)| , δ max ) ,
where u is the steering control action to be designed and δ max is the physical limitation of the steering angle. The state-space matrices of the control-based model ( 6) are given by Âv

(t) = A v (t) + ∆A v (t), Bv (t) = B v (t) + ∆B v (t) with 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 v =     b 1 b 2 0 0     , E v =     0 0 -v x 0     ,
and

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, 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 1 = 2C f M v x , b 2 = 2l f C f I z .
Moreover, the matrices ∆A v and ∆B v represent the system norm-bounded uncertainties caused by the uncertain stiffness.

From the expressions of C f and C r in (3), these uncertain matrices can be represented as

∆A v (t) = H v ∆(t)L v , ∆B v (t) = H v ∆(t)N v ,
with

H v =     2∆Cr M vx 2∆C f M vx -2∆Crlr Iz 2∆C f l f Iz 0 0 0 0     , L v =     -1 -1 lr vx - l f vx 0 0 0 0     , ∆(t) = diag(ζ r (t), ζ f (t)), N v = 0 1 .
To facilitate the real-time implementation, the path following control design is performed in the discrete-time domain. Then, applying the Euler's discretization

ẋ(t) x(κ + 1) -x(κ) T e , ∀κ ∈ Z + ,
we can transform system (6) into its discrete-time counterpart

x(κ + 1) = Â(v x )x(κ) + B(v x )sat(u(κ)) + E(v x )w(κ) (7)
where z(κ) denotes the value of the signal z at the κ-instant, and Σ(v x ) indicates that matrix Σ explicitly depends on the time-varying speed v x . The state-space matrices of ( 7) are given by

Â(v x ) = I + T e Âv (v x ), B(v x ) = T e Bv (v x ), E(v x ) = T e E v (v x ), T e = 0.01,
where T e is the sampling time in second.

Commercial vehicles are usually equipped with an inertial navigation system to measure the yaw rate r and an odometer to measure the vehicle speed v x . The look-ahead lateral deviation y L and the heading error ψ L can be measured by a video camera. The sideslip angle β can be measured by dual antenna GPS systems or Correvit optical sensors. Unfortunately, due to their excessive costs1 , the measurements of β is unavailable for real-time implementation in practice [START_REF] Nguyen | Sensor reduction for driver-automation shared steering control via an adaptive authority allocation strategy[END_REF], [START_REF] Wang | Output-feedback robust control for vehicle path tracking considering different human drivers' characteristics[END_REF]. Then, the output equation of the vehicle system ( 7) is given by

y =   0 1 0 0 0 0 1 0 0 0 0 1   x = Cx.
The controlled output vector z is defined such that it can represent both the path following performance and the driving comfort. (i) The heading error ψ L represents both the tracking performance and the future vehicle position error. (ii) The lateral deviation y L informs about the path following. (iii) The lateral acceleration a y v x r indicates the driving comfort. Then, the controlled output is defined as follows:

z =   ψ L y L a y   =   0 0 1 0 0 0 0 1 0 v x 0 0   x = D(v x )x. (8)

C. Path Following Control Objectives

This paper provides a systematic method to design path following controllers for autonomous vehicles which satisfies the following requirements.

• The control implementation can be done with only low-cost sensors of commercial vehicles. Moreover, the proposed path following controllers must be of simple control structure for real-time application perspective. • A satisfactory control performance should be achieved under uncertain lateral tire forces and physical constraints on both the vehicle states and the steering input. • The closed-loop steering control performance is theoretically guaranteed with Lyapunov-based arguments.

Furthermore, the control gains can be easily computed with available numerical toolboxes. To achieve these challenging control objectives, we propose in the following a new method to design fuzzy SOF controllers in presence of modeling uncertainties and system constraints.

III. CONSTRAINED OUTPUT FEEDBACK CONTROL DESIGN

FOR TAKAGI-SUGENO FUZZY SYSTEMS This section formulates the SOF control problem of fuzzy systems subject to modeling uncertainties and system constraints. Then, the proposed LMI-based control solution is applied to the path following of autonomous ground vehicles.

A. System Description

A class of T-S fuzzy systems subject to modeling uncertainties and actuator saturation with r inference rules can be described as follows [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF]:

RULE R i : IF ξ 1 (κ) is M i 1 and . . . and ξ p (κ) is M i p . THEN x(κ + 1) = Âi (κ)x(κ) + Bi (κ)sat(u(κ)) + E i w(κ), z(κ) = D i x(κ), y(κ) = C i x(κ), (9)
where x(κ) ∈ R nx is the state vector, u(κ) ∈ R nu the control input, w(κ) ∈ R nw the system disturbance, z(κ) ∈ R nz the controlled output, and y(κ) ∈ R ny the measured output, and

Âi (κ) = A i + ∆A i (κ), Bi (κ) = B i + ∆B i (κ).
The vector of premise variables is defined as

ξ(κ) = [ξ 1 (κ) • • • ξ p (κ)] ∈ R p .
The ith local state-space matrices with appropriate dimensions (A i , B i , C i , D i , E i ) are constant. For the fuzzy system (9), R i denotes the ith fuzzy inference rule. M i j , with i ∈ I r and j ∈ I p , is the fuzzy set. The fuzzy membership functions (MFs) are given by

h i (ξ) = p j=1 µ i j (ξ j ) r i=1 p j=1 µ i j (ξ j ) , i ∈ I r ,
where µ i j (ξ j ) represents the grade of membership of ξ j in the respective fuzzy set M i j . Note that the MFs satisfy the following convex sum property:

r i=1 h i (ξ) = 1, 0 ≤ h i (ξ) ≤ 1, ∀i ∈ I r . ( 10
)
Let Ω be the set of membership functions satisfying [START_REF] Xu | Design, analysis, and experiments of preview path tracking control for autonomous vehicles[END_REF], i.e., h = (h 1 (ξ), h 2 (ξ), . . . , h r (ξ)) ∈ Ω. Using the center-ofgravity method for defuzzification, the fuzzy system (9) can be represented in the following compact form [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF]:

x(κ + 1) = Â(h)x(κ) + B(h)sat(u(κ)) + E(h)w(κ), z(κ) = D(h)x(κ), y(κ) = C(h)x(κ), (11)
where the uncertain state-space matrices verify the following norm-bounded structures:

Â(h) = A(h) + ∆A(h), ∆A(h) = H(h)∆(κ)L(h), B(h) = B(h) + ∆B(h), ∆B(h) = H(h)∆(κ)N (h), (12)
with

Π(h) = r i=1 h i (ξ)Π i , Π ∈ {A, B, C, D, E, H, L, N }. (13)
Note that the constant matrices with appropriate dimensions H i , L i , N i , for ∀i ∈ I r , defined in ( 12)-( 13), are known, and the time-varying uncertain matrix satisfies ∆(κ) ∆(κ) I. The input saturation function is defined as

sat(u l (κ)) = sign(u l (κ)) min(|u l (κ)|, ūl ), κ ∈ Z + ,
where the control bounds ūl > 0, for ∀l ∈ I nu , are given. The system disturbance w in [START_REF] Zhao | Design of a control system for an autonomous vehicle based on adaptive-PID[END_REF] is bounded in amplitude and belongs to the following class of function:

W ∞ φ = w : R + → R nw , w(κ) w(κ) ≤ φ, κ ∈ Z + , with a positive scalar φ > 0.
In many practical situations, the disturbance signal w can be measured, e.g., the road curvature ρ r in ( 6) can be measured by the cameras [START_REF] Xu | Design, analysis, and experiments of preview path tracking control for autonomous vehicles[END_REF]. Then, incorporating the real-time disturbance information for feedforward control may improve the control performance during the transient phases [START_REF] Snider | Automatic steering methods for autonomous automobile path tracking[END_REF]. Hence, we consider the following non-PDC control law [START_REF] Nguyen | Gain-scheduled static output feedback control for saturated LPV systems with bounded parameter variations[END_REF]:

u(κ) = F (h)G(h) -1 y(κ) + K(h)w(κ), (14)
where the MF-dependent matrices

F (h) G(h) K(h) = r i=1 h i (ξ) F i G i K i ,
are to be determined.

Remark 1.

If the disturbance w is not available online, the classical non-PDC control scheme can be directly recovered by setting K i = 0, ∀i ∈ I r . Moreover, as can be seen in Theorem 1, the matrix K(h) appears linearly in the design conditions. Hence, the incorporation of K(h) into the SOF controller [START_REF] Chen | Hierarchical coordinated control method for unmanned robot applied to automotive test[END_REF] does not induce any additional difficulty when deriving LMI-based design conditions.

Define the dead-zone nonlinearity ψ(•) : R nu → R nu as

ψ(u(κ)) = u(κ) -sat(u(κ)).
From ( 11) and ( 14), the closed-loop fuzzy system can be expressed as follows:

x(κ + 1) = A cl (h)x(κ) + E cl (h)w(κ) -B(h)ψ(u(κ)), z(κ) = D(h)x(κ), y(κ) = C(h)x(κ), (15)
with

A cl (h) = Â(h) + B(h)F (h)G(h) -1 C(h), E cl (h) = E(h) + B(h)K(h).
For the control design, we consider the following MFdependent Lyapunov candidate function:

V(x, h) = x r i=1 h i (ξ)Q i -1 x x Q(h) -1 x,
with Q i 0, for ∀i ∈ I r . The level set associated with V(x, h) is defined as

L V = {x ∈ R nx : V(x, h) ≤ 1, for ∀h ∈ Ω} . ( 16
)
The following concept of robustly invariant set [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF] is crucial to characterize the local stability of the fuzzy system [START_REF] Hwang | Path tracking of an autonomous ground vehicle with different payloads by hierarchical improved fuzzy dynamic sliding-mode control[END_REF]. Definition 1. The set L V defined in ( 16) is said to be robustly invariant with respect to the closed-loop system (15) if there exist two positive scalars α and τ such that

∆V + α(V(x(κ), h(κ)) -1) + τ (φ -w(κ) w(κ)) < 0, (17)
with

∆V = V(x(κ + 1), h(κ + 1)) -V(x(κ), h(κ)), ∀κ ∈ Z + ,
for ∀x(κ) ∈ L V \{0}, ∀w(κ) ∈ W ∞ φ , and ∀h ∈ Ω. Note that condition [START_REF] Amer | Modelling and control strategies in path tracking control for autonomous ground vehicles: a review of state of the art and challenges[END_REF] guarantees that any closed-loop trajectory of (15) initialized in L V will remain within this set for ∀w(κ) ∈ W ∞ φ and ∀κ ∈ Z + . Moreover, if the disturbances vanish, i.e., w(κ) = 0, then for ∀x(0) ∈ L V , the corresponding trajectory converges to the origin with a predefined decay rate α.

This paper proposes an LMI-based solution for the following control problem.

Problem 1. Determine the MF-dependent matrices F (h), G(h) and K(h) of the non-PDC output feedback controller [START_REF] Chen | Hierarchical coordinated control method for unmanned robot applied to automotive test[END_REF] such that system (15) satisfies the following properties.

i) Property 1: The set L V defined in ( 16) is robustly invariant with respect to the fuzzy system [START_REF] Hwang | Path tracking of an autonomous ground vehicle with different payloads by hierarchical improved fuzzy dynamic sliding-mode control[END_REF]. Moreover, the controlled output of ( 15) is L ∞ -norm bounded, i.e., z z ≤ γ, for γ > 0. ii) Property 2: For any x(0) ∈ L V , the corresponding trajectory of ( 15) remains inside the polyhedral set of the state space described by

S x = x ∈ R nx : X (k) x ≤ 1, k ∈ I q , (18)
where the given matrix X ∈ R q×nx characterizes the state constraint domain of system [START_REF] Hwang | Path tracking of an autonomous ground vehicle with different payloads by hierarchical improved fuzzy dynamic sliding-mode control[END_REF]. Remark in Property 1 that minimizing the L ∞ -norm upper bound γ leads to a better control performance.

The following technical lemma is useful to deal with the dead-zone nonlinearity ψ(•).

Lemma 1.

[50], [START_REF] Nguyen | LMI-based control synthesis of constrained T-S systems subject to L 2 or L∞ disturbances[END_REF] Let us consider a MF-dependent matrix

M (h) = N i=1 h i (ξ)M i ∈ R nu×nx ,
and the following set:

S u = x ∈ R nx : M (h)Q(h) -1 x ū . If x ∈ S u ⊂ R nx , then ψ(u) S(h) -1 u -ψ(u) + M (h)Q(h) -1 x ≥ 0, (19)
for S(h) = N i=1 h i (ξ)S i , and S i ∈ R nu×nu , for ∀i ∈ I r , are any diagonal positive definite matrices.

B. LMI-Based Non-PDC Output Feedback Control Design

The following theorem provides sufficient conditions to design a SOF controller ( 14) that can solve Problem 1.

Theorem 1. Consider the fuzzy system [START_REF] Zhao | Design of a control system for an autonomous vehicle based on adaptive-PID[END_REF] with w ∈ W ∞ φ , and a state constraint domain S x defined in [START_REF] Lee | Synthesis of robust lane keeping systems: Impact of controller and design parameters on system performance[END_REF]. If there exist a positive definite MF-dependent matrix

Q(h) ∈ R nx×nx , a diagonal positive definite MF-dependent matrix S(h) ∈ R nu×nu , MF-dependent matrices M (h) ∈ R nu×nx , F (h) ∈ R nu×ny , G(h) ∈ R ny×ny , K(h) ∈ R nu×nw , and positive scalars , γ, α, τ , ρ such that α -τ φ > 0, (20) Q(h) M (h) (l) ū2 l 0, ∀l ∈ I nu , (21)
Q(h) X (k) Q(h) 1 0, ∀k ∈ I q , (22)
Q(h) D(h)Q(h) γI 0, (23)
  Υ ρH(h) -ρI L(h) 0 -ρI   ≺ 0, (24)
for all h(ξ(κ)), h(ξ(κ + 1)) ∈ Ω, where

Υ =       (α -1)Q(h) Υ 21 -2S(h) 0 0 -τ I Υ 41 Υ 42 Υ 43 Υ 44 Υ 51 F (h) 0 Υ 54 Υ 55       , L(h) =   L(h)Q(h) 0 0 0 0 Θ 1 (h)C(h) Θ 2 (h) N (h)K(h) 0 0 0 0 0 0 Θ 1 (h)   , H(h) =   0 0 0 H(h) 0 0 0 0 H(h) 0 0 0 0 H(h) 0   , (25)
and

Υ 21 = F (h)C(h) + M (h), Θ 1 (h) = N (h)F (h), Υ 41 = A(h)Q(h) + B(h)F (h)C(h), Υ 42 = -B(h)S(h), Υ 43 = E(h) + B(h)K(h), Θ 2 (h) = -N (h)S(h), Υ 44 = -Q(h + ) = - N i=1 h i (ξ(κ + 1))Q i , Υ 51 = C(h)Q(h) -G(h)C(h), Υ 54 = F (h) B(h) , Υ 55 = -G(h) + G(h) .
Then, the non-PDC output feedback controller ( 14) guarantees the two closed-loop properties defined in Problem 1 for the fuzzy system [START_REF] Hwang | Path tracking of an autonomous ground vehicle with different payloads by hierarchical improved fuzzy dynamic sliding-mode control[END_REF].

Proof. Note that inequality [START_REF] Suh | Stochastic model predictive control for lane change decision of automated driving vehicles[END_REF] implies G(h) + G(h) 0. This guarantees that the matrix G(h) is nonsingular, thus the validity of the control expression [START_REF] Chen | Hierarchical coordinated control method for unmanned robot applied to automotive test[END_REF].

Applying Schur complement lemma [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF], we can prove that (24) is equivalent to

Υ + ρH(h)H(h) + ρ -1 L(h) L(h) ≺ 0. (26)
Denote ∆(κ) = diag(∆(κ), ∆(κ), ∆(κ)). Since ∆ ∆ I, using the following matrix fact:

X Y + Y X ρX X + ρ -1 Y Y,
with X = H(h) and Y = ∆L(h), it follows from (26) that

Υ + H(h)∆L(h) + L(h) ∆ H(h) ≺ 0. ( 27
)
From the definitions of Υ, H(h) and L(h) in [START_REF] Wang | Crash mitigation in motion planning for autonomous vehicles[END_REF], inequality ( 27) can be exactly rewritten in the form

      (α -1)Q(h) Υ 21 -2S(h) 0 0 -τ I Υ41 Υ42 Υ43 Υ 44 Υ 51 F (h) 0 Υ54 Υ 55       ≺ 0, (28)
where

Υ41 = Â(h)Q(h) + B(h)F (h)C(h), Υ42 = -B(h)S(h), Υ43 = E(h) + B(h)K(h), Υ54 = F (h) B(h) .
Multiplying inequality [START_REF] Guo | Simultaneous trajectory planning and tracking using an MPC method for cyberphysical systems: A case study of obstacle avoidance for an intelligent vehicle[END_REF] by

    I 0 0 0 0 0 I 0 0 F (h)G(h) -1 0 0 I 0 0 0 0 0 I B(h)F (h)G(h) -1     ,
on the left and its transpose on the right leads to

    (α -1)Q(h) W 1 (h) -2S(h) 0 0 -τ I W 2 (h) Υ42 Υ43 Υ 44     ≺ 0, (29)
with

W 1 (h) = F (h)G(h) -1 C(h)Q(h) + M (h), W 2 (h) = Â(h)Q(h) + B(h)F (h)G(h) -1 C(h)Q(h).
Pre-and postmultiplying (29

) by diag(Q(h) -1 , S(h) -1 , I, I) leads to     (α -1)Q(h) -1 W 3 (h) -2S(h) -1 0 0 -τ I A cl (h) -B(h) Υ43 Υ 44     ≺ 0, (30)
with

W 3 (h) = S(h) -1 F (h)G(h) -1 C(h) + M (h)Q(h) -1 .
Applying Schur complement lemma, we can easily prove that (30) is equivalent to

W Q(h + ) -1 W -   (1 -α)Q(h) -1 -W 3 2S(h) -1 0 0 τ I   ≺ 0, (31)
with

W = A cl (h) -B(h) E cl (h) . Multiplying inequal- ity (31) by x ψ(u)
w on the left and its transpose on the right, and using the control expression ( 14) and the closed-loop expression [START_REF] Hwang | Path tracking of an autonomous ground vehicle with different payloads by hierarchical improved fuzzy dynamic sliding-mode control[END_REF], we can obtain

∆V + 2ψ(u) S(h) -1 u -ψ(u) + M (h)Q(h) -1 x + αV(x) -τ w w < 0. ( 32
)
By Lemma 1, it follows from (32) that

∆V + αV(x) -τ w w < 0, ∀x ∈ L V \{0}. ( 33
)
Note that inequality [START_REF] Amer | Modelling and control strategies in path tracking control for autonomous ground vehicles: a review of state of the art and challenges[END_REF] follows directly from ( 20) and [START_REF] Wu | The heterogeneous systems integration design and implementation for lane keeping on a vehicle[END_REF]. Moreover, pre-and postmultiplying [START_REF] Chu | Active disturbance rejection control applied to automated steering for lane keeping in autonomous vehicles[END_REF] by diag(Q(h) -1 , I), then applying Schur complement lemma [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF], we obtain

D(h) D(h) γQ(h) -1 . ( 34
)
It follows directly from [START_REF] Nguyen | Fuzzy control systems: Past, present and future[END_REF] that

z z = x D(h) D(h)x ≤ γx Q(h) -1 x ≤ γ, ∀x ∈ L V ,
which means that the L ∞ -norm of the controlled output z is bounded by γ. As a result, Property 1 is proved. Pre-and postmultiplying [START_REF] Jeong | Nonlinear hybrid impedance control for steering control of rack-mounted electric power steering in autonomous vehicles[END_REF] by diag(Q(h) -1 , I) yields

Q(h) -1 M (h) (l) Q(h) -1 ū2 l 0, l ∈ I nu . ( 35
)
By Schur complement lemma, it is shown that inequality [START_REF] Nguyen | Constrained output-feedback control for discrete-time fuzzy systems with local nonlinear models subject to state and input constraints[END_REF] is equivalent to [START_REF] Barbosa | Robust path-following control for articulated heavy-duty vehicles[END_REF] holds. Following the similar arguments, we can show that [START_REF] Yang | Optimal fault-tolerant pathtracking control for 4WS4WD electric vehicles[END_REF] guarantees that L V ⊆ S x . Hence, Property 2 is also proved, thus the proof can be concluded.

x Q(h) -1 x ≥ 1 ū2 l x U Ux, ∀x ∈ R nx , ∀l ∈ I nu ,( 36
)
with U = M (h) (l) Q(h) -1 . Since V(x, h) ≤ 1, for ∀h ∈ Ω, it follows from (36) that L V ⊆ S u . Hence, if x ∈ L V , then inequality (
Theorem 1 cannot be directly used for the control design due to the presence of h(ξ(κ)), h(ξ(κ + 1)) ∈ Ω in the matrix inequalities ( 21)- [START_REF] Suh | Stochastic model predictive control for lane change decision of automated driving vehicles[END_REF]. We derive in the following theorem a numerically tractable solution for Problem 1.

Theorem 2. Consider the fuzzy system [START_REF] Zhao | Design of a control system for an autonomous vehicle based on adaptive-PID[END_REF] with w ∈ W ∞ φ , and a state constraint domain S x defined in [START_REF] Lee | Synthesis of robust lane keeping systems: Impact of controller and design parameters on system performance[END_REF]. If there exist positive definite matrices

Q i ∈ R nx×nx , diagonal positive definite matrices S i ∈ R nu×nu , matrices M i ∈ R nu×nx , F i ∈ R nu×ny , G i ∈ R ny×ny , K i ∈ R nu×nw , for i ∈ I r ,
and positive scalars , γ, α, τ , ρ such that the optimization (37) is achievable. Then, the non-PDC static output feedback controller [START_REF] Chen | Hierarchical coordinated control method for unmanned robot applied to automotive test[END_REF] guarantees the closed-loop properties defined in Problem 1 for the fuzzy system [START_REF] Hwang | Path tracking of an autonomous ground vehicle with different payloads by hierarchical improved fuzzy dynamic sliding-mode control[END_REF]. Moreover, the L ∞ -norm of the controlled output is minimized.

min νi, i∈Ir γ (37)
s.t. ( 20), ( 38), ( 39), ( 40) and [START_REF] Nam | Estimation of sideslip and roll angles of electric vehicles using lateral tire force sensors through RLS and Kalman filter approaches[END_REF] where

ν i = ( , γ, α, τ, ρ, Q i , S i , M i , F i , G i , K i ) and Q i M i(l) ū2 l 0, ∀l ∈ I nu , ∀i ∈ I r , (38)
Q i X (k) Q i 1 0, ∀k ∈ I q , ∀i ∈ I r , (39)
Q j D i Q j γI 0, ∀i ∈ I r , ∀j ∈ I r , (40)
Ψ iikq ≺ 0, Ψ ijkq + Ψ jikq ≺ 0, ∀i, j, k, q ∈ I r , i < j. ( 41
)
The quantity Ψ ijkq in ( 41) is defined as

Ψ ijkq =   Υ ijkq ρH i -ρI L ijq 0 -ρI   ,
with

H i =   0 0 0 H i 0 0 0 0 H i 0 0 0 0 H i 0   , L ijq =   L i Q j 0 0 0 0 N i F j C q -N i S j N i K j 0 0 0 0 0 0 N i F j   , Υ ijkq =       Γ 11 Γ 21 -2S j 0 0 -τ I Γ 41 -B i S j Γ 43 -Q k Γ 51 F j 0 F j B i Γ 55      
, and

Γ 11 = (α -1)Q j , Γ 21 = F j C q + M j , Γ 41 = A i Q j + B i F j C q , Γ 43 = E i + B i K j , Γ 51 = C q Q j -G j C q , Γ 55 = -(G j + G j ).
Proof. Multiplying ( 38) by h i (ξ) ≥ 0 and summing up for ∀i ∈ I r , we obtain [START_REF] Jeong | Nonlinear hybrid impedance control for steering control of rack-mounted electric power steering in autonomous vehicles[END_REF]. Similarly, we can prove that ( 39) implies [START_REF] Yang | Optimal fault-tolerant pathtracking control for 4WS4WD electric vehicles[END_REF]. Moreover, since h(ξ) = h(ξ(κ)) ∈ Ω and h(ξ

+ ) = h(ξ(κ + 1)) ∈ Ω, it follows from (41) that r i=1 r k=1 r q=1 h i (ξ) 2 h k (ξ + )h q (ξ)Ψ iikq + r i=1 r i<j r k=1 r q=1 h i (ξ)h j (ξ)h k (ξ + )h q (ξ)(Ψ ijkq + Ψ jikq ) = r i=1 r j=1 r k=1 r q=1 h i (ξ)h j (ξ)h k (ξ + )h q (ξ)Ψ ijkq ≺ 0. ( 42
)
Note that ( 42) is exactly [START_REF] Suh | Stochastic model predictive control for lane change decision of automated driving vehicles[END_REF]. Hence, inequality (41) implies [START_REF] Suh | Stochastic model predictive control for lane change decision of automated driving vehicles[END_REF]. Similarly, it is shown that ( 40) implies [START_REF] Chu | Active disturbance rejection control applied to automated steering for lane keeping in autonomous vehicles[END_REF]. Then, by the results of Theorem 1, the proof can be concluded.

Remark 2. The control design procedure in Theorem 2 is expressed as an optimization problem under a set of LMI constraints with a line search over the parameter . The control feedback gains F i , G i and K i , for ∀i ∈ I r , can be easily computed using Matlab software with Yalmip toolbox.

C. Application to Fuzzy-Model-Based Path Tracking Control 1) Vehicle Fuzzy Modeling: Note that the vehicle lateral dynamics [START_REF] Alcala | Autonomous vehicle control using a kinematic Lyapunov-based technique with LQR-LMI tuning[END_REF] and the controlled output z in (8) depend on the speed terms v x , 1 vx and 1 v2

x , which are measured and bounded

v min ≤ v x ≤ v max , v min = 5 [m/s], v max = 30 [m/s]. If v x , 1 vx and 1 v 2
x are independently considered as premise variables, then the sector nonlinearity approach [29, Chapter 2] leads to a T-S fuzzy representation with 2 3 = 8 fuzzy rules 2 . This fuzzy representation would be complex and conservative for the real-time vehicle control design due to the strong interdependency of these three terms. To overcome this drawback, the following change of premise variable is performed [START_REF] Nguyen | Driver-automation cooperative approach for shared steering control under multiple system constraints: Design and experiments[END_REF]:

1 v x = 1 v 0 + 1 v 1 ξ, (43)
where the new time-varying parameter θ is used to describe the variation of v x between its lower bound v min and upper bound v max , with ξ min ≤ ξ ≤ ξ max , ξ min = -1, ξ max = 1.

Then, using the Taylor's approximation, it follows that

v x v 0 1 - v 0 v 1 ξ , 1 v 2 x 1 v 2 0 1 + 2 v 0 v 1 ξ , (44)
where

v 0 = 2v min v max v min + v max , v 1 = 2v min v max v min -v max .
Substituting ( 43) and ( 44) into ( 7), we can obtain the following vehicle dynamics:

x(κ + 1) = Â(ξ)x(κ) + B(ξ)sat(u(κ)) + E(ξ)w(κ), z(κ) = D(ξ)x(κ), y(κ) = Cx(κ), (45)
which depends linearly on the new parameter ξ. Using the sector nonlinearity approach [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF], the vehicle model ( 45) can be exactly represented in the following T-S fuzzy form:

x(κ + 1) = 2 i=1 h i (ξ)( Âi x(κ) + Bi sat(u(κ)) + E i w(κ)), z(κ) = 2 i=1 h i (ξ)D i x(κ), y(κ) = Cx(κ), (46)
where the scalar membership functions are defined as

h 1 (ξ) = ξ max -ξ 2 , h 2 (ξ) = 1 -h 1 (ξ).
The corresponding state-space matrices are given by

Π 1 = Π(ξ min ), Π 2 = Π(ξ max ), with Π ∈ Â, B, C, D, E .
Remark 3. Using the variable change [START_REF] Nguyen | Unknown input observers for simultaneous estimation of vehicle dynamics and driver torque: Theoretical design and hardware experiments[END_REF] and the Taylor's approximation [START_REF] Li | Integrated longitudinal and lateral tire/road friction modeling and monitoring for vehicle motion control[END_REF] allows to reduce the number of fuzzy rules from eight to two. This significantly decreases not only the design conservatism but also the numerical complexity for real-time control implementation.

2) Vehicle Design Constraints: First, the steering input must be limited to guarantee not only the driving comfort but also the vehicle closed-loop stability during some specific maneuvers [START_REF] Nguyen | Fuzzy steering control for autonomous vehicles under actuator saturation: Design and experiments[END_REF]. For the studied autonomous vehicle, the control input limitation δ max = 10 [deg] is the maximal steering angle. Second, the trust driving region should be also taken into account in the control design to improve the safety and comfort [START_REF] Funke | Collision avoidance and stabilization for autonomous vehicles in emergency scenarios[END_REF], [START_REF] Nguyen | Driver-automation cooperative approach for shared steering control under multiple system constraints: Design and experiments[END_REF]. Here, this region can be characterized by the vehicle state constraints as follows.

• The front wheels can be guaranteed to remain inside the road lane with the condition

- (2d -a) 2 ≤ y L + (l f -l s ) ψ L ≤ (2d -a) 2 , (47)
where a = 1.5 [m] is the vehicle width, and d = 1.5 [m] the lane width. • The following state bounds represent the normal operating conditions of the vehicle:

|β| ≤ 0.05 [rad], |r| ≤ 0.55 [rad/s], |ψ L | ≤ 0.1 [rad], |y L | ≤ 1 [m]. (48)
The state constraints ( 47) and ( 48) can be reformulated in the form [START_REF] Lee | Synthesis of robust lane keeping systems: Impact of controller and design parameters on system performance[END_REF] for the control design.

IV. ILLUSTRATIVE RESULTS AND DISCUSSIONS

This section presents the validation of the proposed fuzzy SOF controller with the high-fidelity CarSim vehicle simulator. To show the practical control performance, we consider dynamic driving scenarios such as race track with variable road curvatures and double lane change (DLC) along with highly variable road friction and speed conditions. The parameters of the CarSim vehicle model considered are

M = 1653 [kg], C f = 95000 [N/rad], C r = 85500 [N/rad], l f = 1.4 [m], l r = 1.646 [m], and I z = 2765 [kgm 2 ]
. The steer ratio between the driver wheel and the front road steer angle is R s = 17.5. The look-ahead distance to compute the tracking errors is l s = 5 [m]. Note that in CarSim vehicle simulator, based on the sensor information of the driver preview points, the look-ahead distance and lateral error y L can be easily obtained. To consider the variable friction in the control design, we assume that the parametric uncertainties in the front and rear tire stiffness coefficients are of 15 %, which represents a highly nonlinear behavior of the lateral tire forces.

Solving the LMI-based conditions in Theorem 2 with a decay rate α = 0.01, the following control solution is obtained: The performance of the proposed fuzzy SOF controller is now evaluated for two dynamic driving scenarios: race course track, and DLC at high speeds. Note that these scenarios are much challenging compared to those considered in [START_REF] Zhang | A novel fuzzy observer-based steering control approach for path tracking in autonomous vehicles[END_REF].

F 1 = 0.

A. Dynamic Driving: Course Track with Variable Curvatures

For the integrated CarSim/Matlab co-simulations, the path tracking on a race course track has been conducted with road curvatures varying in the range ρ r ∈ [-0.02, 0.04] and the road friction of µ = 0.75, i.e., F yf = 2µC f α f and F yr = 2µC r α r . The vehicle traverses the track at dynamic longitudinal speeds in the range v x ∈ [START_REF] Dahmani | Observer-based state feedback control for vehicle chassis stability in critical situations[END_REF]60] [km/h]. For regulating the longitudinal speed, the inbuilt PI controller from CarSim has been used. The speed tracking and the path following for the considered driving scenario are shown in Fig. 2. Observe in Fig. 2(a) that even without requiring the online measurement of the sideslip angle, the proposed fuzzy non-PDC controller can guarantee an efficient path following under the presence of nonlinear tire forces and uncertainties. Furthermore, the low speed tracking error as depicted in Fig. 2(b) ensures that safe speed limits across a curve are maintained. The lane deviation errors, i.e., lateral error and heading error, are shown in Fig. 3. Remark that the controlled lane errors are within the constraint limits defined in [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. For comparison analysis, the lane tracking errors with the inbuilt look-ahead preview controller (PC-CSIM) of the CarSim simulator is also illustrated in Fig. 3. Compared to the proposed fuzzy static output feedback (TS-SOF) controller, the lane tracking errors are much larger and cross the lane boundaries, especially at tight curves. Hence, for the considered road friction conditions, the proposed fuzzy SOF controller is able to keep the vehicle on lane which is not the case when using the preview PC-CSIM controller. Subsequently, the illustrations for the vehicle states β, r, the designed steering control input δ and the lateral acceleration are shown in Fig. 4 for both PC-CSIM and fuzzy SOF controllers. It can be seen that the designed control is always within the required constraint of 10 [deg] for both approaches. Further, the sideslip angle does not exceed 0.05 [rad], which ensures that the tire forces are not saturated.

With the maximum yaw rate limited by 0.55 [rad/s], the vehicle maintains a stable control performance. In comparison to the PC-CSIM controller, the proposed non-PDC design provide a smooth steering actions, leading to less oscillatory behaviors and a more stable performance in maintaining the state constraints even when the lateral acceleration is high under the considered friction level. II. Note that the proposed TS-SOF controller provides much better path tracking performance for all considered evaluation indexes as shown in Table II. We remark that although the PC-CSIM controller can ensure that the design constraints on the states are maintained, yet the lane errors are quite large indicating a poor tracking performance. However, employing the proposed TS-SOF controller, the vehicle states are constrained and the tracking errors are also small. It should be noted that the maximum value of the controlled sideslip angle is less than 3 degrees even at a peak lateral acceleration of 4.18 [m/s 2 ], which indicates that the tire forces are not saturated. Hence, the proposed fuzzy SOF controller can satisfy the path following objectives in the presence of modeling uncertainties with dynamic driving maneuvers.

B. Extreme Maneuver: DLC at High Speed and Low Friction

To further evaluate the control performance, a dynamic DLC maneuver at high speed of 100 [km/h] has been considered. This maneuver was executed for a road surface with variable friction levels, i.e., µ = 1, µ = 0.75 and µ = 0.5. The corresponding vehicle responses during this maneuver are presented in Fig. 5. We can observe from the responses of the steering input and the sideslip angle that with a decrease in friction conditions, the oscillations in the controlled responses increase. For all friction conditions, the maximum steering input does not violate the imposed design constraint. However, when the friction condition decreases too low, the sideslip angle increases and crosses the constraint of 0.05 [rad]. Note that this friction condition with µ = 0.5 is out of the uncertainty range that can be covered by the proposed fuzzy SOF controller. As can be seen from Fig. [m]. We remark that for all the considered speeds and the friction conditions, the vehicle remains within the lane based on computed values of e RM S . Moreover, due to the extreme DLC at high speeds, the lateral acceleration of the vehicle is very high, i.e., greater than 5 [m/s 2 ]. However, even for such lateral acceleration levels, the proposed fuzzy SOF design still ensures that all the constraints on the steering input and the vehicle states are satisfied with a satisfactory tracking performance. Such control performance behaviors can be exhibited by the proposed design for other extreme driving conditions such as sudden lane change, driving around extreme curve at high speeds, etc. For illustrations, the results of the controlled driving along a 8-shaped track is depicted in Fig. 6 at a speed of 60 [km/h] with a high surface friction of µ = 1. This track consists of sharp curves and can be visualized as combinations of the S-shaped real-world driving scenario. Note that although the controlled steering and the sideslip angles are constrained within limits, yet a steady-state error on the path tracking is observed. This steady-state error arises as the vehicle navigates a sharp curve at high speed. However, the vehicle remains inside the road lane. In general, such a curve transition with high speeds can introduce oscillations in the designed steering input, which may lead to subsequent saturation. However, using the proposed fuzzy SOF control design, the steering input is smooth without any saturation phenomena. Despite a slight increase of the lane tracking error, the smoothness of the steering control in this case is crucial for comfort reasons. V. CONCLUDING REMARKS A new LMI-based control solution for path tracking of autonomous vehicles has been proposed. Using a fuzzy modelbased control framework, the vehicle speed variation and the uncertain behaviors of lateral tire forces are effectively handled. In addition, the physical contraints on the steering input and the vehicle state are considered in the control design though the robust set-invariance property and Lyapunov stability arguments. To reduce the design conservatism, fuzzy Lyapunov functions and a non-PDC control scheme are exploited for theoretical developments. The practical performance of the new SOF path following controller is demonstrated with CarSim/Matlab co-simulations under dynamic driving maneuvers. For future works, we extend the proposed fuzzy SOF scheme to deal with coupled longitudinal-lateral control issues. Experimental validations with real autonomous vehicles under extensive driving conditions will also be considered.

Fig. 1 .

 1 Fig. 1. Vehicle lateral dynamics modeling.

Fig. 2 .

 2 Fig. 2. Tracking performance. (a) Path tracking with the designed fuzzy SOF controller, (b) speed tracking using PI controller from the CarSim simulator.

Fig. 3 .

 3 Fig. 3. Path tracking errors. (a) Lateral error y L , (b) heading error ψ L .

Fig. 4 .

 4 Fig. 4. Vehicle closed-loop behaviors. (a) Sideslip angle β, (b) yaw rate r, (c) designed steering control δ, (d) lateral acceleration ay.

1 Fig. 5 .

 15 Fig. 5. The designed control for a vehicle executing a DLC under various road friction conditions at 100 km/h. (a) Steering control input, (b) lateral acceleration, (c) sideslip angle, (d) lateral tire force.

Fig. 6 .

 6 Fig. 6. Path tracking performance obtained with a 8-shaped track with high vehicle speed and surface friction. (a) Sideslip angle and steering input, (b) Path tracking performance.

  ∞ -gain performance is γ min = 0.2050. Remark that the control gains and the Lyapunov matrices obtained for two linear subsystems of the vehicle fuzzy model[START_REF] Nguyen | Driver-automation cooperative approach for shared steering control under multiple system constraints: Design and experiments[END_REF] are clearly different. This represents the advantage of using the proposed nonquadratic Lyapunov function and non-PDC control scheme to handle a large speed variation.

					4219 -0.1664 -0.6867 ,
		F 2 = -2.8799 -0.1978 -0.8417 ,
					18.1239 -0.5172 -2.5855 
		G 1 =	 -0.0957 0.2480	0.6858	 ,
				-0.8100 0.6855	2.5230
					22.8436 -0.5761 -2.7658 
		G 2 =		0.1206	0.2427	0.5523	 ,
				-0.3020 0.6578	2.0223
	K 1 = -0.0829, K 2 = -1.5143.
	The corresponding Lyapunov matrices are given by
			0.3506	0.2718 -0.0827 -0.3231 
	Q 1 =	   -0.0827 -0.7307 0.2553 0.2718 31.3393 -0.7307 -3.7276 0.7211    ,
		-0.3231 -3.7276 0.7211	2.6953
			0.3301	0.3164 -0.0857 -0.3319 
	Q 2 =	   -0.0857 -0.6205 0.2543 0.3164 26.5060 -0.6205 -3.2835 0.7116    ,
		-0.3319 -3.2835 0.7116	2.6574
	and the minimum L

TABLE II PATH

 II TRACKING PERFORMANCE COMPARISON FOR µ = 0.75.

	Controller y L RM S	r RM S	|β|max	|ay|max	|δ|max
	PC-CSIM	1.125	0.150	0.041	5.17	7.69
	TS-SOF	0.155	0.142	0.039	4.18	6.32

  5(b), such surface conditions lead to high lateral accelerations greater than 6 [m/s 2 ], which indicates a very critical operating condition of the vehicle. With the violation of the sideslip constraint, the tire forces approach the saturation region, which may lead to a loss of vehicle stability. To ascertain the performance of the proposed fuzzy SOF controller with low-friction roads, the results obtained for variable speeds in [km/h] are further presented in Table III with e RM S = y L RM S + (l f -l s )ψ L RM S

TABLE III DLC

 III FOR VARIABLE FRICTION CONDITIONS AT DIFFERENT SPEEDS.

	Friction	vx	e RM S	r RM S	|β|max	|ay|max	|δ|max
	µ = 1	100	0.962	0.132	0.048	7.21	3.81
	µ = 0.85	90	0.866	0.124	0.039	6.48	3.67
	µ = 0.75	80	0.765	0.116	0.032	5.68	3.58
	µ = 0.5	75	0.728	0.114	0.042	5.05	3.86

For instance, the cost of a Correvit optical sensor used to measure the sideslip angle is about 15ke.

Note that the vehicle fuzzy model[START_REF] Wang | Lateral control of autonomous vehicles based on fuzzy logic[END_REF] in[START_REF] Zhang | A novel fuzzy observer-based steering control approach for path tracking in autonomous vehicles[END_REF] is not appropriately represented with four fuzzy rules. Indeed, there are three premise variables therein, i.e., vx,[START_REF] Broggi | Autonomous vehicles control in the VisLab intercontinental autonomous challenge[END_REF] vx and 1 m , thus the sector nonlinearity approach should yield an eight-rule vehicle fuzzy model.

This work is supported in part by the French Ministry of Higher Education and Research, in part by the National Center for Scientific Research (CNRS), in part by the Nord-Pas-de-Calais Region under the project ELSAT 2020, in part by the Brazilian agencies CNPq (Grant Number: 307933/2018-0) and FAPEMIG (Grant Number: PPM-00053-17). A.-T. Nguyen and T.-M. Guerra are with the LAMIH laboratory UMR


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

Thierry-Marie Guerra

Víctor Campos

Anh-Tu Nguyen

Member, IEEE Thierry-Marie Guerra

A.-T Nguyen

T.-M Guerra

Anh-Tu Nguyen

Simultaneous Estimation of State and Unknown Input with L∞ Guarantee on Error-Bounds for Fuzzy Descriptor Systems

Keywords: Fuzzy models, fuzzy descriptor systems, fuzzy observers, unknown inputs, Lyapunov method

This paper proposes a new l∞ observer design for fuzzy descriptor systems with unknown inputs. The descriptor form is treated using a singular redundancy system representation. To keep the consistency of the resulting fuzzy observer structure, we make use of a virtual variable playing the role of the one-step ahead state estimate. As a result, the observer gain can be constructed with free-structure decision variables to reduce the design conservatism. Using a membership-function-dependent Lyapunov function, the observer design is reformulated as a convex optimization problem with a single line search parameter. In particular, the error bounds of both the state and the unknown input estimations can be minimized through the guaranteed l∞ performance level. The effectiveness of our result is demonstrated with a challenging real-world application on robot manipulators.

I. INTRODUCTION

T AKAGI-Sugeno (T-S) fuzzy model-based approach has become an effective tool to deal with nonlinear systems [START_REF] Nguyen | Fuzzy control systems: Past, present and future[END_REF]. Over the last decades, researchers have paid a great deal of attention to the analysis and synthesis issues of T-S fuzzy systems [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]- [START_REF] Nguyen | Static output feedback design for a class of constrained Takagi-Sugeno fuzzy systems[END_REF]. Since T-S fuzzy systems are constructed from local linear models blending with nonlinear membership functions (MFs), design conditions for fuzzy controllers/observers can be derived using Lyapunov method and linear matrix inequality (LMI) formulations, see [START_REF] Nguyen | Fuzzy control systems: Past, present and future[END_REF], [START_REF] Tanaka | A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions[END_REF]- [START_REF] Su | A novel approach to output feedback control of fuzzy stochastic systems[END_REF] and related references therein.

A large class of physical systems can be naturally represented using descriptor models [START_REF] Tanaka | A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions[END_REF], [START_REF] Xu | Robust Control and Filtering of Singular Systems[END_REF]. The descriptor forms appear in various engineering applications including robotics [START_REF] Spong | Robot Modeling and Control[END_REF], healthcare devices [START_REF] Guelton | An alternative to inverse dynamics joint torques estimation in human stance based on a Takagi-Sugeno unknown-inputs observer in the descriptor form[END_REF], transportation and power systems [START_REF] Xu | Robust Control and Filtering of Singular Systems[END_REF] and so on. Fundamental results on analysis and design of descriptor systems have been extensively proposed for linear cases, see [START_REF] Xu | Robust Control and Filtering of Singular Systems[END_REF], [START_REF] Chadli | Novel bounded real lemma for discretetime descriptor systems: Application to H∞ control design[END_REF]- [START_REF] Wang | H -/H∞ fault detection observer in finite frequency domain for linear parameter-varying descriptor systems[END_REF] and related references. Various extensions to nonlinear/fuzzy descriptor systems have been also reported [START_REF] Li | Fuzzy-model-based D-stability and nonfragile control for discrete-time descriptor systems with multiple delays[END_REF], [START_REF] Hassan | H∞ observers design for a class of nonlinear time-delay systems in descriptor form[END_REF]- [START_REF] Guerra | Observer design for Takagi-Sugeno descriptor models: An LMI approach[END_REF]. Note that for fuzzy-model-based approaches, exploiting the descriptor forms allows reducing both the number of fuzzy rules and that of LMI-based design conditions [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. As a result, the computational burden and the design conservatism can be significantly reduced [START_REF] Tanaka | A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions[END_REF], [START_REF] Guerra | Observer design for Takagi-Sugeno descriptor models: An LMI approach[END_REF].

Up to now, existing results on T-S fuzzy systems are mostly concerned with known inputs [START_REF] Chadli | Robust observer design for unknown inputs Takagi-Sugeno models[END_REF], especially in cases of fuzzy descriptor forms [START_REF] Tanaka | A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions[END_REF], [START_REF] Li | Fuzzy-model-based D-stability and nonfragile control for discrete-time descriptor systems with multiple delays[END_REF], [START_REF] Guerra | Observer design for Takagi-Sugeno descriptor models: An LMI approach[END_REF]. However, in many practical situations, all input signals cannot be known [START_REF] Marx | Design of observers for T-S descriptor systems with unknown inputs and application to fault diagnosis[END_REF]. Moreover, unmodeled dynamics, uncertain disturbances and faults in engineering systems can be all considered as unknown inputs (UIs) [START_REF] Zhang | Reduced-order observer design for switched descriptor systems with unknown inputs[END_REF]. Therefore, designing UI observers is of crucial importance from both theoretical and practical viewpoints, especially in the areas of fault diagnosis [START_REF] Wang | H -/H∞ fault detection observer in finite frequency domain for linear parameter-varying descriptor systems[END_REF], [START_REF] Marx | Design of observers for T-S descriptor systems with unknown inputs and application to fault diagnosis[END_REF], [START_REF] Zhang | Reduced-order observer design for switched descriptor systems with unknown inputs[END_REF], [START_REF] Jia | Fault reconstruction and faulttolerant control via learning observers in Takagi-Sugeno fuzzy descriptor systems with time delays[END_REF], and fault-tolerant control [START_REF] Jia | Fault reconstruction and faulttolerant control via learning observers in Takagi-Sugeno fuzzy descriptor systems with time delays[END_REF]- [START_REF] Yin | Sliding mode observer-based FTC for Markovian jump systems with actuator and sensor faults[END_REF]. Unfortunately, due to the technical difficulties in dealing with the descriptor form, the design of UI observers for fuzzy descriptor systems still remains challenging [START_REF] Jia | Fault reconstruction and faulttolerant control via learning observers in Takagi-Sugeno fuzzy descriptor systems with time delays[END_REF]. For instance, some matrix-rank conditions are required for the UI decoupling in [START_REF] Marx | Design of observers for T-S descriptor systems with unknown inputs and application to fault diagnosis[END_REF], [START_REF] Jia | Fault reconstruction and faulttolerant control via learning observers in Takagi-Sugeno fuzzy descriptor systems with time delays[END_REF], which can be restrictive for fuzzy descriptor observer design. The singular redundancy representation approach in [START_REF] Guelton | An alternative to inverse dynamics joint torques estimation in human stance based on a Takagi-Sugeno unknown-inputs observer in the descriptor form[END_REF] requires a special Luenberger-like observer structure, leading to a conservative design. This paper aims to avoid these technical difficulties. Our main contributions are summarized as follows.

1) A singular redundancy approach is exploited to deal with the descriptor form for observer design in presence of UIs. We introduce a virtual variable, playing the role of the one-step ahead state estimate, to guarantee the consistency of the resulting observers. Hence, together with specific convexification techniques, the proposed observer gains can be of free structures to reduce the conservatism. Moreover, our fuzzy observer design does not require any specific rank condition for UI decoupling. 2) Using Lyapunov-based arguments, the observer design procedure is recast as a convex optimization problem with a single line search parameter, which can be effectively solved with semidefinite programming [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. In particular, the estimation errors of both the state and the UI can be minimized through a guaranteed l ∞ performance level.

Notation: N is the set of non-negative integers, and I p = {1, 2, ..., p} ⊂ N. For a matrix X, X denotes its transpose, X 0 means that X is positive definite, HeX = X +X , and X denotes its maximum singular value. For a vector x ∈ R n , we denote its 2-norm as x = √ x x. For a sequence of vectors {x k } k∈N (respectively matrices {X k } k∈N ), we denote

x l∞ = sup k≥0 x k (respectively X ∞ = sup k≥0 X k ). Then, {x k } k∈N ∈ l ∞ if x l∞ < ∞ and {X k } k∈N ∈ L ∞ if X ∞ < ∞. diag(X 1 , X 2 ) denotes a block-diagonal matrix composed of X 1 , X 2 .
I is the identity matrix of appropriate dimension. " " stands for the terms deduced by symmetry.

II. PROBLEM FORMULATION

This section formulates the l ∞ observer design of fuzzy descriptor systems using a singular redundancy approach.

A. Description of Fuzzy Descriptor Systems

T-S fuzzy systems are expressed by fuzzy IF-THEN rules, which locally represent linear input-output relations of nonlinear systems. A class of T-S fuzzy descriptor systems with r inference rules can be described as follows [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]:

RULE R i : IF ξ 1k is M i 1 and . . . and ξ pk is M i p THEN E i x k+1 = A i x k + B i u k + D i d k y k = C i x k (1)
where , i ∈ I r , where µ i j (ξ j ) represents the membership grade of ξ j in the respective fuzzy set M i j . Note that the MFs satisfy the following property:

x k ∈ R nx is the state vector, u k ∈ R nu the known input, d k ∈ R n d the unknown input, and y k ∈ R ny the measured output. ξ k = [ξ 1k • • • ξ pk ] ∈ R p is
r i=1 h i (ξ) = 1, 0 ≤ h i (ξ) ≤ 1, ∀i ∈ I r . (2)
Let Ω be the set of MFs satisfying [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. Using the center-ofgravity method for defuzzification, the fuzzy descriptor system (1) can be represented in the following compact form [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]:

E(h)x k+1 = A(h)x k + B(h)u k + D(h)d k y k = C(h)x k (3)
where

Π(h) = r i=1 h i (ξ)Π i , Π ∈ {A, B, C, D, E}
and h = (h 1 (ξ), h 2 (ξ), . . . , h r (ξ)) ∈ Ω. We assume that E(h) is a nonsingular matrix for ∀h ∈ Ω. Note that various nonlinear systems in mechatronics and robotics engineering can be modeled as in [START_REF] Mozelli | A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems[END_REF], where E(h) includes the inertia matrix [START_REF] Spong | Robot Modeling and Control[END_REF], and is thus invertible [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF], [START_REF] Guelton | An alternative to inverse dynamics joint torques estimation in human stance based on a Takagi-Sugeno unknown-inputs observer in the descriptor form[END_REF]. Assume also that D(h) is of full-column rank for ∀h ∈ Ω, and {d k } k∈N ∈ l ∞ .

Remark 1. For a given nonlinear descriptor system, the sector nonlinearity approach [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] can be used to derive an exact fuzzy model (3) of this system on a compact set of the state space.

Let us define xk = x k x k+1 . To deal with the descriptor matrix E(h) on the left-hand side, we rewrite system (3) in the following equivalent form:

Ē xk+1 = Ā(h)x k + B(h)u k + D(h)d k y k = C(h)x k (4)
with

Ē = I 0 0 0 , Ā(h) = 0 I A(h) -E(h) , B(h) = 0 B(h) , D(h) = 0 D(h) , C(h) = C(h) 0 .
Remark 2. The singular redundancy form (4) has been exploited for continuous-time fuzzy descriptor systems [START_REF] Tanaka | A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions[END_REF], [START_REF] Guelton | An alternative to inverse dynamics joint torques estimation in human stance based on a Takagi-Sugeno unknown-inputs observer in the descriptor form[END_REF]. However, control and observer designs based on this form have not been yet reported for the discrete-time counterparts. Note that the discrete-time case requires a significantly different treatment compared to the continuous-time results [START_REF] Xu | Robust Control and Filtering of Singular Systems[END_REF].

As in the continuous-time case, the representation (4) would naturally lead to the following fuzzy observer form [START_REF] Guelton | An alternative to inverse dynamics joint torques estimation in human stance based on a Takagi-Sugeno unknown-inputs observer in the descriptor form[END_REF]:

I 0 0 0 xk+1 xk+2 = 0 I A(h) -E(h) xk xk+1 + 0 B(h) u k + M 1 (h) M 2 (h) C(h) 0 x k -xk x k+1 -xk+1 (5)
where

M (h) = M 1 (h) M 2 (h)
is the membershipfunction-dependent observer gain, and xk is the estimate of the state x k . However, it follows from the first row of ( 5) that

xk+1 ≡ xk+1 + M 1 (h)C(h)(x k -xk ). (6)
To guarantee the consistency of equation ( 6) for ∀k ∈ N, and for any initial condition of the system state x 0 ∈ R nx , one should impose M 1 (h) = 0. This may lead to a conservative observer design [START_REF] Guerra | Observer design for Takagi-Sugeno descriptor models: An LMI approach[END_REF]. To avoid any structure imposition on the observer gain, we consider the following fuzzy observer form:

Ē xk+1 = Ā(h) xk + B(h)u k + G(h) -1 L(h) (y k -ŷk ) ŷk = C(h) xk (7)
with xk = x k ψ k . The membership-function-dependent observer gains to be determined

G(h) = r i=1 h i (ξ k )G i , L(h) = r i=1 h i (ξ k )L i
are of appropriate dimensions. The existence of G(h) -1 is further discussed in Theorem 1.

Remark 3. The key idea of the observer form ( 7) is to exploit the new virtual variable ψ k as an estimate of x k+1 to guarantee the consistency of this form without requiring any special structure of the observer gains G(h) and L(h).

Let us define the state estimation errors as

e k = x k -xk , ēk = xk -xk = e k x k+1 -ψ k .
From ( 4) and ( 7), the error dynamics is given by

Ēē k+1 = A(h)ē k + D(h)d k (8)
where

A(h) = Ā(h) -G(h) -1 L(h) C(h).
We are now in the position to formulate the observer design problem.

Problem 1. Given a fuzzy descriptor system (3). Determine the observer gains G(h) and L(h) such that the error dynamics (8) verifies the following properties. (P1) If d k = 0, ∀k ∈ N, the error dynamics ( 8) is globally exponentially stable with respect to the origin. (P2) If d k = 0, ∀k ∈ N, the estimation error e k is uniformly bounded for any initial condition e 0 and any input sequence {d k } k∈N ∈ l ∞ . In addition, we have the property

lim k→∞ sup e k < γ d l∞ , γ > 0 (9)
where the l ∞ -gain γ is specified in Theorem 1. Moreover, for e 0 = 0, it follows that e k < γ d l∞ , ∀k ∈ N.

The error dynamics (8) verifying the properties (P1)-(P2) is said to be globally uniformly l ∞ -stable with a performance level γ. A similar concept of l ∞ -stablity can be found in [START_REF] Chakrabarty | State and unknown input observers for nonlinear systems with bounded exogenous inputs[END_REF].

III. LMI-BASED SYNTHESIS OF FUZZY DESCRIPTOR OBSERVERS WITH l ∞ PERFORMANCE GUARANTEE
This section provides a numerically tractable solution to design an l ∞ observer for the descriptor fuzzy system (3). We begin with the following theorem providing sufficient conditions to guarantee the global uniform l ∞ -stability of the error dynamics [START_REF] Su | A novel approach to output feedback control of fuzzy stochastic systems[END_REF].

Theorem 1. Consider the error dynamics [START_REF] Su | A novel approach to output feedback control of fuzzy stochastic systems[END_REF]. Given a scalar α ∈ (0, 1), if there exist membership-function-dependent symmetric matrices

P (h) ∈ R nx×nx , R(h) ∈ R nx×nx , matrices Q(h) ∈ R nx×nx , G 1 (h) ∈ R nx×nx , G 2 (h) ∈ R nx×nx , G 3 (h) ∈ R nx×nx , G 4 (h) ∈ R nx×nx , L 1 (h) ∈ R nx×ny , L 2 (h) ∈ R nx×ny ,
and positive scalars τ , γ such that

Σ + Φ + Φ ≺ 0 (10) P (h) Q(h) R(h) 0 (11)
P (h) I γI 0 (12)
for any h, h + ∈ Ω, where

Σ =       (α -1)P (h) 0 0 0 0 P (h + ) 0 0 Q(h + ) R(h + ) 0 0 0 0 -αγI       Φ =       Γ Υ -G 1 (h) -G 2 (h) G 2 (h)D(h) Ξ Φ -G 3 (h) -G 4 (h) G 4 (h)D(h) τ Γ τ Υ -τ G 1 (h) -τ G 2 (h) τ G 2 (h)D(h) τ Ξ τ Φ -τ G 3 (h) -τ G 4 (h) τ G 4 (h)D(h) 0 0 0 0 0       Γ = G 2 (h)A(h) -L 1 (h)C(h), Υ = G 1 (h) -G 2 (h)E(h) Ξ = G 4 (h)A(h) -L 2 (h)C(h), Φ = G 3 (h) -G 4 (h)E(h) h + = (h 1 (ξ k+1 ), h 2 (ξ k+1 ), . . . , h r (ξ k+1 )).
Then, the error dynamics ( 8) is globally uniformly l ∞ -stable with performance level γ. Moreover, the fuzzy observer structure corresponding to the initial system (3) is given by

E(h)x k+1 = A(h)x k + B(h)u k + E(h) I K(h) (y k -ŷk ) ŷk = C(h)x k (13)
with

K(h) = G(h) -1 L(h) and
G(h) = G 1 (h) G 2 (h) G 3 (h) G 4 (h) , L(h) = L 1 (h) L 2 (h) . (14)
Proof. To begin with, let us define

P (h) = P (h) Q(h) R(h) , K(h) = K 1 (h) K 2 (h) . (15)
It is important to note that inequality [START_REF] Spong | Robot Modeling and Control[END_REF] implies

P (h + ) -τ G(h) + G(h) ≺ 0. (16)
Since P (h + ) 0 due to [START_REF] Guelton | An alternative to inverse dynamics joint torques estimation in human stance based on a Takagi-Sugeno unknown-inputs observer in the descriptor form[END_REF], inequality [START_REF] Marx | Design of observers for T-S descriptor systems with unknown inputs and application to fault diagnosis[END_REF] guarantees the nonsingularity of G(h), thus the existence of G(h) -1 .

For the observer design, let us consider the following membership-function-dependent Lyapunov function:

V k = ē k Ē P (h) Ēē k . ( 17
)
Due to the special structure of Ē, it follows that Ē P (h) Ē = I 0 0 0

P (h) Q(h) R(h) I 0 0 0 = P (h) 0 0 0 .
Hence, V k can be rewritten as V k = e k P (h)e k , which is positive definite for ∀k ∈ N since P (h) 0. From the matrix partitions in ( 14)- [START_REF] Hassan | H∞ observers design for a class of nonlinear time-delay systems in descriptor form[END_REF], inequality [START_REF] Spong | Robot Modeling and Control[END_REF] can be represented as

  (α -1) Ē P (h) Ē 0 P (h + ) 0 0 -αγI   + Ψ + Ψ ≺ 0 ( 18
)
where

Ψ = G(h) τ G(h) 0 A(h) -I D(h) .
From the dynamics equation ( 8), we note that

2Λ k A(h)ē k -Ēē k+1 + D(h)d k = 0, ∀k ∈ N (19) with Λ k = ē k G(h)+τ ē k+1
Ē G(h). Pre-and postmultiplying [START_REF] Zhang | Reduced-order observer design for switched descriptor systems with unknown inputs[END_REF] with ē k ē k+1 Ē d k and its transpose while taking into account [START_REF] Jia | Fault reconstruction and faulttolerant control via learning observers in Takagi-Sugeno fuzzy descriptor systems with time delays[END_REF], we obtain

ē k+1 Ē P (h + ) Ēē k+1 + (α -1)ē k Ē P (h) Ēē k -αγd k d k ≺ 0, ∀k ∈ N. (20)
Condition [START_REF] Gao | A survey of fault diagnosis and fault-tolerant techniques-Part I: Fault diagnosis with model-based and signal-based approaches[END_REF] can be represented in the compact form

∆V k + α(V k -γd k d k ) < 0, ∀k ∈ N ( 21
)
where V k is defined in [START_REF] Guerra | Observer design for Takagi-Sugeno descriptor models: An LMI approach[END_REF], and ∆V k = V k+1 -V k is its difference along the trajectories of the error dynamics [START_REF] Su | A novel approach to output feedback control of fuzzy stochastic systems[END_REF]. It follows from ( 21) that

V k < (1 -α)V k-1 + αγ d k-1 2 , ∀k ≥ 1. (22)
By recursivity and since α ∈ (0, 1), we derive from ( 22) that

V k < (1 -α) k V 0 + αγ k-1 i=0 (1 -α) i d k-1-i 2 < (1 -α) k V 0 + αγ d 2 l∞ k-1 i=0 (1 -α) i < (1 -α) k V 0 + γ d 2 l∞ , ∀k ≥ 1. (23)
It is clear that inequality [START_REF] Chakrabarty | State and unknown input observers for nonlinear systems with bounded exogenous inputs[END_REF] guarantees the property (P1) defined in Problem 1 when d k = 0, for ∀k ∈ N. By Schur complement lemma [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF], it follows from ( 12) and ( 23) that

e k 2 < γV k < γ(1 -α) k V 0 + γ 2 d 2 l∞ , ∀k ≥ 1.
(24) Remark that inequality (24) implies [START_REF] Xu | Robust Control and Filtering of Singular Systems[END_REF]. Moreover, we have V 0 = 0, for e 0 = 0. Then, it follows from (24) that e k 2 < γ 2 d 2 l∞ , ∀k ≥ 1. As a consequence, the property (P2) is verified.

To recover the observer structure [START_REF] Wang | Observer design for discrete-time descriptor systems: an LMI approach[END_REF], let us put [START_REF] Li | Fuzzy-model-based D-stability and nonfragile control for discrete-time descriptor systems with multiple delays[END_REF] in its explicit form. Then, we can directly obtain

ψ k = xk+1 -K 1 (h)(y k -ŷk ) (25) E(h)ψ k = A(h)x k + B(h)u k + K 2 (h)(y k -ŷk ). (26)
Eliminating the virtual variable ψ k in (25) and (26), the observer structure ( 13) is straightforwardly recovered. The proof can be now concluded.

Remark 4. Note from (25) that the virtual variable ψ k plays a similar role as xk+1 in [START_REF] Nguyen | Static output feedback design for a class of constrained Takagi-Sugeno fuzzy systems[END_REF]. If d k = 0, for ∀k ∈ N, and the estimation error ēk in (8) converges to the origin, then xk → x k and ψ k → xk+1 when k → ∞.

Remark 5. The proposed observer design offers several extra degrees of freedom to enable less conservative results. First, the use of the singular redundancy form (4) of system (3) together with the virtual estimation variable ψ k in (8) allows introducing slack variables with free structures into the design problem. Second, the observer gain K(h) is constructed from G(h) and L(h) which are independent to P (h). Moreover, introducing the extra variable G(h), the Lyapunov matrix P (h) is decoupled from any product with the state-space matrices Ā(h), C(h), D(h), Ē(h). Third, all decision variables can be membership-function-dependent for the observer design. Remark 6. Consider the error dynamics [START_REF] Su | A novel approach to output feedback control of fuzzy stochastic systems[END_REF]. By minimizing the l ∞ -gain level γ, it follows directly from ( 9) that the estimation error e k can be minimized.

Remark 7. The observer design in Theorem 1 is directly applied to the standard T-S fuzzy systems by setting E(h) = I. Hence, the proposed method can be considered as a generalization of the l ∞ observer design for fuzzy descriptor systems. Conditions ( 10)-( 12) are expressed in terms of membershipfunction-dependent matrix inequalities. Hence, these cannot be directly used to design l ∞ fuzzy observers. Theorem 2 aims to convert these inequalities into a finite set of numerically tractable design conditions. Theorem 2. Consider the error dynamics [START_REF] Su | A novel approach to output feedback control of fuzzy stochastic systems[END_REF]. For a given scalar α ∈ (0, 1), if there exist symmetric matrices

P i ∈ R nx×nx , R i ∈ R nx×nx , matrices Q i ∈ R nx×nx , G 1i ∈ R nx×nx , G 2i ∈ R nx×nx , G 3i ∈ R nx×nx , G 4i ∈ R nx×nx , L 1i ∈ R nx×ny , L 2i ∈ R nx×ny , i ∈ I r ,
and positive scalars τ , γ such that the following optimization problem is achievable:

minimize ζi, i∈Ir γ (27)
subject to Σ P li + He(Φ φii ) ≺ 0, i, l ∈ I r (28) Σ P li + He(Φ φij + Φ φji ) ≺ 0, i, j, l ∈ I r , i < j (29)

P i Q i R i 0, i ∈ I r ( 30
)
P i I γI 0, i ∈ I r ( 31
)
where

ζ i = (τ, γ, P i , R i , Q i , L 1i , L 2i , G 1i , G 2i , G 3i , G 4i
) and

Σ P il =       (α -1)P i 0 0 0 0 P l 0 0 Q l R l 0 0 0 0 -αγI       Φ φij =       Γ ij Υ ij -G 1i -G 2i G 2i D j Ξ ij Φ ij -G 3i -G 4i G 4i D j τ Γ ij τ Υ ij -τ G 1i -τ G 2i τ G 2i D j τ Ξ ij τ Φ ij -τ G 3i -τ G 4i τ G 4i D j 0 0 0 0 0       Γ ij = G 2i A j -L 1i C j , Ξ ij = G 4i A j -L 2i C j Υ ij = G 1i -G 2i E j , Φ ij = G 3i -G 4i E j .
Then, the error dynamics ( 8) is globally uniformly l ∞ -stable with performance level γ. Moreover, the fuzzy observer structure corresponding to the initial system (3) is given in [START_REF] Wang | Observer design for discrete-time descriptor systems: an LMI approach[END_REF].

Proof. Based on the convex combination form of the statespace matrices A(h), C(h), D(h), E(h), and the decision matrices

P (h), R(h), Q(h), L 1 (h), L 2 (h), G 1 (h), G 2 (h), G 3 (h), G 4 (h), inequality (10)
can be exactly represented as

r l=1 r i=1 h l (ξ k+1 )h 2 i (ξ k )(Σ P li + Φ φii + Φ φii ) + r l=1 N i=1 N i<j h l (ξ k+1 )h i (ξ k )h j (ξ k )Θ lij ≺ 0 (32)
where Θ lij = Σ P li + He(Φ φij + Φ φji ). Since h i (ξ) ≥ 0, ∀i ∈ I r , it is clear that (28) and (29) guarantee condition (32). Moreover, condition [START_REF] Guelton | An alternative to inverse dynamics joint torques estimation in human stance based on a Takagi-Sugeno unknown-inputs observer in the descriptor form[END_REF] can be also rewritten as

r i=1 h i (ξ) P i Q i R i 0. (33)
We can easily see that (30) ensures (33). Similarly, condition [START_REF] Chadli | Novel bounded real lemma for discretetime descriptor systems: Application to H∞ control design[END_REF] is also verified due to (31). The rest of the proof follows the results of Theorem 1.

Remark 8. Note that the positive scalar τ is introduced in the observer design through the equality [START_REF] Jia | Fault reconstruction and faulttolerant control via learning observers in Takagi-Sugeno fuzzy descriptor systems with time delays[END_REF] to reduce the design conservatism. A similar idea has been exploited in [START_REF] Mozelli | A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems[END_REF] for the control design of non-descriptor fuzzy systems. The design conditions in Theorem 2 are expressed in terms of linear matrix inequalities with a line search over the scalar τ . A gridding method can be performed to search for τ . Then, the optimization problem (27) under LMI constraints (28)-(31) can be effectively solved with available numerical solvers [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF].

IV. UNKNOWN INPUT ESTIMATION WITH GUARANTEE ON ULTIMATE NORM-BOUNDED ERROR

The reconstruction of the UI of fuzzy descriptor systems is discussed in this section. Moreover, we also provide a characterization on the ultimate norm upper-bound of the UI estimation error as in [START_REF] Chakrabarty | State and unknown input observers for nonlinear systems with bounded exogenous inputs[END_REF].

To begin with, we derive from (3) that

D(h)d k = E(h)x k+1 -A(h)x k -B(h)u k . (34)
Since D(h) is of full-column rank for ∀h ∈ Ω, its pseudoinverse D(h) † exists and D(h) † D(h) = I. Premultiplying (34) with D(h) † yields

d k = D(h) † (E(h)x k+1 -A(h)x k -B(h)u k ).
The estimate dk of d k is proposed in the form

dk = D(h) † (E(h)x k+1 -A(h)x k -B(h)u k ). (35)
Hence, the UI estimation error e dk = d k -dk is given by

e dk = D(h) † (E(h)e k+1 -A(h)e k ).
It follows from ( 35) that

e dk ≤ D(h) † E(h) e k+1 + D(h) † A(h) e k ≤ D(h) † ( E(
where

η = D(h) † ∞ ( E(h) ∞ + A(h) ∞ ).
The following result summarizes the above discussions.

Corollary 1. Consider the error dynamics [START_REF] Su | A novel approach to output feedback control of fuzzy stochastic systems[END_REF]. If the optimization problem (27) in Theorem 2 is feasible. Then, there exists a fuzzy descriptor observer [START_REF] Wang | Observer design for discrete-time descriptor systems: an LMI approach[END_REF] such that (i) The error dynamics ( 8) is globally uniformly l ∞ -stable with a guaranteed performance level γ, (ii) The UI estimation error is ultimately bounded as shown in (38), where dk is defined in (35).

Remark that if the l ∞ -gain level γ is arbitrarily small and

{ Φ(h(ξ k ))} k∈N ∈ L ∞ , with Φ ∈ {A, D † , E}.
Then, the state and the UI can be estimated with an arbitrary degree of accuracy as respectively shown in (37) and (38).

V. ILLUSTRATIVE EXAMPLE

For illustration purposes, let us consider a nonlinear twolink robot manipulator [START_REF] Spong | Robot Modeling and Control[END_REF] whose dynamics is given by

M (θ(t)) θ(t) -N (θ(t)) = u(t) + d(t) (39)
where

M (θ) = m 1 l 2 + m 2 L 2 1 + J 1 m 2 lL 2 cos(θ 1 -θ 2 ) m 2 L 2 2 + J 2 N (θ) = (m 1 L 1 + m 2 l)g sin(θ 1 ) m 2 L 2 g sin(θ 2 )
and θ = θ 1 θ 2 is the position vector, u = u 1 u 2 is the known torque input, d = d 1 d 2 is the unknown input at the joints. The robot nomenclature is given in Table I.

A discrete-time representation of the robot model (39) can be obtained by the Euler discretization method as where x k = θ 1k θ 2k θ1k θ2k and

E(ξ k )x k+1 = A(ξ k )x k + B(u k + d k ), y k = Cx k (40)
E(ξ k ) = I 0 0 M (ξ k ) , A(ξ k ) = I T s I T s R(ξ k ) M (ξ k ) B = 0 T s I , C = I 0 M (ξ k ) = m 1 l 2 + m 2 L 2 1 + J 1 m 2 lL 2 ξ 1k m 2 L 2 2 + J 2 R(ξ k ) = (m 1 L 1 + m 2 l)gξ 2k 0 0 m 2 L 2 gξ 3k ξ 1k = cos(θ 1k -θ 2k ), ξ 2k = sin(θ 1k ) θ 1k , ξ 3k = sin(θ 2k ) θ 2k .
For system (40), the vector of premise variables is defined as

ξ k = ξ 1k ξ 2k ξ 3k , with ξ i min ≤ ξ ik ≤ ξ i max , ∀i ∈ I 3 , ξ 1 min = -1, ξ 1 max = 1 ξ 2 min = ξ 3 min = -0.2172, ξ 2 max = ξ 3 max = 1.
Applying sector nonlinearity approach [2, Chapter 2], an equivalent fuzzy descriptor representation (3) with eight fuzzy rules (r = 8) can be obtained for the nonlinear system (40). The state-space matrices and the nonlinear MFs corresponding to this fuzzy representation are omitted for brevity. Theorem 2 is employed with α = 0.05 and a logarithmically scaled grid over τ ∈ 10 -5 , 10 5 with 100 points. The suboptimal values of γ found for each τ are given in Fig. 1. To illustrate the interest of using the virtual variable ψ k for the observer design, the case in which L 1i = 0, ∀i ∈ I 8 , i.e., design conditions corresponding to the observer structure [START_REF] Nguyen | Static output feedback design for a class of constrained Takagi-Sugeno fuzzy systems[END_REF], is also presented. We assume that γ is infinite when the optimization problem (27) is unfeasible. Note that the classical observer structure (5) always leads to more conservative results compared to the new observer [START_REF] Li | Fuzzy-model-based D-stability and nonfragile control for discrete-time descriptor systems with multiple delays[END_REF]. The design conditions in Theorem 2 are feasible for τ ∈ 1.1233, 10 5 , with a minimum at τ = 3853.5 for γ = 0.071. The corresponding observer solution is used for estimation purposes. For illustrations, we select x 0 = π π 1 -1 , x0 = π π 0 0 , u k = 0 0 as the known input, and the UI vector d k is as

d k = 0 0 , k ≤ 20 5 cos(0.05k) 5 cos 0.05k + π 3 , k > 20.
The above choice of d k allows us to compare the estimation performance for two cases: with (Case 1) and without (Case 2) the presence of unknown inputs. The estimation performance is summarized in Fig. 2. Observe in Fig. 2(a) that for Case 1, the state estimation error e k asymptotically converges toward the origin from the initial condition. We remark in Case 2 that the system state can be estimated with a satisfactory accuracy and the l ∞ -gain performance is verified. Moreover, as represented in Figs. 2(b) and (c), both unknown inputs are effectively reconstructed with the estimate given in (35).

VI. CONCLUSIONS

A new l ∞ observer design has been proposed for fuzzy descriptor systems. Based on Lyapunov stability arguments, the design conditions are derived in terms of LMI constraints over a single line search parameter. In particular, exploiting a singular redundancy representation and a virtual estimation variable, all involved decision variables for the observer design can be of free structure to reduce the design conservatism. The estimation performance of both the state and the UI is guaranteed with a suboptimal l ∞ -gain level. We show the application of our result to the observer problem of a nonlinear robot manipulator. This work focuses on the case where the premise variables are online measurable. Designing UI observers for fuzzy descriptor systems with unmeasurable premise variables is not trivial, which is currently investigated.

Fig. 1 .

 1 Fig. 1. Suboptimal values of γ as a function of the scalar parameter τ in Theorem 2 and Theorem 2 with L 1i = 0, ∀i ∈ I 8 , for the nonlinear two-link robot manipulator with α = 0.05.

Fig. 2 .

 2 Fig. 2. Estimation performance. (a) State estimation with (Case 1) and without (Case 2) the presence of unknown inputs. (b) and (c) Estimation of the unknown inputs in Case 2.

  the vector of measurable premise variables. (A i , B i , C i , D i , E i ) are known constant matrices with appropriate dimensions, which represent the ith local linear model. R i denotes the ith fuzzy inference rule. M i j , with i ∈ I r and j ∈ I p , is the fuzzy set. The fuzzy MFs are given by h i (ξ) =

	p j=1 µ i j (ξj )
	r i=1	p j=1 µ i j (ξj )

  h) e k+1 + A(h) e k . (36)

	If the optimization problem (27) is feasible. Then, we have
	lim k→∞	sup e k < γ d l∞ , lim k→∞	sup e k+1 < γ d l∞ . (37)
	Combining (36) and (37), the ultimate norm upper-bound of
	the estimation error d k -dk is characterized by
		lim k→∞	sup e dk ≤ γη d l∞