Cuong M Nguyen

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

Sébastien Delprat
email: sebastien.delprat@uphf.fr.

A.-T Nguyen

A.-T Nguyen

Cascade Takagi-Sugeno Fuzzy Observer Design for Nonlinear Uncertain Systems with Unknown Inputs: A Sliding Mode Approach

Keywords: Takagi-Sugeno fuzzy models, cascaded observers, sliding mode technique, nonlinear observer, unknown input, uncertainty

This paper investigates the design of Takagi-Sugeno (TS) fuzzy model-based observers for nonlinear systems with parametric uncertainties and unknown inputs. To address this challenging problem, two observers are constructed in cascade. Based on the sliding mode technique, the first observer allows to examine a new system whose both state and output equations are subject to uncertainties but without unknown inputs. The second Luenberger-type observer is designed for the new system where the effects of uncertainties on the estimation error can be canceled. The TS fuzzy observer design is recast as optimization problems under linear matrix inequalities, which can be effectively solved using convex optimization technique. The new cascade observer structure enables a simultaneous estimation of the system states, the unknown inputs and the uncertainties of the original nonlinear system. The effectiveness and advantage of the proposed estimation method is demonstrated via two numerical examples including a nonlinear vehicle application.

I. INTRODUCTION

Physical systems usually involve complex nonlinear dynamics. Takagi-Sugeno (TS) fuzzy model-based approaches have become one of the most popular and promising tools to deal with such nonlinear systems [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF]. For smooth nonlinear systems, TS fuzzy modeling provides a systematic framework to obtain exact representations in a compact set or approximations with any degree of accuracy [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. In particular, TS fuzzy model-based conditions for stability analysis and controller/observer design of nonlinear systems can be expressed in terms of linear matrix inequalities (LMIs), which can be effectively solved using convex optimization techniques [START_REF] Nguyen | Fuzzy control systems: Past, present and future[END_REF]. Therefore, TS fuzzy methods have been successfully applied to many engineering applications, see for instance [START_REF] Dong | Control synthesis of continuous-time TS fuzzy systems with local nonlinear models[END_REF]- [START_REF] Zhang | A novel fuzzy observer-based steering control approach for path tracking in autonomous vehicles[END_REF].

Nonlinear observer design is a fundamental concern in control theory, which has attracted lots of attention [START_REF] Khalil | Nonlinear Systems[END_REF]. Specifically, many works on TS fuzzy model-based observer design for nonlinear systems have been reported [START_REF] Pan | A unified framework for asymptotic observer design of fuzzy systems with unmeasurable premise variables[END_REF]- [START_REF] Xie | Fault estimation observer design for discrete-time Takagi-Sugeno fuzzy systems based on homogenous polynomially parameter-dependent Lyapunov functions[END_REF]. Since physical systems are inherently subject to uncertainties [START_REF] Khalil | Nonlinear Systems[END_REF], the class of nonlinear uncertain systems is of crucial importance in control theory. The system uncertainties can emerge from modeling errors and/or uncertain parameters. Accordingly, a considerable effort has been devoted to develop robust observers for nonlinear systems where modeling uncertainties are taken into account [START_REF] Boulkroune | Design of a unified adaptive fuzzy observer for uncertain nonlinear systems[END_REF], [START_REF] Franco | Adaptive estimation for uncertain nonlinear systems with measurement noise: A sliding-mode observer approach[END_REF]- [START_REF] Li | Robust H∞ sliding mode observer design for a class of Takagi-Sugeno fuzzy descriptor systems with time-varying delay[END_REF]. In addition to uncertainties, unknown inputs (UIs) are another important issue in robust control since UIs can represent faults, disturbances, or unmeasurable inputs in dynamical systems [START_REF] Nguyen | Avoiding unmeasured premise variables in designing unknown input observers for Takagi-Sugeno fuzzy systems[END_REF]- [START_REF] Feng | Fault estimation based on sliding mode observer for Takagi-Sugeno fuzzy systems with digital communication constraints[END_REF]. Therefore, many observer designs for uncertain systems with UIs have been developed in the literature. Using a sliding mode approach, the problem of fault estimation for nonlinear uncertain systems is studied in [START_REF] Yan | Nonlinear robust fault reconstruction and estimation using a sliding mode observer[END_REF], where the estimation error depends on the bound of the uncertainty. A robust sliding mode observer is developed in [START_REF] Lee | Robust H∞ sliding mode descriptor observer for fault and output disturbance estimation of uncertain systems[END_REF] to estimate both states and disturbances of uncertain systems using a descriptor system approach. Fault reconstruction for a class of linear parameter varying (LPV) systems is investigated in [START_REF] Alwi | Fault reconstruction using a LPV sliding mode observer for a class of LPV systems[END_REF] by designing sliding mode observers (SMOs). A SMO scheme for a class of affine LPV systems is proposed in [START_REF] Chen | Sliding mode observers for a class of linear parameter varying systems[END_REF] where the assumption on factorization of the fault distribution matrix in [START_REF] Alwi | Fault reconstruction using a LPV sliding mode observer for a class of LPV systems[END_REF] is removed. A sensor fault estimation scheme for uncertain TS fuzzy systems is proposed in [START_REF] Brahim | Robust fault detection for uncertain T-S fuzzy system with unmeasurable premise variables: Descriptor approach[END_REF], which is robust to external disturbances. In [START_REF] Hosseini | Optimal reset unknown input observer design for fault and state estimation in a class of nonlinear uncertain systems[END_REF], an optimal reset law is used to design unknown input observers for a class of nonlinear uncertain systems where states and faults can be simultaneously estimated. Despite recent advances in observer design for nonlinear uncertain systems with UIs, this research topic still remains challenging, especially for TS fuzzy systems with unmeasured nonlinearities [START_REF] Nguyen | Avoiding unmeasured premise variables in designing unknown input observers for Takagi-Sugeno fuzzy systems[END_REF]. To our knowledge, there is no existing work addressing the problem of simultaneously estimating system states, uncertainties and UIs of nonlinear systems.

Sliding mode technique has been widely used as an effective approach to deal with systems subject to UIs [START_REF] Nguyen | Sliding mode observer for estimating states and faults of linear time-delay systems with outputs subject to delays[END_REF]- [START_REF] Davila | Second-order sliding-mode observer for mechanical systems[END_REF]. Due to its robustness with respect to modeling uncertainties and disturbances, this technique has been popularly used for both control and estimation of uncertain dynamical systems [START_REF] Yi | Sliding mode observer for nonlinear uncertain systems[END_REF]- [START_REF] Su | Event-triggered fuzzy control for nonlinear systems via sliding mode approach[END_REF]. In sliding mode observers, UIs can be estimated by constructing discontinuous signals, which allows a finite-time convergence of output estimation errors [START_REF] Edwards | Sliding Mode Control: Theory and Applications[END_REF]. This has motivated us to exploit SMO structure to solve the observer design problem for TS fuzzy models with uncertainties and UIs.

This paper presents a new method to design TS fuzzy sliding mode based observers for nonlinear systems with UIs, parametric uncertainties and unmeasured nonlinearities. Using the sliding mode technique, two cascaded observers are constructed. A SMO is first designed which allows to consider a new system without UIs but both state and output equations are subject to uncertainties. Then, a Luenberger-type observer is designed where the effects of the uncertainties on the estimation error can be canceled. This allows the estimations of the states of the original systems as well as the modeling uncertainties and the UIs. The main contributions of the paper can be summarized as follows.

• We propose a new sliding mode based observer design for nonlinear systems with uncertainties and UIs, where two TS fuzzy observers are constructed in cascade interconnection. Based on Lyapunov stability theory, the observer design is reformulated as optimization problems under LMI constraints, effectively solved with numerical solvers. • Compared to relevant existing results [START_REF] Yan | Nonlinear robust fault reconstruction and estimation using a sliding mode observer[END_REF], [START_REF] Alwi | Fault reconstruction using a LPV sliding mode observer for a class of LPV systems[END_REF], [START_REF] Chen | Sliding mode observers for a class of linear parameter varying systems[END_REF], [START_REF] Edwards | Sliding Mode Control: Theory and Applications[END_REF], the main advantage of the proposed cascade observer scheme is the ability to simultaneously estimate the state vector, the unknown input and the uncertainty. Note that the results in [START_REF] Yan | Nonlinear robust fault reconstruction and estimation using a sliding mode observer[END_REF] can only provide the estimation of the state and the unknown input. Meanwhile, the uncertainty is not considered in [START_REF] Alwi | Fault reconstruction using a LPV sliding mode observer for a class of LPV systems[END_REF], [START_REF] Chen | Sliding mode observers for a class of linear parameter varying systems[END_REF], [START_REF] Edwards | Sliding Mode Control: Theory and Applications[END_REF]. Hence, the results in [START_REF] Alwi | Fault reconstruction using a LPV sliding mode observer for a class of LPV systems[END_REF], [START_REF] Chen | Sliding mode observers for a class of linear parameter varying systems[END_REF], [START_REF] Edwards | Sliding Mode Control: Theory and Applications[END_REF] can only provide the estimation of the state and the sum of disturbance and uncertainty. Note also that we consider a class of nonlinear uncertain systems with a state-dependent dynamic matrix as well as unmeasured nonlinearities. However, the system dynamic matrix in [START_REF] Yan | Nonlinear robust fault reconstruction and estimation using a sliding mode observer[END_REF] is constant, and there is no unmeasured nonlinearity in [START_REF] Alwi | Fault reconstruction using a LPV sliding mode observer for a class of LPV systems[END_REF], [START_REF] Chen | Sliding mode observers for a class of linear parameter varying systems[END_REF]. In [START_REF] Edwards | Sliding Mode Control: Theory and Applications[END_REF], neither time-varying system matrix nor nonlinearity is investigated. • The effectiveness and advantage of the new cascade observer scheme is demonstrated via two numerical examples including a practical application on the estimation of vehicle nonlinear dynamics. The paper is organized as follows. Section II formulates the observer design problem for a class of nonlinear systems subject to both UIs and parametric uncertainties. Based on Lyapunov stability arguments, Section III presents the LMI-based design of two TS fuzzy observers in cascade interconnection. The sliding motion analysis is also provided. In Section IV, we demonstrate the effectiveness and advantage of the new results through two examples. Section V provides some concluding remarks and related future works.

Notation. For a positive integer r, we denote I r = {1, 2, . . . , r}. For i ∈ I r , we denote σ r (i) = [0, . . . , 0, ith 1 , 0, . . . , 0] ∈ R r a vector of the canonical basis of R r . For two vectors x, y ∈ R n , x i denotes its ith entry, and column{x, y} = x y . For a matrix X, X denotes its transpose, X 0 means X is positive definite, and He(X) = X + X . diag(X 1 , X 2 ) denotes a block-diagonal matrix composed of X 1 , X 2 . I denotes an identity matrix of appropriate dimension, and I n denotes an identity matrix of dimension n. In block matrices, the symbol stands for the terms deduced by symmetry. Arguments are omitted when their meanings are clear.

II. PROBLEM FORMULATION

Consider the following nonlinear system with unknown inputs and parametric uncertainties:

ẋ = (A(ζ) + ∆A(ζ))x + f (u, y) + Gg(x) + Dd, y = Cx, x ∈ D x , (1)
where D x is a state-space compact set, x ∈ R n is the state vector, u ∈ R m is the control input, y ∈ R p is the output vector, and ζ ∈ R n ζ is the vector of measurable premise variables. The unknown input d ∈ R q is bounded, i.e., ||d|| ≤ Λ d , for some

Λ d > 0. The constant matrices G ∈ R n×ng , D ∈ R n×q , C ∈ R p×n are known; f (u, y) ∈ R
∆A(ζ) = M (ζ)F (t)N (ζ), (2)
where M (ζ) and N (ζ) are known, and the unknown time-varying matrix F (t) satisfies F F ≤ I. Without loss of generality, for observer design we assume that the nonlinear system (1) is stable or can be stabilized by a controller.

Remark 1. Physical systems often operate in bounded regions [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. Thus, it is reasonable to assume that the state vector x belongs to the compact set D x , see for instance the vehicle nonlinear dynamics [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF] discussed in Section IV.

Let us denote w = F N (ζ)x ∈ R nw , then system (1) can be rewritten as

ẋ = A(ζ)x + M (ζ)w + f (u, y) + Gg(x) + Dd, y = Cx, (3)
where w represents a bounded uncertainty, i.e., ||w|| ≤ Λ w , for some positive scalar Λ w > 0.

Remark 2. Our study is motivated by the vehicle nonlinear dynamics, described by the form (1)-( 2). This form can be rewritten as in [START_REF] Nguyen | Fuzzy control systems: Past, present and future[END_REF], which is the class of systems we consider in this paper. In other words, we deal with any system which can be represented in the form (3), and the specific uncertain system structure (1)-( 2) is only one source for (3).

Applying the sector nonlinearity approach [2, Chapter 2] with the vector of premise variables ζ and x ∈ D x , the nonlinear system (3) can be equivalently represented by the following TS fuzzy model with nonlinear consequents:

ẋ = N i=1 h i (ζ)(A i x + M i w) + f (u, y) + Gg(x) + Dd, y = Cx, N = 2 n ζ , (4)
where the local matrices A i and M i are constant. For x ∈ D x , the membership functions h i (ζ) satisfy the convex sum property

N i=1 h i (ζ) = 1, 0 ≤ h i (ζ) ≤ 1, ∀i ∈ I N . (5)
Furthermore, we have

A(ζ) M (ζ) = N i=1 h i (ζ) A i M i . Remark 3. For ζ = [ζ 1 . . . ζ j . . . ζ n ζ ] ∈ R n ζ ,
µ 0 j (ζ j ) = ζ j -ζ j ζ -ζ j , µ 1 j (ζ j ) = ζ j -ζ j ζ -ζ j . Note that µ 0 j (ζ j ) + µ 1 j (ζ j ) = 1, ζ j = ζ j µ 0 j (ζ j ) + ζ j µ 1 j (ζ j ).
We name µ 0 j (ζ j ) as "j-negative" (or M 0 j ) and µ 1 j (ζ j ) as "j-positive" (or M 1 j ). The membership functions h i (ζ) and the local constant matrices A i , M i can be determined via N = 2 n ζ fuzzy IF-THEN rules as follows (M i j can be either

M 0 j or M 1 j , ∀j ∈ I n ζ , i ∈ I N ): RULE i : IF ζ 1 is M i 1 and . . . ζ j is M i j and . . . ζ n ζ is M i n ζ THEN ẋ = A i x + M i w + f (u, y) + Gg(x) + Dd, y = Cx, (6)
where

A i = A(ζ i 1 , . . . , ζ i j , . . . , ζ i n ζ ), M i = M (ζ i 1 , . . . , ζ i j , . . . , ζ i n ζ ), and
h i (ζ) = n ζ j=1 µ i j (ζ j ). (7)
Note that, if

ζ j is M 0 j : ζ i j = ζ j , µ i j (ζ j ) = µ 0 j (ζ j ); if ζ j is M 1 j : ζ i j = ζ j , µ i j (ζ j ) = µ 1 j (ζ j ).
Using the center-of-gravity method for defuzzification [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF], the TS fuzzy system (6) can be represented in the compact form [START_REF] Dong | Control synthesis of continuous-time TS fuzzy systems with local nonlinear models[END_REF]. For h i (ζ) given by [START_REF] Nguyen | Takagi-Sugeno fuzzy unknown input observers to estimate nonlinear dynamics of autonomous ground vehicles: Theory and real-time verification[END_REF], it can be verify that 4) is an equivalent TS fuzzy model of system [START_REF] Nguyen | Fuzzy control systems: Past, present and future[END_REF]. See the examples in Section IV for more details.

N i=1 h i (ζ) = 1. Furthermore, (
For observer design, we consider the following assumptions for system (1), equivalently represented in the TS fuzzy form (4) within the state-space compact set D x .

Assumption 1. The following rank conditions hold:

rank(CD) = rank(D), (8) rank(C) = p > q = rank(D).
Assumption 2. The nonlinear function g(x) is differentiable with respect to x and satisfies

g ij ≤ ∂g i ∂x j (x) ≤ ḡij , ∀i ∈ I ng , ∀j ∈ I n , (9)
with

g ij = min x∈Dx ∂g i ∂x j (x) , g ij = max x∈Dx ∂g i ∂x j (x) .
Remark 4. Note that under condition [START_REF] Zhang | A novel fuzzy observer-based steering control approach for path tracking in autonomous vehicles[END_REF] in Assumption 1, there exists a coordinate transformation T for system (1) such that C and D have the following form [START_REF] Edwards | Sliding Mode Control: Theory and Applications[END_REF]:

C = [0 I p ], D = 0 D 2 n -p p . (10)
We further assume that D 2 can be partitioned as

D 2 = 0 D 22 p -q q . ( 11
)
To simplify the presentation, hereafter we consider system (1) where C and D have the form ( 10)- [START_REF] Boulkroune | Design of a unified adaptive fuzzy observer for uncertain nonlinear systems[END_REF]. Note that condition p > q in Assumption 1 will allow us to have the partition [START_REF] Tapia | Nonlinear sliding mode control design: An LMI approach[END_REF] where ν eq1 ∈ R p-q . Then, we can consider system [START_REF] Yi | Sliding mode observer for nonlinear uncertain systems[END_REF] for the second observer design of the proposed cascade observer scheme. Note that Assumption 2 is useful to deal with the mismatching nonlinear term caused by the unmeasured nonlinearity g(x) in the estimation error using the differential mean value theorem [START_REF] Pan | A unified framework for asymptotic observer design of fuzzy systems with unmeasurable premise variables[END_REF], [START_REF] Zemouche | Observers for a class of Lipschitz systems with extension to H∞ performance analysis[END_REF].

Remark 5. In the following, for brevity we will use A(ζ) and

N i=1 h i (ζ)A i interchangeably as A(ζ) = N i=1 h i (ζ)A i . Similarly, we will use M (ζ) and N i=1 h i (ζ)M i interchangeably as M (ζ) = N i=1 h i (ζ)M i .
Study objective: Based on a cascade observer approach, this paper aims at designing an observer scheme to simultaneously estimate the state vector x, the unknown input d and the uncertainty w of system (1), equivalently represented by the TS fuzzy model (4).

III. CASCADE TAKAGI-SUGENO FUZZY OBSERVER DESIGN

This section presents the design of two TS fuzzy observers, whose cascade interconnection is depicted in Fig. 1.

A. First Design: TS Fuzzy Sliding Mode Observer

Consider the following TS fuzzy SMO for system (1):

ẋ = N i=1 h i (ζ)(A i x + L i e y ) + f (u, y) + Gg(x) -Jν, ( 12
)
where x is the estimate of x, the observer gains L i , for i ∈ I N , are to be designed, and

χ = I n-p 0 x, e y = y -C x, x = χ y , J = 0 I p .
The discontinuous signal ν(t) is defined as

ν = -µ P -1 y ey
||ey|| , e y = 0, 0, otherwise

where µ > 0 and P y ∈ R p×p are to be determined. From ( 1) and ( 12), the dynamics of the state estimation error e = x -x is defined as

ė = (A(ζ) -L(ζ)C)e + M (ζ)w + G(g(x) -g(x)) + Dd + Jν, (14)
where

L(ζ) = N i=1 h i (ζ)L i .
The following differential mean value theorem (DMVT) is useful to deal with the mismatching nonlinear term g(x) -g(x).

Lemma 1 ([40]

). Let g(x) : R n → R ng and a, b ∈ R n . If g(x) is differentiable with respect to x on the convex hull of a and b, then there exist vectors

c i = α i a + (1 -α i )b, with α i ∈ (0, 1), ∀i ∈ I ng , such that g(a) -g(b) =   ng i=1 n j=1 σ ng (i)σ n (j) ∂g i ∂x j (c i )   (a -b).
Under Assumption 2 and by Lemma 1, we can rewrite the nonlinear term g(x) -g(x) as

g(x) -g(x) = A g (x -x) = A g e x 0
,

where e x = [I n-p 0]e, and

A g = ng i=1 n j=1 σ ng (i)σ n (j) ∂g i ∂x j (c i ), with c i = α i x + (1 -α i )
x and α i ∈ (0, 1). Note that A g is a matrix with parameters varying in a bounded convex set H g , whose the set of vertices is given by

V Hg = ρ = (ρ ij ) ∈ R ng×n : ρ ij ∈ g ij , g ij ,
where g ij and g ij are determined as in [START_REF] Khalil | Nonlinear Systems[END_REF]. Let us denote

A 1 g = A g [I n-p 0] , A 10 g = [A 1 g 0].
Then, it follows that g(x)-g(x) = A 10 g e. Note that the matrix A 1 g (respectively A 10 g ) is with parameters varying in a bounded convex set H 1 g (respectively H 10 g ), whose the set of vertices is V H 1 g (respectively V H 10 g ). As a result, the error dynamics ( 14) can be rewritten as

ė = A(ζ) + GA 10 g -L(ζ)C e + M (ζ)w + Dd + Jν. ( 15
)
Based on the H ∞ filtering approach, we design the observer gain L(ζ) such that the following condition holds:

V (e) + e e -λ 2 w w < 0,

where V (e) = e P e, with P 0, is a Lyapunov function candidate, and λ > 0 is the level of disturbance attenuation. The following theorem provides sufficient conditions to guarantee condition [START_REF] Li | Robust H∞ sliding mode observer design for a class of Takagi-Sugeno fuzzy descriptor systems with time-varying delay[END_REF] with a minimal value of λ.

Theorem 1. Consider the error dynamics [START_REF] Nguyen | Robust observer and observer-based control designs for discrete one-sided Lipschitz systems subject to uncertainties and disturbances[END_REF]. Condition ( 16) is guaranteed with a minimal λ if there exist matrices of appropriate dimensions P 1 0, P y 0, X i , for i ∈ I N , and a scalar λ > 0 that solve the convex optimization problem min( λ) subject to

Ω i = He P (A i + GA 10 g ) -X i C + I P M i -λI ≺ 0, (17)
for ∀i ∈ I N , ∀A 10 g ∈ V H 10 g , with P = diag(P 1 , P y ), and the positive scalar µ in (13) satisfies

µ ≥ ||P y D 2 ||Λ d . (18)
Furthermore, the observer gain matrices in [START_REF] Liu | Takagi-Sugeno fuzzy model based fault estimation and signal compensation with application to wind turbines[END_REF] are obtained as L i = P -1 X i , and λ = √ λ.

Proof. The time-derivative of V (e) along the solution of system ( 15) is given by

V (e) = 2e P (A(ζ) + GA 10 g -L(ζ)C)e + M (ζ)w + 2e y P y D 2 d -2µ||e y || ≤ 2e P (A(ζ) + GA 10 g -L(ζ)C)e + M (ζ)w -2||e y ||(µ -||P y D 2 ||Λ d ). (19)
Since X i = P L i , for i ∈ I N and λ = λ 2 , it follows from ( 16), ( 18) and ( 19) that

V (e)+e e -λ 2 w w

≤ e He P (A(ζ) + GA 10 g ) -X(ζ)C + I e + 2e P M (ζ)w -λw w = N i=1 h i (ζ)[e w ]Ω i e w . (20)
Due to the convex sum property (5) of the membership functions h i (ζ), for i ∈ I N , it follows from inequality (20) that condition [START_REF] Li | Robust H∞ sliding mode observer design for a class of Takagi-Sugeno fuzzy descriptor systems with time-varying delay[END_REF] holds if [START_REF] Nguyen | Avoiding unmeasured premise variables in designing unknown input observers for Takagi-Sugeno fuzzy systems[END_REF] holds. Moreover, λ is minimized by the minimization of λ.

Remark 6. Note that if condition (16) holds, then e is bounded, i.e., ||e(t)|| ≤ Λ e , for ∀t ≥ 0. This means that x only estimates x within a bound. The second observer will be designed in Section III-C to improve the estimation precision of x.

Remark 7. The differential mean value theorem has been shown as an effective tool to deal with unmeasured nonlinearities in various estimation problems, see for instance [START_REF] Nguyen | Takagi-Sugeno fuzzy unknown input observers to estimate nonlinear dynamics of autonomous ground vehicles: Theory and real-time verification[END_REF], [START_REF] Nguyen | Avoiding unmeasured premise variables in designing unknown input observers for Takagi-Sugeno fuzzy systems[END_REF], [START_REF] Zemouche | Observers for a class of Lipschitz systems with extension to H∞ performance analysis[END_REF] and related references. Recently, an alternative approach to DMVT has been proposed in [START_REF] Quintana | An exact handling of the gradient for overcoming persistent problems in nonlinear observer design via convex optimization techniques[END_REF], [START_REF] Coutinho | Unknown input observer for nonlinear systems with arbitrary relative degree[END_REF]. However, the approach in [START_REF] Quintana | An exact handling of the gradient for overcoming persistent problems in nonlinear observer design via convex optimization techniques[END_REF], [START_REF] Coutinho | Unknown input observer for nonlinear systems with arbitrary relative degree[END_REF] requires the derivation of the polynomial forms and their corresponding TS models, which could be possibly confuse the readers if integrated into our TS fuzzy cascaded observer scheme. As a result, we prefer to use the classical DMVT approach to ensure the readability of this paper.

B. Sliding Motion Analysis

We partition e, G, L i , A i and M i , for i ∈ I N , as follows:

e = e x e y , G = G 1 G 2 , L i = L i1 L i2 , A i = A i11 A i12 A i21 A i22 , M i = M i1 M i2 , (21)
where

e y ∈ R p , L i2 ∈ R p×p , G 2 ∈ R p×ng , A i22 ∈ R p×p , and M i2 ∈ R p×nw .
Then, it follows from ( 15) and ( 21) that

ėx = A 11 (ζ) + G 1 A 1 g e x + (A 12 (ζ) -L 1 (ζ))e y + M 1 (ζ)w, (22)
ėy = A 21 (ζ) + G 2 A 1 g e x + (A 22 (ζ) -L 2 (ζ))e y + M 2 (ζ)w + D 2 d + ν, (23)
with

A 11 (ζ) A 12 (ζ) L 1 (ζ) M 1 (ζ) A 21 (ζ) A 22 (ζ) L 2 (ζ) M 2 (ζ) = N i=1 h i (ζ) A i11 A i12 L i1 M i1 A i21 A i22 L i2 M i2 .
The following theorem provides a sufficient condition for the occurrence of the sliding motion.

Theorem 2. Consider system ( 22)- [START_REF] Chen | Sliding mode observers for a class of linear parameter varying systems[END_REF]. If the positive scalar µ in ( 13) is selected as

µ = max 1≤i≤N ||P y A 21i || + max A 1 g ∈V H 1 g ||P y G 2 A 1 g || + max 1≤i≤N ||P y (A 22i -L 2i )|| Λ e + max 1≤i≤N ||P y M 2i ||Λ w + ||P y D 2 ||Λ d + μ, (24)
with μ > 0. Then, the sliding motion occurs on the surface S = {e : Ce = 0}, where e y = 0 and ėy = 0.

Proof. For sliding motion analysis, we consider the Lyapunov function candidate V y (e y ) = e y P y e y . Differentiating the function V y (e y ) along the solution of system [START_REF] Chen | Sliding mode observers for a class of linear parameter varying systems[END_REF], it follows that

Vy (e y ) = 2e y P y {(A

21 (ζ) + G 2 A 1 g )e x + (A 22 (ζ) -L 2 (ζ))e y + M 2 (ζ)w + D 2 d} -2µ||e y || ≤ -2||e y ||(µ -||P y A 21 (ζ)||||e x || -||P y G 2 A 1 g ||||e x || -||P y (A 22 (ζ) -L 2 (ζ))||||e y || -||P y M 2 (ζ)||||w|| -||P y D 2 ||||d||). (25)
If condition [START_REF] Brahim | Robust fault detection for uncertain T-S fuzzy system with unmeasurable premise variables: Descriptor approach[END_REF] holds, then it follows from [START_REF] Hosseini | Optimal reset unknown input observer design for fault and state estimation in a class of nonlinear uncertain systems[END_REF] that

Vy (e y ) ≤ -2μ||e y || ≤ - 2μ λ max (P y ) V y (e y ), (26)
where λ max (P y ) is the maximal eigenvalue of P y . Note that ( 26) is the reachability condition, hence the sliding motion occurs on S = {e : Ce = 0}, where e y = 0 and ėy = 0.

Remark 8. If the unknown input d is bounded by a known time-varying function instead of a known constant, i.e., ||d|| ≤ ϕ(t, u, y), where ϕ(t, u, y) > 0, for ∀(t, u, y), the sliding motion on the surface S = {e : Ce = 0} is guaranteed by choosing a time-varying parameter µ(t, u, y) instead of a constant scalar µ as follows:

µ(t, u, y) = max 1≤i≤N ||P y A 21i || + max A 1 g ∈V H 1 g ||P y G 2 A 1 g || + max 1≤i≤N ||P y (A 22i -L 2i )|| Λ e + max 1≤i≤N ||P y M 2i ||Λ w + ||P y D 2 ||ϕ(t, u, y) + μ.
This can be proved in a similar manner to the proof of Theorem 2. It is important to note that a constant bound for the unknown input d is preferred over a time-varying bound as it simplifies the design of µ, thus the overall observer design.

Under the sliding motion, it follows from ( 22) and ( 23) that

ėx = A 11 (ζ) + G 1 A 1 g e x + M 1 (ζ)w, (27)
0 = A 21 (ζ) + G 2 A 1 g e x + M 2 (ζ)w + D 2 d + ν eq , (28)
where ν eq is the equivalent control.

Remark 9. The stability of system ( 27)-( 28) is guaranteed by Theorem 1. Indeed, condition [START_REF] Nguyen | Avoiding unmeasured premise variables in designing unknown input observers for Takagi-Sugeno fuzzy systems[END_REF] implies that

P 1 (A 11 (ζ) + G 1 A 1 g ) + (A 11 (ζ) + G 1 A 1 g ) P 1 ≺ 0.
We further partition ν eq , G 2 , A i21 , M i2 , for i ∈ I N , as follows:

ν eq = ν eq1 ν eq2 , G 2 = G 21 G 22 , A i21 = A i211 A i212 , M i2 = M i21 M i22 , (29)
where

ν eq2 ∈ R q , G 22 ∈ R q×ng , A i212 ∈ R q×(n-p) and M i22 ∈ R q×nw .
Then, it follows from ( 28) and ( 29) that

ν eq1 = -(A 211 (ζ) + G 21 A 1 g )e x -M 21 (ζ)w, (30)
ν eq2 = -(A 212 (ζ) + G 22 A 1 g )e x -M 22 (ζ)w -D 22 d, (31)
with

A 211 (ζ) M 21 (ζ) A 212 (ζ) M 22 (ζ) = N i=1 h i (ζ) A i211 M i21 A i212 M i22 .
From ( 27) and [START_REF] Tan | Extended results on robust state estimation and fault detection[END_REF], let us consider the following system:

ėx = A 11 (ζ)e x + G 1 (g(x) -g(x) + M 1 (ζ)w, ν eq1 = -A 211 (ζ)e x -G 21 (g(x) -g(x)) -M 21 (ζ)w, (32)
where e x is the state vector, and ν eq1 is the measured output.

Remark 10. With the sliding mode observer [START_REF] Liu | Takagi-Sugeno fuzzy model based fault estimation and signal compensation with application to wind turbines[END_REF], the estimation error e x is only guaranteed to be bounded, i.e., x only estimates x within a bound. However, the SMO design allows to consider system [START_REF] Yi | Sliding mode observer for nonlinear uncertain systems[END_REF] with the following properties:

• the unknown input d is not present in (32),

• both state and output equations of system (32) are subject to the uncertainty w. System (32) enables the design of the second observer to H ∞ asymptotically estimate e x , i.e., the effect of w on the estimation error e can be minimized. Moreover, the uncertainty w and the unknown input d can also be recovered under an extra rank condition stated in Assumption 3.

C. Second Design: Luenberger-Type TS Fuzzy Observer

Consider the following TS fuzzy Luenberger-type observer structure for system [START_REF] Yi | Sliding mode observer for nonlinear uncertain systems[END_REF]:

ėx = A 11 (ζ)ê x + N j=1 h j (ζ)K j (ν eq1 + A 211 (ζ)ê x ) +   G 1 + N j=1 h j (ζ)K j G 21   (g(x + ēx ) -g(x)) , ( 33
)
where êx is the estimate of e x , the observer gains K j , for j ∈ I N , are to be determined, and ēx = column{ê x , 0}. The dynamics of the estimation error ẽx = e x -êx is given by

ėx = A 11 (ζ) + G 1 A 1 g + K(ζ) A 211 (ζ) + G 21 A 1 g ẽx + (M 1 (ζ) + K(ζ)M 21 (ζ)) w, (34)
where

K(ζ) = N j=1 h j (ζ)K j .
Hereafter, the observer gain K(ζ) is designed such that the following condition holds:

Ve (ẽ x ) + ẽ x ẽx -γ 2 w w < 0, (35)
where V e (ẽ x ) = ẽ x Qẽ x , with Q 0, is a Lyapunov function candidate, and γ > 0 is the level of disturbance attenuation to be minimized.

Remark 11. Note from (35) that if γ is minimized to be arbitrarily small, i.e., γ ≈ 0, then the effect of w on the estimation error ẽx can be negligible. Using the H ∞ filtering based approach, condition [START_REF] Ferreira De Loza | High-order sliding-mode observer-based input-output linearization[END_REF] guarantees the H ∞ asymptotic stability of system [START_REF] Jiang | Takagi-Sugeno model-based sliding mode observer design for finite-time synthesis of semi-Markovian jump systems[END_REF] without requiring to solve a set of algebraic equations to cancel the effect of w on ẽx .

The following lemma is useful to design the TS fuzzy observer [START_REF] Ferreira De Loza | Unmatched uncertainties compensation based on high-order sliding mode observation[END_REF].

Lemma 2 ([43]

). Consider the inequality

N i=1 N j=1 h i (ζ)h j (ζ)Ψ ij ≺ 0, (36)
where Ψ ij , for i, j ∈ I N , are symmetric matrices of appropriate dimensions, and the functions h i (ζ) satisfy the convex sum property [START_REF] Pletschen | Nonlinear state estimation for suspension control applications: a Takagi-Sugeno Kalman filtering approach[END_REF]. Condition (36) holds if

Ψ ii ≺ 0, ∀i ∈ I N , 2 N -1 Ψ ii + Ψ ij + Ψ ji ≺ 0, ∀i, j ∈ I N , i = j.
The following theorem provides sufficient conditions to design the observer (33) while minimizing the value of γ in [START_REF] Ferreira De Loza | High-order sliding-mode observer-based input-output linearization[END_REF].

Theorem 3. Consider the error estimation dynamics [START_REF] Jiang | Takagi-Sugeno model-based sliding mode observer design for finite-time synthesis of semi-Markovian jump systems[END_REF]. Condition ( 35) is guaranteed with a minimal γ if there exist matrices of appropriate dimensions Q 0, Y i (1 ≤ i ≤ N ), and a scalar γ > 0 that solve the convex optimization problem min(γ) subject to

Φ ii ≺ 0, ∀i ∈ I N , (37) 2
N -1 Φ ii + Φ ij + Φ ji ≺ 0, ∀i, j ∈ I N , i = j, (38)
with

Φ ij = He Q A 11i + G 1 A 1 g + Y j A 211i + G 21 A 1 g + I QM 1i + Y j M 21i -γI , for ∀A 1 g ∈ V H 1 g .
Moreover, the observer gains in [START_REF] Ferreira De Loza | Unmatched uncertainties compensation based on high-order sliding mode observation[END_REF] are obtained as K j = Q -1 Y j , for j ∈ I N , and γ = √ γ.

Proof. Since Y (ζ) = QK(ζ) and γ = γ 2 , we have

Ve (ẽ x ) + ẽ x ẽx -γ 2 w w = ẽ x He Q(A 11 (ζ) + G 1 A 1 g ) + Y (ζ) A 211 (ζ) + G 21 A 1 g + I ẽx + 2ẽ x (QM 1 (ζ) + Y (ζ)M 21 (ζ))w -γw w = N i=1 N j=1 h i (ζ)h j (ζ)[ẽ x w ]Φ ij ẽx w . ( 39
)
Applying Lemma 2, it follows that condition [START_REF] Ferreira De Loza | High-order sliding-mode observer-based input-output linearization[END_REF], or equivalently [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF], is verified if conditions ( 37) and ( 38) hold. Moreover, the value of γ = √ γ is minimized by the minimization of γ.

Remark 12. Since y = [0 I p ]x is already available from the output, we only need to estimate χ = [I n-p 0]x. We have χ = χ + e x , where e x is bounded. If êx can be designed to H ∞ asymptotically estimate e x , then χ is H ∞ asymptotically estimated by

χ = χ + êx . ( 40
)
Note that, if γ can be minimized to be sufficiently small γ ≈ 0, the estimation of χ by χ is similar to asymptotic estimation.

The following assumption is necessary for the estimation of the uncertainty w and the unknown input d.

Assumption 3. The matrix M 21 (ζ), obtained from the partitions ( 21) and ( 29) of the uncertainty structure matrix M (ζ) in ( 2), is full column rank.

Remark 13. Note from ( 2), ( 21) and ( 29) that M 21 (ζ) depends on the uncertainty structure of a given system. Hence, Assumption 3 should be checked on a case-by-case basis to be able to estimate both w and d.

Let us denote

M † 21 (ζ) = (M 21 (ζ) M 21 (ζ)) -1 M 21 (ζ) . It follows from (30) that w = -M † 21 (ζ) (ν eq1 + A 211 (ζ)e x + G 21 (g(x) -g(x))) . (41)
From ( 41), we can estimate the uncertainty w as

ŵ = -M † 21 (ζ) (ν eq1 + A 211 (ζ)ê x + G 21 ∆ g ) , (42)
with ∆ g = g(x) -g(x) and x = column{ χ, y}. Similarly, it follows from ( 31) that

d = -D -1 22 (ν eq2 + A 212 (ζ)e x + G 22 (g(x) -g(x)) + M 22 (ζ)w) . (43)
From ( 43), the unknown input d can be estimated by

d = -D -1 22 (ν eq2 + A 212 (ζ)ê x + G 22 ∆ g + M 22 (ζ) ŵ) . ( 44
)
Since χ is an H ∞ asymptotic estimation of χ, from ( 41) and ( 42), ŵ provides an H ∞ asymptotic estimation for w. A similar remark can be done for the estimation of d with (44). Note that, if the minimal γ is sufficiently small γ ≈ 0, the estimations of w and d are similar to asymptotic estimation.

Remark 14. When g(x) ≡ 0, by denoting

f i = [w d] ,
and

D(ζ) = [M (ζ) D]
, system (3) can be rewritten into a similar form with system (1)-( 2) in [START_REF] Alwi | Fault reconstruction using a LPV sliding mode observer for a class of LPV systems[END_REF], [START_REF] Chen | Sliding mode observers for a class of linear parameter varying systems[END_REF]. Note that, it is assumed in [START_REF] Alwi | Fault reconstruction using a LPV sliding mode observer for a class of LPV systems[END_REF] that D(ζ) can be factorized into a constant and a varying component. This assumption is necessary for the existence of the coordinate transformation in the next step to transform ( 1)-( 2) in [START_REF] Alwi | Fault reconstruction using a LPV sliding mode observer for a class of LPV systems[END_REF] to the required form. If D(ζ) cannot be factorized into a constant and a varying component, the result in [START_REF] Alwi | Fault reconstruction using a LPV sliding mode observer for a class of LPV systems[END_REF] as well as its direct extension cannot be applied. Similarly, the result in [START_REF] Yan | Nonlinear robust fault reconstruction and estimation using a sliding mode observer[END_REF] as well as its direct extension cannot be applied in this case. However, the proposed observer scheme in this paper is still applicable. On the other hand, the SMO scheme in [START_REF] Chen | Sliding mode observers for a class of linear parameter varying systems[END_REF] would strictly require p > q + n w while our observer scheme only requires p > q. Thus, the scheme in this paper still works in many cases where the scheme in [START_REF] Chen | Sliding mode observers for a class of linear parameter varying systems[END_REF] as well as its direct extension is not applicable. The advantage of our result over the results in [START_REF] Yan | Nonlinear robust fault reconstruction and estimation using a sliding mode observer[END_REF], [START_REF] Alwi | Fault reconstruction using a LPV sliding mode observer for a class of LPV systems[END_REF], [START_REF] Chen | Sliding mode observers for a class of linear parameter varying systems[END_REF] is illustrated in Example 2 in Section IV.

The proposed cascade TS fuzzy observer design is summarized in Algorithm 1.

Algorithm 1: Cascade Observer Design Procedure Input: Nonlinear uncertain system (1).

Output: Cascade TS fuzzy observer ( 12), ( 33) and ( 40), UI estimator (44), and uncertainty estimator [START_REF] Coutinho | Unknown input observer for nonlinear systems with arbitrary relative degree[END_REF]. 1 Verify the conditions in Assumptions 1 and 2.

• If YES, then go to Step 2.

• If NO, then the proposed method is unapplicable to the considered system. 2 Transform system (1) into the desired form ( 10)-( 11) (if necessary), and obtain the equivalent TS fuzzy model (4).

3 Solve the optimization problem in Theorem 1 to compute the gain L(ζ) for the sliding mode observer [START_REF] Liu | Takagi-Sugeno fuzzy model based fault estimation and signal compensation with application to wind turbines[END_REF], and determine P y for the discontinuous signal ν. 4 Use condition [START_REF] Brahim | Robust fault detection for uncertain T-S fuzzy system with unmeasurable premise variables: Descriptor approach[END_REF] to determine µ for ν. 5 Solve the optimization problem in Theorem 3 to compute the gain K(ζ) for the Luenberger observer [START_REF] Ferreira De Loza | Unmatched uncertainties compensation based on high-order sliding mode observation[END_REF]. 6 Estimate the unmeasurable state x with the expression of χ in [START_REF] Zemouche | Observers for a class of Lipschitz systems with extension to H∞ performance analysis[END_REF]. 7 Verify the rank condition in Assumption 3.

• If YES, then go to Step 8.

• If NO, then we cannot estimate the UI and the uncertainty. 8 Estimate the uncertainty w with [START_REF] Coutinho | Unknown input observer for nonlinear systems with arbitrary relative degree[END_REF], and the unknown input d with (44).

IV. ILLUSTRATIVE EXAMPLES

To illustrate the effectiveness and advantage of the proposed results over existing results in the literature, we will consider two numerical examples including a practical example of vehicle nonlinear dynamics and an academic example. Simulations of both examples are carried out via MATLAB with a sampling time of 0.001 [s]. All the convex optimization problems are solved using MATLAB Robust Control Toolbox.

Example 1. We consider the estimation problem of the vehicle nonlinear dynamics. Under normal driving conditions without longitudinal slip and with small angles assumption, the vehicle dynamics can be described as follows [START_REF] Nguyen | Takagi-Sugeno fuzzy unknown input observers to estimate nonlinear dynamics of autonomous ground vehicles: Theory and real-time verification[END_REF], [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF]:

vy = - 2(C f + C r )v y m v v x + 2(C r l r -C f l f )y r m v v x -y r v x + 2C f δ m v - C dy ρ a A f y v 2 y 2m v , ẏr = 2(C r l r -C f l f )v y I z v x - 2(C f l 2 f + C r l 2 r )y r I z v x + 2C f l f δ I z , vx = y r v y - C dx ρ a A f x v 2 x 2m v + T w m v R t , (45)
where the vehicle nomenclature is given in Table I. Here, the numerical parameters are taken from the INSA autonomous vehicle in our LAMIH-CNRS laboratory.

C dx Longitudinal drag coefficient 0.32 [-] A f y Lateral frontal area 2.01 [m 2 ] A f x Longitudinal frontal area 1.97 [m 2 ]
Note that the yaw rate y r and the longitudinal speed v x can be measured online. However, the measurement of the lateral speed v y is not available due to cost reasons. The steering angle u = δ is considered as a known control input, while the wheel torque d = T w is an unknown input. Furthermore, it is not possible to perfectly identify the tire cornering stiffness in practice [START_REF] Zhang | A novel observer design for simultaneous estimation of vehicle steering angle and sideslip angle[END_REF]. For illustrations, we assume that the rear cornering stiffness C r is an uncertain parameter, i.e., C r = C r0 + ∆C r , where C r0 = 56636 is the nominal value and ∆C r represents the parametric uncertainty: ∆C r = C r∆ C r0 F (t), with C r∆ ∈ (0, 1) and |F (t)| ≤ 1. In this paper, we consider the case with 10% of parametric uncertainty, i.e., C r∆ = 0.1, and we assume that F (t) = cos(t) for simulation purposes. The vehicle system (45) can be reformulated in the form (3) where

x = v y y r v x , ζ = y r 1 vx
, and

A(ζ) =    - 2(C f +Cr0) mvvx 2(Cr0lr-C f l f ) mvvx -y r 2(Cr0lr-C f l f ) Izvx - 2(C f l 2 f +Cr0l 2 r ) Izvx 0 y r 0 0    , f (u, y) =    2C f δ mv 2C f l f δ Iz - C dx ρaA f x v 2 x 2mv    , G =   - C dy ρaA f y 2mv 0 0   , D =   0 0 1 mvRt   , M (ζ) =   2Cr0Cr∆ mvvx -2Cr0Cr∆lr Izvx 0   , g(x) = v 2 y , C = [0 I 2 ], N = [-1 l r 0]. (46)
Considering the physical limitations during normal driving conditions [START_REF] Nguyen | Takagi-Sugeno fuzzy unknown input observers to estimate nonlinear dynamics of autonomous ground vehicles: Theory and real-time verification[END_REF], the vehicle state-space compact set is defined as

D x = (v y , y r , v x ) ∈ R 3 : v y ∈ [v y , v y ], y r ∈ [y r , y r ], v x ∈ [v x , v x ] ,
where v y = -1.

h 1 (ζ) = (y r -y r )(1/v x -1/v x ) (y r -y r )(1/v x -1/v x ) , h 2 (ζ) = (y r -y r )(1/v x -1/v x ) (y r -y r )(1/v x -1/v x ) , h 3 (ζ) = (y r -y r )(1/v x -1/v x ) (y r -y r )(1/v x -1/v x ) , h 4 (ζ) = (y r -y r )(1/v x -1/v x ) (y r -y r )(1/v x -1/v x ) .
It is clear that

4 i=1 h i (ζ) = 1, h i (ζ) ≥ 0, for i ∈ I 4 , and y r = y r (h 1 (ζ) + h 3 (ζ)) + y r (h 2 (ζ) + h 4 (ζ)), ( 47
) 1 v x = h 1 (ζ) + h 2 (ζ) v x + h 3 (ζ) + h 4 (ζ) v x , (48)
Substituting ( 47) and (48

) into A(ζ) and M (ζ), it follows that A(ζ) = 4 i=1 h i (ζ)A i and M (ζ) = 4 i=1 h i (ζ)M i , with A 1 = A(y r , 1/v x ), A 2 = A(y r , 1/v x ), A 3 = A(y r , 1/v x ), A 4 = A(y r , 1/v x ), M 1 = M (y r , 1/v x ), M 2 = M (y r , 1/v x ), M 3 = M (y r , 1/v x ), M 4 = M (y r , 1/v x ).
Since rank(CD) = rank(D) and rank(D) = 1 < 2 = rank(C), Assumption 1 holds. As shown in ( 46), the vehicle model (45) already has C and D in the form ( 10)-( 11), thus a coordinate transformation is not required. Note that Assumption 2 also holds where g(x) = v 2 y is differentiable with respect to v y and 2v y ≤ ∂g ∂vy = 2v y ≤ 2v y . Accordingly, the sets of vertices are obtained as V H 1 g = 2v y , 2v y , V H 10 Note that, since p = 2 = q + n w , the SMO scheme in [START_REF] Chen | Sliding mode observers for a class of linear parameter varying systems[END_REF] as well as its direct extension are not applicable to system (45). Solving the convex optimization problem in Theorem 1, we obtain the observer gain for the sliding mode observer [START_REF] Liu | Takagi-Sugeno fuzzy model based fault estimation and signal compensation with application to wind turbines[END_REF] as with the attenuation level λ = 0.0781. With regard to the discontinuous signal ν in (13), the matrix P y is computed as P y = 0.0449 0.0000 0.0000 1.4127 , and µ is computed by [START_REF] Brahim | Robust fault detection for uncertain T-S fuzzy system with unmeasurable premise variables: Descriptor approach[END_REF] as µ = 23. For the observer implementation, to avoid the chattering, the discontinuous signal ν is approximated by

L 1 =   0.
ν δ = -µ P -1 y e y ||e y || + δ , δ = 0.3.
Solving the convex optimization problem in Theorem 3, we can compute the observer gain for observer [START_REF] Ferreira De Loza | Unmatched uncertainties compensation based on high-order sliding mode observation[END_REF] as K i = 1.08, for i ∈ I 4 , with the attenuation level γ = 8.1370 × 10 -8 . e x is estimated by êx , and χ = v y is estimated by χ = χ + êx accordingly. Furthermore, the unknown input d is estimated with (44), and the uncertainty w is estimated with [START_REF] Coutinho | Unknown input observer for nonlinear systems with arbitrary relative degree[END_REF].

Hereafter, a realistic driving scenario is considered to demonstrate the estimation performance of the proposed cascade observer scheme. Note that due to the lack of vehicle sensors, the validation tests have been performed with the nonlinear vehicle model (45). However, for each driving scenario, the real data of the steering angle δ and the wheel torque T w , collected from the INSA autonomous vehicle, are used as vehicle inputs. In this driving scenario, the vehicle follows a random driving trajectory, then enters and exits a roundabout. The corresponding vehicle trajectory, longitudinal speed, and steering angle are depicted in Figs. 2(a),(b) and (c), respectively. The convergence to the sliding surface of the SMO error e y is illustrated in Fig. 2(d). The estimation performance of the proposed cascade TS fuzzy observer is demonstrated in Fig. 3. The estimation of the SMO error e x is illustrated in Fig. 3(a). Meanwhile, Fig. 3(b) shows the estimation of the unmeasured lateral speed v y , Fig. 3(c) shows the estimation of the wheel torque T w considered as unknown input d, and Fig. 3(d) shows the estimation of the parametric uncertainty w. We can see that the proposed observer scheme provides a satisfactory estimation performance despite the presence of the parametric uncertainty.

A(ζ) =     -3 1 1 0 1 -2 0 0 -x 3 -1 -3 1 x 2 0 0 -1     , f (u, y) =     u 0 0 0     , G =     0 0.1 0 0     , D =     0 0 0 1     , M (ζ) =     -0.3 0.1 0.2 0 -0.1x 3 -0.1 0.1x 2 0     , g(x) = x 3 1 , C = [0 I 3 ], N = [I 2 0]. (49)
For illustrations, let u ≡ 1, d = 0.1 sin(0.2t), and F = diag(cos(t), sin(t)). The state-space compact set is defined as

D x = [-1, 1] × [-1, 1] × [-1, 1] × [-1, 1].
Using the sector nonlinearity method [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] with the vector of measured premise variables ζ = [x 2 x 3 ] , an equivalent four-rule TS fuzzy model of system (49) can be obtained for x ∈ D x . The membership functions h i (ζ) are given by

h 1 (ζ) = (1 -x 3 )(1 -x 2 ) 4 , h 2 (ζ) = (x 3 + 1)(1 -x 2 ) 4 , h 3 (ζ) = (1 -x 3 )(x 2 + 1) 4 , h 4 (ζ) = (x 3 + 1)(x 2 + 1) 4 .
It is easy to verify that

4 i=1 h i (ζ) = 1, h i (ζ) ≥ 0, ∀i ∈ I 4 , and
x 2 = -h 1 (ζ) -h 2 (ζ) + h 3 (ζ) + h 4 (ζ), (50)
x 3 = -h 1 (ζ) -h 3 (ζ)) + h 2 (ζ) + h 4 (ζ). (51)
Substituting ( 50) and (51 Then, the results in [START_REF] Yan | Nonlinear robust fault reconstruction and estimation using a sliding mode observer[END_REF], [START_REF] Alwi | Fault reconstruction using a LPV sliding mode observer for a class of LPV systems[END_REF] as well as their direct extensions cannot be applied. On the other hand, since p = 3 = q + n w , the SMO scheme in [START_REF] Chen | Sliding mode observers for a class of linear parameter varying systems[END_REF] and its direct extension are not applicable to system (49). Solving the convex optimization problem in Theorem 1, we obtain the observer gain for the sliding mode observer [START_REF] Liu | Takagi-Sugeno fuzzy model based fault estimation and signal compensation with application to wind turbines[END_REF] as [START_REF] Brahim | Robust fault detection for uncertain T-S fuzzy system with unmeasurable premise variables: Descriptor approach[END_REF] as µ = 10. To avoid the chattering, the discontinuous signal ν is approximated by ν δ = -µ P -1 y e y ||e y || + δ , δ = 0.03.

) into A(ζ) and M (ζ), it follows that A(ζ) = 4 i=1 h i (ζ)A i and M (ζ) = 4 i=1 h i (ζ)M i , with A 1 = A(-1, -1), A 2 = A(-1, 1), A 3 = A(1, -1), A 4 = A(1, 1), M 1 = M (-1, -1), M 2 = M (-1, 1), M 3 = M (1, -1)
L 1 =     0.
Solving the convex optimization problem in Theorem 3, we can compute the observer gain for the Luenberger-type observer [START_REF] Ferreira De Loza | Unmatched uncertainties compensation based on high-order sliding mode observation[END_REF] as

K 1 = K 3 = [1 1], K 2 = K 4 = [2 1]
, with the attenuation level γ = 3.6297 × 10 -7 . Since γ ≈ 0, the estimations of e x , x 1 , d, w are similar to asymptotic estimation.

V. CONCLUDING REMARKS

We have studied the observer design for a class of nonlinear systems with parametric uncertainties and unknown inputs based on TS fuzzy models and sliding mode technique. To this end, two cascaded observers have been constructed. A sliding mode observer is first designed which allows to examine a new system whose both state and output equations are subject to only uncertainties, without unknown inputs. Then, a Luenberger-type observer is designed for the new system where the effect of uncertainties on the estimation error can be eliminated. Accordingly, the overall cascade observer scheme can provide the estimations of the system state as well as the unknown input and the uncertainty. Two numerical examples including an application on the estimation of vehicle nonlinear dynamics have been provided to demonstrate the effectiveness and advantage of the new results. For future works, we focus on the experimental validation of the proposed method with the INSA autonomous vehicle and real vehicle sensors. Investigating systems with output measurements subject to sensor faults is also a promising extension of this paper.

Fig. 1 .

 1 Fig. 1. TS fuzzy cascade observer structure.

  5 [m/s], v y = 1.5 [m/s], y r = -0.6 [rad/s] and y r = 0.6 [rad/s], and v x = 5 [m/s], v x = 30 [m/s]. Using the sector nonlinearity method [2] with the vector of measured premise variables ζ = [y r 1 vx ] , an equivalent four-rule TS fuzzy model of the vehicle nonlinear system (45) can be obtained for x ∈ D x . To this end, we define the membership functions h i (ζ), for i ∈ I 4 , as

  g = [2v y 0 0], [2v y 0 0] . Since M 21 (ζ) = -2C r0 C r∆ l r /(I z v x ), where v x ∈ [5, 30], M 21 (ζ) is full column rank, which verifies Assumption 3.

Fig. 2 .

 2 Fig. 2. Example 1: (a) vehicle trajectory X-Y , (b) longitudinal speed, (c) steering angle δ, (d) sliding motion of ey = [e 2 e 3 ] .

Fig. 3 .

 3 Fig. 3. Example 1: (a) estimation of the SMO error ex, (b) estimation of the lateral speed vy, (c) estimation of the wheel torque d = Tw, (d) estimation of the uncertainty w.

Example 2 .

 2 Let us consider a nonlinear systems subject to uncertainty and unknown input which can be represented in the form (3) where x = x 1 x 2 x 3 x 4 , ζ = x 2 x 3 , and

, M 4 = 1 ≤ 3 . 10 g

 41310 M (1, 1).As rank(CD) = rank(D) and rank(D) = 1 < 3 = rank(C), Assumption 1 holds. System (49) already has C and D in the form (10)-[START_REF] Boulkroune | Design of a unified adaptive fuzzy observer for uncertain nonlinear systems[END_REF], thus a coordinate transformation is not required. Assumption 2 holds as g(x) = x3 1 is differentiable with respect to x 1 and 0 ≤ ∂g ∂x1 = 3x2 The sets of vertices are obtained as V H 1 g = {0, 3}, V H 1x 3 -0.1 has non-zero determinant, M 21 (ζ) is full column rank and Assumption 3 is verified. It is important to note that, in this example, M (ζ) and accordingly [M (ζ) D] cannot be factorized into a constant and a varying component.

Fig. 4 .

 4 Fig. 4. Example 2: (a) sliding motion of ey = [e 2 e 3 e 4 ] (b) estimation of the SMO error ex, (c) estimation of the unmeasurable state x 1 , (d) estimation of the unknown input d, (e) estimation of the uncertainty w 1 , (f) estimation of the uncertainty w 2 .

  n and g(x) ∈ R ng are nonlinear functions; and A(ζ) ∈ R n×n is the nominal system matrix. The parametric uncertainty matrix ∆A(ζ) is represented as

  since system (1) operates in a compact set, we have ζ j ∈ [ζ j , ζ j ], ∀j ∈ I n ζ . Let us define the following membership grades µ 0 j (ζ j ) and µ 1 j (ζ j ) of ζ j :

  with the attenuation level λ = 0.1055. With regard to the discontinuous signal ν in (13), the matrix P y is computed as

			2287	0.2557 -0.1939 		0.1850 -0.4010 -0.0951 
		8.0747 -0.6136 -0.5473 -0.6415 2.0668 0.4986    , L 2 =	   -0.9885 2.0164 8.0624 -1.6174 -0.4579 0.9675    ,
		-0.2441 0.3592		2.9191	-0.2340 0.8682	2.8680
			0.2254	0.1570 0.4116			0.1814 -0.4994 0.5098	
	L 3 =	   -0.6208 2.0496 0.9564 8.0754 -0.7237 0.4758	   ,	L 4 =	   -0.9789 2.0431 0.4828 8.0914 -1.7615 0.6510	   ,
			0.1711	0.9317 2.8724		0.1441	0.4175 2.9164
						0.3734	0.0714 -0.0054	
				P y =		0.0714	1.0912 -0.1638	 ,
					-0.0054 -0.1638 1.0076
	and µ is computed by

This work was supported by the French Ministry of Higher Education and Research, the National Center for Scientific Research (CNRS), and the Hautsde-France region under the ELSAT 2020 project.

DATA AVAILABILITY STATEMENT

Data sharing is not applicable to this article as no new data were created or analyzed in this study.