Z Wang

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

Avoiding Unmeasured Premise Variables in

Designing Unknown Input Observers for Takagi-Sugeno Fuzzy Systems Anh-Tu Nguyen * , Member, IEEE, Juntao Pan, Thierry-Marie Guerra, and Zhenhua Wang

Abstract-This paper investigates the design of unknown input (UI) observers for a large class of nonlinear systems using Takagi-Sugeno (TS) fuzzy modeling. To avoid the well-known issue on the unmeasured premise variables in fuzzy observer design, we reformulate the nonlinear systems in a TS fuzzy form with local nonlinear models. A particular feature of these so-called N-TS fuzzy models is that all the unmeasured nonlinearities are isolated in a nonlinear consequent. Together with a judicious use of the differential mean value theorem, the N-TS fuzzy reformulation enables an effective framework to design fuzzy UI observers. Based on an UI decoupling technique, no specific information on the UI is required for fuzzy observer design. The asymptotic estimations of both the state and the UI are guaranteed with fuzzy Lyapunov arguments. The observer gains can be effectively computed following an LMI-based design procedure. Numerical illustrations are given to demonstrate the interests of the proposed method over related existing results.

Index Terms-Takagi-Sugeno models, fuzzy observers, unmeasured premise variables, unknown inputs, Lyapunov method.

I. INTRODUCTION

S TATE estimation of nonlinear systems is a basic concern in control theory and applications. Since the pionner work on linear time-invariant systems [START_REF] Luenberger | Observers for multivariable systems[END_REF], numerous extensions have been studied for nonlinear cases, see [START_REF] Besançon | Nonlinear Observers and Applications[END_REF] for an overview. Moreover, unmodeled dynamics, uncertain disturbances and faults in engineering systems, or attack signals in secure communication or cyber-physical systems can be all considered as unknown inputs (UIs) [START_REF] Nguyen | Simultaneous estimation of state and unknown input with ∞ guarantee on error-bounds for fuzzy descriptor systems[END_REF]. Hence, observer design for nonlinear systems affected by UIs has played a key role for fault diagnosis [START_REF] Chen | Fuzzy nonlinear unknown input observer design with fault diagnosis applications[END_REF]- [START_REF] Zhao | A novel approach to state and unknown input estimation for Takagi-Sugeno fuzzy models with applications to fault detection[END_REF], robust 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] Lan | Integrated design of fault-tolerant control for nonlinear systems based on fault estimation and T-S fuzzy modeling[END_REF]. Motivated by these issues, we are interested in designing UI observers for nonlinear systems, described as

x k+1 = Ψ(x k , u k ) + Dd k , y k = Cx k , (1)
where

x k ∈ D x ⊆ R nx is the system state, u k ∈ D u ⊆ R nu is the known input, d k ∈ R n d is the unknown input, y k ∈ R
lim k→∞ |x k -xk | = 0, lim k→∞ |d k -dk | = 0.
Takagi-Sugeno (TS) fuzzy model-based technique [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] has been shown as a promising framework to deal with this problem [START_REF] Rotondo | Robust unknown input observer for state and fault estimation in discrete-time TS systems[END_REF], [START_REF] Jia | Fault reconstruction and faulttolerant control via learning observers in Takagi-Sugeno fuzzy descriptor systems with time delays[END_REF], [START_REF] Chadli | Robust observer design for unknown inputs TS models[END_REF]. The major interests of TS fuzzy approaches come from two main factors. First, TS fuzzy modeling can be used to approximate any smooth nonlinear system with any preciseness [START_REF] Nguyen | Fuzzy control systems: Past, present and future[END_REF]. In particular, the sector nonlinearity approach [10, Ch. 2] offers a systematic way to derive an equivalent TS fuzzy representation of the original nonlinear system in a compact set of the state space. Second, based on Lyapunov method, design conditions for fuzzy observers/controllers are derived in the form of linear matrix inequalities (LMIs), which are efficiently solved via convex optimization techniques, see for instance [START_REF] Li | Observer-based fuzzy control for nonlinear networked systems under unmeasurable premise variables[END_REF]- [START_REF] Wang | H∞ observer design for fuzzy system with immeasurable state variables via a new Lyapunov function[END_REF]. To design a fuzzy observer for the nonlinear system (1), existing approaches have been mainly based on the following classical TS fuzzy representation:

x k+1 = n i=1 η i (z k ) (A i x k + B i u k + Dd k ) , (2)
where the matrices of appropriate dimensions (A i , B i ) are constant, n is the number of fuzzy rules, z k is the vector of premise variables. The membership functions (MFs) are such that

η i (z k ) ≥ 0, n i=1 η i (z k ) = 1.
The existing TS fuzzy approaches to design fuzzy UI observers usually require some a priori information on the UIs. First, assuming that the UIs are of polynomial forms [START_REF] Koenig | Unknown input proportional multiple-integral observer design for linear descriptor systems: application to state and fault estimation[END_REF], fuzzy proportional-integral observers [START_REF] Han | State and disturbance observers-based polynomial fuzzy controller[END_REF] are only suitable to estimate low-frequency UI signal. Second, the forms of the UIs are assumed to be known and generated by an exogenous system [START_REF] Wu | Disturbance rejection fuzzy control for nonlinear parabolic PDE systems via multiple observers[END_REF]. Third, norm-bound conditions on the UI and/or its time derivatives are also required in [START_REF] Jia | Fault reconstruction and faulttolerant control via learning observers in Takagi-Sugeno fuzzy descriptor systems with time delays[END_REF], [START_REF] Zhang | Disturbance observerbased integral sliding-mode control for systems with mismatched disturbances[END_REF]. However, such a priori information on the UIs is not always available for many practical situations. To avoid this drawback, UI decoupling approaches have been proposed [START_REF] Zhao | A novel approach to state and unknown input estimation for Takagi-Sugeno fuzzy models with applications to fault detection[END_REF], [START_REF] Chadli | Robust observer design for unknown inputs TS models[END_REF], [START_REF] Vu | State/disturbance observer and controller synthesis for the T-S fuzzy system with an enlarged class of disturbances[END_REF]. However, the system output in [START_REF] Chadli | Robust observer design for unknown inputs TS models[END_REF] must be of the form y k = Cx k + F d k , with a full-row rank matrix F , leading to a restrictive design [START_REF] Rotondo | Robust unknown input observer for state and fault estimation in discrete-time TS systems[END_REF]. Moreover, the approaches in [START_REF] Zhao | A novel approach to state and unknown input estimation for Takagi-Sugeno fuzzy models with applications to fault detection[END_REF], [START_REF] Vu | State/disturbance observer and controller synthesis for the T-S fuzzy system with an enlarged class of disturbances[END_REF] can be only applied to TS fuzzy systems with measurable premise variables.

Two cases are distinguished for fuzzy observer design depending on the real-time availability of vector z k [START_REF] Bergsten | Observers for Takagi-Sugeno fuzzy systems[END_REF]. For the first case, z k can be fully measured from the output y k . Then, the MFs η i (z k ), for i = 1, 2, . . . , n, are directly used to construct the fuzzy observers. The main goal here is to reduce the design conservativeness via various matrix transformations and fuzzy Lyapunov functions [START_REF] Nguyen | Simultaneous estimation of state and unknown input with ∞ guarantee on error-bounds for fuzzy descriptor systems[END_REF], [START_REF] Xie | Relaxed fuzzy observer design of discrete-time nonlinear systems via two effective technical measures[END_REF], [START_REF] Li | Weighted fuzzy observerbased fault detection approach for discrete-time nonlinear systems via piecewise-fuzzy Lyapunov functions[END_REF]. However, the design approaches in this case can be only applied to a restrictive class of nonlinear systems. To overcome this major drawback, most of research efforts have been devoted to the second case, much more challenging, for which the premise variables z k are unmeasured or only partially measured. The technical difficulty consists in dealing with a mismatch nonlinear term, function of η i (z k ) -η i (ẑ k ), where ẑk is an estimate of z k . Up to now, the most common approach is based on the Lipschitz property of the MFs [START_REF] Chen | Fuzzy nonlinear unknown input observer design with fault diagnosis applications[END_REF], [START_REF] Chadli | Robust observer design for unknown inputs TS models[END_REF], [START_REF] Bergsten | Observers for Takagi-Sugeno fuzzy systems[END_REF]. Despite its simplicity, this approach generally leads to over-conservative results [START_REF] Ichalal | How to cope with unmeasurable premise variables in Takagi-Sugeno observer design: dynamic extension approach[END_REF]. Differential mean value theorem (DMVT) has been exploited in [START_REF] Guerra | H∞ LMIbased observer design for nonlinear systems via Takagi-Sugeno models with unmeasured premise variables[END_REF] to handle the unmeasurable MFs. However, the normbounded uncertainty approach used to deal with the unknown time-varying terms issued from the application of DMVT may lead to a complex and conservative design framework.

It is important to note that existing results on TS fuzzy observer design are mostly concerned with known inputs, see [START_REF] Ichalal | How to cope with unmeasurable premise variables in Takagi-Sugeno observer design: dynamic extension approach[END_REF], [START_REF] Guerra | H∞ LMIbased observer design for nonlinear systems via Takagi-Sugeno models with unmeasured premise variables[END_REF] and related works. Moreover, fuzzy UI observers and fuzzy fault detection/diagnosis have been mainly proposed for TS systems with measurable premise variables [START_REF] Nguyen | Simultaneous estimation of state and unknown input with ∞ guarantee on error-bounds for fuzzy descriptor systems[END_REF], [START_REF] Rotondo | Robust unknown input observer for state and fault estimation in discrete-time TS systems[END_REF], [START_REF] Jia | Fault reconstruction and faulttolerant control via learning observers in Takagi-Sugeno fuzzy descriptor systems with time delays[END_REF], [START_REF] Han | State and disturbance observers-based polynomial fuzzy controller[END_REF], [START_REF] Vu | State/disturbance observer and controller synthesis for the T-S fuzzy system with an enlarged class of disturbances[END_REF], [START_REF] Li | Weighted fuzzy observerbased fault detection approach for discrete-time nonlinear systems via piecewise-fuzzy Lyapunov functions[END_REF], [START_REF] Zhang | Reduced-order observer design for switched descriptor systems with unknown inputs[END_REF]. Some few extensions based on the conservative Lipschitz assumption to deal with the unmeasured premise variables can be found in [START_REF] Chen | Fuzzy nonlinear unknown input observer design with fault diagnosis applications[END_REF], [START_REF] Chadli | Robust observer design for unknown inputs TS models[END_REF]. DMVT has been recently exploited in [START_REF] Guerra | Decoupling unknown input observer for nonlinear quasi-LPV systems[END_REF] to design UI observers for continuous-time systems with unmeasured nonlinearities. Unfortunately, the proposed observer structure requires the information of the time-derivative of the measured premise variables, which can be unmeasurable. In particular, asymptotic estimation convergence cannot be guaranteed in [START_REF] Guerra | Decoupling unknown input observer for nonlinear quasi-LPV systems[END_REF]. Motivated by the above practical and theoretical issues, this letter proposes a new method to design fuzzy UI observers while avoiding the issue of unmeasured premise variables. Specifically, our main contributions are as follows.

i) We reformulate system (1) as a N-TS fuzzy model, for which all the unmeasured nonlinearities in (1) are isolated in a local nonlinear consequent. With a judicious use of the differential mean value theorem, this particular feature of the resulting N-TS model enables an effective framework to design UI observers with reduced conservatism. ii) The asymptotic estimations of both the state and the UI are guaranteed using fuzzy Lyapunov-based arguments.

No specific information on the UI is required for UI observer design as in most of existing results [START_REF] Jia | Fault reconstruction and faulttolerant control via learning observers in Takagi-Sugeno fuzzy descriptor systems with time delays[END_REF], [START_REF] Han | State and disturbance observers-based polynomial fuzzy controller[END_REF]- [START_REF] Zhang | Disturbance observerbased integral sliding-mode control for systems with mismatched disturbances[END_REF]. Moreover, the observer gains can be effectively computed following an LMI-based design procedure.

Notation. The set of nonnegative integers is denoted by Z + and I r = {1, 2, . . . , r} ⊂ Z + . 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 a vector x, x i denotes its ith entry. For two vectors x, y ∈ R n , the convex hull of these vectors is denoted as

co(x, y) = {λx + (1 -λ)y : λ ∈ [0, 1]}. For a matrix X, X
denotes its transpose, X 0 means X is positive definite, and HeX = X + X . When the existence is guaranteed, X † denotes the Moore-Penrose pseudo-inverse of matrix X, i.e., X † = X X -1 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. In block matrices, the symbol stands for the terms deduced by symmetry.

II. PROBLEM STATEMENT

This section formulates a new framework to design UI observers for fuzzy systems with unmeasured premise variables.

A. Fuzzy System Description

For fuzzy observer design, inspired by the N-TS fuzzy modeling [START_REF] Dong | Output feedback fuzzy controller design with local nonlinear feedback laws for discrete-time nonlinear systems[END_REF], [START_REF] Coutinho | A multipleparameterization approach for local stabilization of constrained Takagi-Sugeno fuzzy systems with nonlinear consequents[END_REF], we propose to reformulate the nonlinear system (1) as

x k+1 = A(ξ k )x k + Dd k + f (ξ k , u k ) + G(ξ k )φ(x k , u k ), (3)
where ξ k ∈ R n ξ is the vector of measured premise variables. The nonlinear functions f :

D x × D u → R nx and φ : D x × D u → R n φ
are differentiable with respect to the state x k . The elements of the vector-valued function f (•) and the matrixvalued functions A(•) and G(•) are measurable whereas those of φ(•) cannot be measured from the output. Using the fuzzy modeling technique [10, Ch. 2], system (3) can be represented by the following TS fuzzy model with nonlinear consequents:

RULE R i : IF ξ 1 is M i 1 and . . . and ξ n ξ is M i n ξ THEN x k+1 = A i x k + Dd k + f (ξ k , u k ) + G i φ(x k , u k ) (4)
where the constant matrices (A i , G i ) are known. R i denotes the ith fuzzy inference rule, and r is the number of inference rules. M i j , with i ∈ I r and j ∈ I n ξ , is the fuzzy set. The fuzzy MFs are given by h

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

r i=1 h i (ξ k ) = 1, 0 ≤ h i (ξ k ) ≤ 1, ∀i ∈ I r . (5)
Let Ω be the set of membership functions satisfying (5), i.e.,

h = h 1 (ξ k ), h 2 (ξ k ), . . . , h r (ξ k ) ∈ Ω. We denote also h + = h 1 (ξ k+1 ), h 2 (ξ k+1 ), . . . , h r (ξ k+1 )
∈ Ω. Applying the center-of-gravity method for defuzzification, the fuzzy system (4) can be represented in the compact form [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] [START_REF] Zhang | Sensor fault estimation of switched fuzzy systems with unknown input[END_REF] where

x k+1 = A(h)x k + Dd k + f (ξ k , u k ) + G(h)φ(x k , u k ),
A(h) G(h) = r i=1 h i (ξ k ) A i G i . Remark 1.
All the unmeasurable premise variables of system (1) are isolated in the nonlinear term φ(x, u) of (3). Then, the MFs h i (ξ k ), i ∈ I r , of system (6) are measurable, and can be directly used to construct fuzzy observers. This particular feature of the N-TS fuzzy modeling (3)-( 6) enables an effective fuzzy UI observer design for nonlinear system (1). Moreover, through the system reformulation (3), a part of the nonlinearities of system (1) are retained in f (ξ k , u k ) and φ(x k , u k ). Hence, the resulting N-TS fuzzy model [START_REF] Zhang | Sensor fault estimation of switched fuzzy systems with unknown input[END_REF] requires fewer fuzzy rules than the classical TS fuzzy model (2), i.e., r < n. In many situations, especially for complex nonlinear systems, this can lead to several advantages: (i) simpler observer structure, (ii) less computational burden and less conservative results for fuzzy observer design [START_REF] Dong | Output feedback fuzzy controller design with local nonlinear feedback laws for discrete-time nonlinear systems[END_REF].

The following assumptions are considered for system [START_REF] Zhang | Sensor fault estimation of switched fuzzy systems with unknown input[END_REF].

Assumption 1. We assume that matrix C is of full row rank, matrix D is of full column rank and rank(CD) = rank(D).

The rank condition in Assumption 1 is common in UI observer design framework for UI decoupling [START_REF] Jia | Fault reconstruction and faulttolerant control via learning observers in Takagi-Sugeno fuzzy descriptor systems with time delays[END_REF], [START_REF] Koenig | Unknown input proportional multiple-integral observer design for linear descriptor systems: application to state and fault estimation[END_REF], [START_REF] Vu | State/disturbance observer and controller synthesis for the T-S fuzzy system with an enlarged class of disturbances[END_REF].

Assumption 2. The function φ(x k , u k ) satisfies the condition

θ ij ≤ ∂φ i ∂x j (x, u) ≤ θ ij , for ∀(i, j) ∈ I n φ × I nx , (7)
with

θ ij = min µ∈Dx×Du ∂φ i ∂x j (µ) , θ ij = max µ∈Dx×Du ∂φ i ∂x j (µ) .
Remark 2. The boundedness condition ( 7) is not restrictive for fuzzy observer design since TS fuzzy models are generally defined within a compact set D x [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. Moreover, the input u k of engineering systems is physically bounded, i.e., u k ∈ D u . Note that in many situations the bounds of u k are not required for the proposed UI observer design, i.e., when the Jacobian of φ(x k , u k ) is independent to u k . This is not the case of the recent DMVT-based TS fuzzy observer design in [START_REF] Guerra | H∞ LMIbased observer design for nonlinear systems via Takagi-Sugeno models with unmeasured premise variables[END_REF], where the bounds of u k are always required for observer design.

B. Problem Definition

We consider the following UI observer structure:

z k+1 = N (h)z k + L(h)y k + M Φ(ξ k , u k , xk ), (8) xk
= z k -F y k , z 0 = 0, (9)
dk = (CD) † y k+1 -C Φ(ξ k , u k , xk ) -CA(h)x k , (10)
with

Φ(ξ k , u k , xk ) = f (ξ k , u k ) + G(h)φ(x k , u k ) and M = I + F C. ( 11
)
The existence of the pseudo-inverse matrix (CD) † is guaranteed by the rank condition in Assumption 1. The MFdependent matrices

N (h) ∈ R nx×nx , L(h) ∈ R nx×ny , M ∈ R nx×nx and F ∈ R nx×ny are to be designed such that L(h) N (h) = N i=1 h i (ξ k ) L i N i .
Let us define e k = xk -x k . It follows from ( 9) and ( 11) that

e k = z k -F y k -x k , or e k = z k -M x k . (12)
Then, the estimation error dynamics can be defined from ( 6), ( 8), ( 9) and ( 12) as

e k+1 = z k+1 -M x k+1 = N (h)e k + M (h)x k + M G(h)∆ φ -M Dd k , ( 13
)
where

∆ φ = φ(x k , u k ) -φ(x k , u k ) and M (h) = N (h)M + L(h)C -M A(h).
The "mismatch" term ∆ φ caused by the unmeasured nonlinearities raises major challenge for fuzzy observer design [START_REF] Chadli | Robust observer design for unknown inputs TS models[END_REF], [START_REF] Guerra | H∞ LMIbased observer design for nonlinear systems via Takagi-Sugeno models with unmeasured premise variables[END_REF]. To effectively deal with this term and guarantee an asymptotic error convergence, the following differential mean value theorem is useful to convert ∆ φ into a function of the estimation error e k .

Lemma 1.

[30] Let h(x) : R nx → R q and a, b ∈ R nx . If h(x) is differentiable on co(a, b), then there exist constant vectors

c i ∈ co(a, b), c i = a, c i = b, for ∀i ∈ I q , such that h(a) -h(b) =   q i=1 nx j=1 σ q (i)σ nx (j) ∂h i ∂x j (c i )   (a -b).
Applying Lemma 1 to the nonlinear function φ(x k , u k ), it follows that there exist χ i ∈ co(x k , xk ), i ∈ I n φ , such that

∆ φ =   n φ i=1 nx j=1 σ n φ (i)σ nx (j) ∂φ i ∂x j (χ i , u k )   (x k -xk ).( 14
)
Let us denote ρ ij = ∂φi ∂xj (χ i , u k ), for ∀(i, j) ∈ I n φ × I nx , and

ρ = ρ 11 , . . . , ρ 1nx , . . . , ρ n φ nx . (15)
Due to condition [START_REF] Zhao | A novel approach to state and unknown input estimation for Takagi-Sugeno fuzzy models with applications to fault detection[END_REF], the parameter ρ belongs to a bounded convex set S φ , whose set of 2 n φ nx vertices is given by

V φ = {ρ . = ρ 11 , . . . , ρ 1nx , . . . , ρ n φ nx : ρ ij ∈ [θ ij , θ ij ]}.
From ( 13) and ( 14), the error dynamics can be rewritten as

e k+1 = N (h, ρ)e k + M (h)x k -M Dd k , (16)
where N (h, ρ) = r i=1 h i (ξ k )N i (ρ lj ) and

N i (ρ lj ) = N i + n φ l=1 nx j=1 σ n φ (l)σ nx (j)ρ lj M G i . ( 17
)
The observer design problem of interest is stated as follows.

Problem 1. Consider a nonlinear system in the N-TS fuzzy form [START_REF] Zhang | Sensor fault estimation of switched fuzzy systems with unknown input[END_REF]. Determine the matrices N (h), L(h), M and F of the UI observer ( 8)- [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] such that the state estimate xk and the UI estimate dk converge asymptotically to the system state x k and the unknown input d k , respectively.

Using Lyapunov-based arguments, this paper proposes a tractable solution for the above observer problem.

III. UI OBSERVER DESIGN FOR FUZZY SYSTEMS WITH UNMEASURED PREMISE VARIABLES

This section develops a new framework to design UI observers for nonlinear systems in the N-TS fuzzy form [START_REF] Zhang | Sensor fault estimation of switched fuzzy systems with unknown input[END_REF].

A. On Fuzzy Unknown Input Observer Design

The following theorem provides the theoretical basis to achieve our UI observer design goal.

Theorem 1. Consider the N-TS fuzzy system (6), there is an asymptotic UI observer in the form ( 8)-( 10) if there exist MF-dependent positive definite matrix

P (h) ∈ R nx×nx , MF-dependent matrices N (h) ∈ R nx×nx , L(h) ∈ R nx×ny , matrices R ∈ R nx×nx , M ∈ R nx×nx and F ∈ R nx×ny with M = I + F C,
and the following conditions are satisfied for h, h + ∈ Ω, ρ ∈ S φ :

N (h)M + L(h)C -M A(h) = 0, (18) M D = 0, (19) P (h) RN (h, ρ) R + R -P (h + ) 0, (20)
with P (h + ) = r l=1 h l (ξ k+1 )P l . Proof. From ( 18) and ( 19), the error dynamics ( 16) can be represented as

e k+1 = N (h, ρ)e k . ( 21
)
Pre-and postmultiplying [START_REF] Zhang | Disturbance observerbased integral sliding-mode control for systems with mismatched disturbances[END_REF] with I -N (h, ρ) , one has

N (h, ρ) P (h + )N (h, ρ) -P (h) ≺ 0. ( 22
)
For the asymptotic stability analysis of the error dynamics ( 21), we consider the fuzzy Lyapunov function candidate

V(e k ) = e k P (h)e k , P (h) = r i=1 h i (ξ k )P i , (23)
with P i 0, for i ∈ I r . The variation of the fuzzy Lyapunov function [START_REF] Li | Weighted fuzzy observerbased fault detection approach for discrete-time nonlinear systems via piecewise-fuzzy Lyapunov functions[END_REF] along the trajectory of system ( 21) is defined as

∆V = V(e k+1 ) -V(e k ) = e k N (h, ρ) P (h + )N (h, ρ) -P (h) e k . ( 24
)
It follows from ( 22) and ( 24) that ∆V < 0, for ∀e k = 0. Now, we prove that the UI estimate dk in [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] asymptotically converges to d k . To this end, note from (6) that

d k = (CD) † (y k+1 -CΦ k -CA(h)x k ) , (25)
with

Φ k = f (ξ k , u k )+G(h)φ(x k , u k ).
The UI estimation error ε k = d k -dk can be obtained from ( 10) and [START_REF] Guerra | H∞ LMIbased observer design for nonlinear systems via Takagi-Sugeno models with unmeasured premise variables[END_REF] as

ε k = (CD) † C (A(h)e k + G(h)∆ φ ) .
Using expression [START_REF] Li | Observer-based fuzzy integral sliding mode control for nonlinear descriptor systems[END_REF], the UI estimation error ε k is rewritten as

ε k = (CD) † CA (h, ρ)e k , (26)
where

A (h, ρ) = r i=1 h i (ξ k )A i (ρ lj ) and A i (ρ lj ) = A i + n φ l=1 nx j=1 σ n φ (l)σ nx (j)ρ lj G i . ( 27
)
Note from the algebraic equation ( 26) that if e k → 0, then ε k → 0, when k → ∞. This concludes the proof.

Due to the nonlinear matrix equality [START_REF] Han | State and disturbance observers-based polynomial fuzzy controller[END_REF], the nonlinear matrix inequality [START_REF] Zhang | Disturbance observerbased integral sliding-mode control for systems with mismatched disturbances[END_REF], and their MF-dependency, Theorem 1 cannot be solved directly for UI observer design. Based on this result, we derive hereafter tractable conditions to design UI observer ( 8)- [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] for system [START_REF] Zhang | Sensor fault estimation of switched fuzzy systems with unknown input[END_REF]. The following technical lemma is useful for the theoretical developments. Lemma 2. [START_REF] Rao | Generalized Inverse of Matrices and Its Applications[END_REF] Consider matrix W ∈ R m×n with m ≥ n, and matrix Z ∈ R k×n . Matrix X of the form X = ZW † + Y I -WW † , with an arbitrary matrix Y ∈ R k×m , is a solution of XW = Z when the condition ZW † W = Z holds.

B. LMI-Based Fuzzy Unknown Input Observer Design

The following theorem provides an LMI-based solution for the N-TS fuzzy UI observer design in Problem 1.

Theorem 2. Consider the N-TS fuzzy system (6), there is an asymptotic UI observer in the form ( 8)-( 10) if there exist positive definite matrices

P i ∈ R nx×nx , matrices R ∈ R nx×nx , Z ∈ R nx×ny , X i ∈ R nx×ny , i ∈ I r , such that P i (R + (RU + ZV )C)A i (ρ lj ) -X i C R + R -P k 0, (28)
for i, k ∈ I r and ρ lj ∈ V φ , l ∈ I n φ , j ∈ I nx , where

U = -D(CD) † , (29)
V = I -(CD)(CD) † , (30)
Z = RY, (31)
X i = RH i . (32)
Furthermore, the matrices M ∈ R nx×nx , F ∈ R nx×ny , N i ∈ R nx×nx , and L i ∈ R nx×ny , for i ∈ I r , of the fuzzy UI observer ( 8)-( 10) are computed as

F = U + Y V, (33)
N i = M A i -H i C, (34)
L i = H i (I + CF ) -M A i F, (35)
and matrix M given by [START_REF] Chadli | Robust observer design for unknown inputs TS models[END_REF].

Proof. Note that condition (28) implies R + R 0, which guarantees the existence of R -1 . Moreover, from [START_REF] Chadli | Robust observer design for unknown inputs TS models[END_REF] and [START_REF] Wu | Disturbance rejection fuzzy control for nonlinear parabolic PDE systems via multiple observers[END_REF], we can obtain

F (CD) = -D. (36)
Under the matrix rank condition in Assumption 1, according to Lemma 2, the solution F of the matrix equation ( 36) is directly obtained in (33) with U and V defined in ( 29) and [START_REF] Zemouche | Observers for a class of Lipschitz systems with extension to H∞ performance analysis[END_REF], respectively. We define

H i = L i + N i F, ∀i ∈ I r . (37)
From ( 11), ( 18) and (37), we can obtain (34). Note also from (34) and (37) that

L i = H i -N i F = H i -(M A i -H i C)F = H i (I + CF ) -M A i F, ∀i ∈ I r ,
which is exactly expression (35). Moreover, it follows from [START_REF] Chadli | Robust observer design for unknown inputs TS models[END_REF] and (34) that

N i = M A i -H i C = (I + F C)A i -H i C = (I + (U + Y V )C)A i -H i C, ∀i ∈ I r . (38)
Using expressions [START_REF] Chadli | Robust observer design for unknown inputs TS models[END_REF], ( 27), (33), (38), followed by the variable changes ( 31)- [START_REF] Löfberg | Yalmip: A toolbox for modeling and optimization in Matlab[END_REF], inequality (28) is rewritten as

P i RN i (ρ lj ) R + R -P k 0, (39)
for i, k ∈ I r and ρ lj ∈ V φ , l ∈ I n φ , j ∈ I nx , with N i (ρ lj ) defined in [START_REF] Koenig | Unknown input proportional multiple-integral observer design for linear descriptor systems: application to state and fault estimation[END_REF]. Since h, h + ∈ Ω and due the convexity property of S φ , it follows that condition (39) implies [START_REF] Zhang | Disturbance observerbased integral sliding-mode control for systems with mismatched disturbances[END_REF].

The proof can be concluded via the result of Theorem 1.

Remark 3. Condition (28) in Theorem 2 is expressed in an LMI form, which can be effectively solved with numerical solvers, e.g., YALMIP toolbox with SDPT3 solver [START_REF] Löfberg | Yalmip: A toolbox for modeling and optimization in Matlab[END_REF].

The UI observer design algorithm is summarized as follows.

Observer Design Procedure Input: Nonlinear system in the N-TS fuzzy form [START_REF] Zhang | Sensor fault estimation of switched fuzzy systems with unknown input[END_REF]. Output: Fuzzy UI observer ( 8)- [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] such that xk → x k and dk → d k , when k → ∞.

Step 1: Check the matrix rank conditions in Assumption 1.

• If YES, then go to Step 2.

• If NO, then the proposed algorithm is unapplicable to the considered system.

Step 2: Compute the matrices U and V from ( 29) and [START_REF] Zemouche | Observers for a class of Lipschitz systems with extension to H∞ performance analysis[END_REF].

Step 3: Solve LMI-based condition [START_REF] Dong | Output feedback fuzzy controller design with local nonlinear feedback laws for discrete-time nonlinear systems[END_REF] to get R, Z, P i and X i , for i ∈ I r .

Step 4: Compute Y from [START_REF] Rao | Generalized Inverse of Matrices and Its Applications[END_REF] as Y = R -1 Z, and H i from [START_REF] Löfberg | Yalmip: A toolbox for modeling and optimization in Matlab[END_REF] as

H i = R -1 X i , for i ∈ I r .
Step 5: Compute the observer gains M , F , N i and L i from ( 11), ( 33), ( 34) and ( 35), respectively.

To illustrate the proposed fuzzy UI observer design, a numerical example is presented in the following section.

IV. NUMERICAL DISCUSSIONS

We revisit the following nonlinear discrete-time system [START_REF] Dong | Output feedback fuzzy controller design with local nonlinear feedback laws for discrete-time nonlinear systems[END_REF]:

x 1(k+1) = (1 + T )x 1k + T x 2k + sin b b T x 3k -0.1T x 4k + T (1 + x 2 1k )u k + T η(x k ) x 2(k+1) = T x 1k + (1 -2T )x 2k x 3(k+1) = T x 1k + (1 -0.3T )x 3k + T x 2 1k x 2k x 4(k+1) = (1 -T )x 4k + sin b b T x 3k + T η(x k ) + T d k y k = x 1k (40)
where

η(x k ) = sin(x 3k ) -sin b b x 3k , x 1k ∈ [-a, a], x 3k ∈ [-b, b] and b = π 2 .
The fixed step of discretization is T = 0.5. The premise variable is defined as ξ k = x 2 1k . Then, system (40) can be rewritten in the form (3) with

A(ξ k ) =     1 + T T 0 -0.1T T 1 -2T 0 0 T T ξ k 1 -0.3T 0 0 0 0 1 -T     , f (ξ k , u k ) =     T (1 + ξ k )u k 0 0 0     , G(ξ k ) = G =     T 0 0 T     , C = 1 0 0 0 , φ(x k ) = sin(x 3k ).
Using the sector nonlinearity approach [10, Ch. 2] with the measurable premise variable ξ k = x 2 1k , the following 2-rule N-TS fuzzy model can be obtained:

x k+1 = 2 i=1 h i (ξ k )A i x k + f (ξ k , u k ) + Gφ(x k ), (41)
where

A 1 =     1 + T T 0 -0.1T T 1 -2T 0 0 T T a 2 1 -0.3T 0 0 0 0 1 -T     , A 2 =     1 + T T 0 -0.1T T 1 -2T 0 0 T 0 1 -0.3T 0 0 0 0 1 -T     .
The corresponding membership functions are given by h

1 (ξ k ) = x 2 1 a 2 and h 2 (ξ k ) = a 2 -x 2 1
a 2 . We define the parameter vector ρ as in [START_REF] Xie | Relaxed fuzzy observer design of discrete-time nonlinear systems via two effective technical measures[END_REF] with ρ 1i = 0, for i ∈ {1, 2, 4}, and

ρ 13 = cos(x 3k ). Since x 3k ∈ [-π 2 , π 2
], then the bounded convex set S φ has two vertices, i.e., V φ = {ρ 13 : ρ 13 ∈ {0, 1}}. The design procedure in Section III can be now applied to design a N-TS fuzzy UI observer for system (40), equivalently represented as (41).

Note that the nonlinearity sin(x 3k ) cannot be measured, which raises technical difficulties for TS fuzzy observer design. For instance, the recent fuzzy UI observer designs with measured premise variables in [START_REF] Jia | Fault reconstruction and faulttolerant control via learning observers in Takagi-Sugeno fuzzy descriptor systems with time delays[END_REF], [START_REF] Vu | State/disturbance observer and controller synthesis for the T-S fuzzy system with an enlarged class of disturbances[END_REF] cannot be used. Due to restrictive rank conditions, the fuzzy UI observer method in [START_REF] Chadli | Robust observer design for unknown inputs TS models[END_REF] is also not applicable in this case. Indeed, TS fuzzy UI observer design, guaranteeing an asymptotic estimation performance for the discrete-time nonlinear system (40), is not yet available in the literature. To provide an idea on the conservatism of the proposed UI observer design, we compare the following methods.

• Method 1: proposed result in Theorem 2.

• Method 2: similar as Method 1 using Lipschitz property in place of DMVT, which can be considered as an extension of [28, Theorem 1]. • Method 3: extension to the discrete-time case of the TS fuzzy UI observer design using Lipschitz property in [START_REF] Chen | Fuzzy nonlinear unknown input observer design with fault diagnosis applications[END_REF]. • Method 4: extension to the discrete-time case with UI of the TS fuzzy observer design using DMVT in [START_REF] Ichalal | How to cope with unmeasurable premise variables in Takagi-Sugeno observer design: dynamic extension approach[END_REF]. For each method, the goal is to search for the largest value of parameter a, denoted by a * , for which an UI observer can be found for system (40). Table I summarizes the numerical comparisons between four considered methods. Remark that the proposed method provides a significant improvement in reducing the conservatism over other ones. Note also that using N-TS fuzzy modeling in Theorem 2 and [START_REF] Dong | Output feedback fuzzy controller design with local nonlinear feedback laws for discrete-time nonlinear systems[END_REF], the number of fuzzy rules is decreased from four to two, leading to a reduction on the computational burden, represented by the number of scalar decision variables N var and the number of rows N row of all involved LMIs, see Remark 1. For illustrations, we consider the UI observer design for system (40) with a = 9301. Fig. 1 depicts the response of the proposed observer with respect to x 0 = 1 -1 0 -1 and a randomly chosen UI signal d k = 5e -0.1k sin(7k). Remark that both the states and the UI are asymptotically estimated with a high accuracy. Moreover, differently from the UI estimation approach in [START_REF] Nguyen | Simultaneous estimation of state and unknown input with ∞ guarantee on error-bounds for fuzzy descriptor systems[END_REF], the proposed observer does not lead to an excessive initial UI estimation error ε 0 . V. CONCLUDING REMARKS An UI observer design has been proposed for nonlinear systems represented in a N-TS fuzzy form. All the unmeasured nonlinearities are contained in a nonlinear consequent part. Via the differential mean value theorem, this N-TS fuzzy reformulation allows for an effective framework to design fuzzy UI observers while avoiding the major issue on unmeasured premise variables. Both the state and the UI can be asymptotically estimated using UI decoupling technique and fuzzy Lyapunov arguments. No a priori information on the UI is required for fuzzy UI observer design. The observer gains can be easily computed from an LMI-based design procedure. Numerical illustrations are given to demonstrate the interests of the proposed fuzzy UI observer design. Future works focus on designing N-TS fuzzy UI observers for nonlinear systems, whose output is also affected by unmeasured nonlinearities.

Fig. 1 .

 1 Fig. 1. Asymptotic estimation performance obtained with system (40).

  ny is the system output. The nonlinear function Ψ : D x ×D u → R nx is differentiable with respect to the state x k . The matrices of appropriate dimensions C and D are constant. The problem is to design a dynamical observer, generating a state estimate xk and an UI estimate dk of system[START_REF] Luenberger | Observers for multivariable systems[END_REF], such that

TABLE I NUMERICAL

 I COMPARISON BETWEEN DIFFERENT METHODS.

	Design	Method 1 Method 2 Method 3 Method 4
	a *	9301.0	6341.7	371.7	7406.2
	Nb. rules	2	2	4	4
	Nvar	56	57	93	92
	Nrow	72	44	784	1040

This work was supported in part by the French Ministry of Higher Education and Research, in part by the CNRS center, in part by the Hautsde-France region under Project ELSAT 2020, in part by the Natural Science Foundation of Ningxia Hui Autonomous Region under Grant 2018AAC03107.

A.-T. Nguyen and T.-M.


Anh-Tu Nguyen

Senior Member, IEEE Thierry-Marie Guerra

Juntao Pan

Takagi-Sugeno Fuzzy Unknown Input Observers to Estimate Nonlinear Dynamics of Autonomous Ground Vehicles: Theory and Real-Time Verification

Keywords: Vehicle dynamics, nonlinear observers, vehicle state estimation, steering angle estimation, torque estimation, Takagi-Sugeno fuzzy systems

We address the simultaneous estimation problem of the lateral speed, the steering input and the effective engine torque, which play a fundamental role in vehicle handling, stability control and fault diagnosis of autonomous ground vehicles. Due to the involved longitudinal-lateral coupling dynamics and the presence of unknown inputs (UIs), a new nonlinear observer design technique is proposed to guarantee the asymptotic estimation performance. To this end, we make use of a specific Takagi-Sugeno (TS) fuzzy representation with nonlinear consequents to exactly model the nonlinear vehicle dynamics within a compact set of the vehicle state. This TS fuzzy modeling not only allows reducing significantly the realtime computational effort in estimating the vehicle variables but also enables an effective way to deal with unmeasured nonlinearities. Moreover, via a generalized Luenberger observer structure, the UI decoupling can be achieved without requiring a priori UI information. Using Lyapunov stability arguments, the UI observer design is reformulated as an optimization problem under linear matrix inequalities, which can be effectively solved with standard numerical solvers. The effectiveness of the proposed TS fuzzy UI observer design is demonstrated with realtime hardware-in-the-loop experiments.

I. INTRODUCTION

R EAL-time knowledge on the vehicle dynamics and the driver-related variables is essential for active safety control [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF], vehicle fault detection and diagnosis [START_REF] Shi | Fault diagnosis of an autonomous vehicle with an improved SVM algorithm subject to unbalanced datasets[END_REF], and driver-vehicle monitoring systems [START_REF] Nguyen | Driver-automation cooperative approach for shared steering control under multiple system constraints: Design and experiments[END_REF] of autonomous ground vehicles. Unfortunately, the onboard vehicle sensors are not always available onboard due to economical and/or technical the human driver variables cannot be directly measured by physical sensors [START_REF] Wang | Online sensing of human steering intervention torque for autonomous driving actuation systems[END_REF]. Hence, developing estimation algorithms to reconstruct the vehicle dynamics from the online information of low-cost sensors has received an ever-increasing interest worldwide [START_REF] Wang | Online sensing of human steering intervention torque for autonomous driving actuation systems[END_REF]- [START_REF] Hashemi | Opinion dynamics-based vehicle velocity estimation and diagnosis[END_REF].

The information on the lateral speed or the sideslip angle is a crucial index for vehicle handling and lateral stability. However, commercial physical sensors to measure the lateral speed are expensive, which cannot be directly used in practice [START_REF] Piyabongkarn | Development and experimental evaluation of a slip angle estimator for vehicle stability control[END_REF]. Hence, a great deal of research efforts has been devolved to the lateral speed estimation [START_REF] Zhang | Robust energy-topeak sideslip angle estimation with applications to ground vehicles[END_REF], [START_REF] Piyabongkarn | Development and experimental evaluation of a slip angle estimator for vehicle stability control[END_REF]- [START_REF] Chen | Vehicle sideslip angle and road friction estimation using online gradient descent algorithm[END_REF]. An industrially amenable kinematic-based approach has been proposed in [START_REF] Selmanaj | Vehicle sideslip estimation: A kinematic based approach[END_REF]. Without requiring tire-road friction parameters or other dynamical vehicle properties, this method can lead to the drift phenomena induced by bias errors [START_REF] Zhang | Robust energy-topeak sideslip angle estimation with applications to ground vehicles[END_REF]. Kalman filtering methods have been widely exploited for estimating sideslip angle, especially within nonlinear estimation context [START_REF] Doumiati | Onboard real-time estimation of vehicle lateral tire-road forces and sideslip angle[END_REF]- [START_REF] Viehweger | Vehicle state and tyre force estimation: demonstrations and guidelines[END_REF]. Despite their effectiveness, these methods require a fine tuning task and a priori information on the noises affected to the vehicles to achieve a satisfactory estimation performance. To overcome these drawbacks, robust observers based on dynamical vehicle properties have been proposed [START_REF] Zhang | Robust energy-topeak sideslip angle estimation with applications to ground vehicles[END_REF], [START_REF] Du | Sideslip angle estimation and stability control for a vehicle with a non-linear tyre model and a varying speed[END_REF], [START_REF] Nguyen | Unknown input observers for simultaneous estimation of vehicle dynamics and driver torque: Theoretical design and hardware experiments[END_REF]. However, the design of robust observers becomes challenging in presence of nonlinear dynamics and/or unknown disturbances, which is unavoidable for critical driving situations [START_REF] Selmanaj | Vehicle sideslip estimation: A kinematic based approach[END_REF]. By exploiting the key features of different low-cost sensors, fusion-databased techniques have been proved as an effective technique to estimate the vehicle sideslip angle [START_REF] Boada | Vehicle sideslip angle measurement based on sensor data fusion using an integrated ANFIS and an unscented Kalman filter algorithm[END_REF]- [START_REF] Cheng | Fusion algorithm design based on adaptive SCKF and integral correction for side-slip angle observation[END_REF]. However, such techniques induce additional sensor costs and complexities for estimation algorithms [START_REF] Selmanaj | Vehicle sideslip estimation: A kinematic based approach[END_REF]. It is important to emphasize that the longitudinal-lateral coupling has been ignored in most of model-based estimation methods [START_REF] Doumiati | Onboard real-time estimation of vehicle lateral tire-road forces and sideslip angle[END_REF]. The main reason is due to the challenges involved in the corresponding nonlinear observer design [START_REF] Zhang | A novel observer design for simultaneous estimation of vehicle steering angle and sideslip angle[END_REF].

Since the steering angle directly controls the vehicle direction, this variable is a fundamental input for path tracking control, path planning, active safety control and also detection of driver failure of intelligent vehicles [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF], [START_REF] Benloucif | Cooperative trajectory planning for haptic shared control between driver and automation in highway driving[END_REF]. The steering angle can be precisely measured with absolute rotary encoders, which are quite expensive and fragile to most of passenger vehicles [START_REF] Zhang | A novel observer design for simultaneous estimation of vehicle steering angle and sideslip angle[END_REF]. The low-cost sensors may lead to faulty measurement results or provide erroneous steering signals. Hence, the estimation of the steering angle has attracted increasing research attention. Based on the information from the global positioning system (GPS) and the micro-electromechanical system (MEMS), an unscented Kalman filter has been proposed in [START_REF] Si | High-precision estimation of steering angle of agricultural tractors using GPS and low-accuracy MEMS[END_REF] to estimate the steering angle for agricultural tractors. A linear extended observer has been proposed in [START_REF] Yang | Front sensor and GPS-based lateral control of automated vehicles[END_REF] for steering angle estimation, which is then exploited to reconstruct the road curvature signal for vehicle lateral control. The effective engine torque is also crucial for various automotive applications, e.g., brake torque control, speed control, adaptive cruise control [START_REF] Moon | Design, tuning, and evaluation of a fullrange adaptive cruise control system with collision avoidance[END_REF]. However, it is difficult to directly measure this vehicle variable for commercial cars due to both economic and technical reasons [START_REF] Franco | Real-time brake torque estimation for internal combustion engines[END_REF]. Then, the effective engine torque is generally supplied in the form of look-uptables (LUTs) as a function of in-cylinder air flow rate, engine speed, injected fuel, etc., which are established and calibrated by steady-state engine tests [START_REF] Franco | Real-time brake torque estimation for internal combustion engines[END_REF], [START_REF] Hedrick | Longitudinal vehicle controller design for IVHS systems[END_REF]. To reduce the costs for development, testing, maintenance, and robustness related to LUTs-based estimations, various model-based algorithms have been developed to estimate the effective engine torque. High-order sliding mode observer has been proposed in [START_REF] Ahmed | Estimating SI engine efficiencies and parameters in second-order sliding modes[END_REF] to estimate the engine friction torque and load torque. Based on the principles of Kalman filtering, a method has been developed in [START_REF] Helm | Combustion torque estimation and misfire detection for calibration of combustion engines by parametric Kalman filtering[END_REF] for estimating the engine combustion torque. A two-step observer design, requiring different intake air measurement sensors and real-time engine speed, has been proposed to estimate the real-time engine torque [START_REF] Hong | Torque observers design for SI engines with different intake air measurement sensors[END_REF]. Na et al. [START_REF] Na | Vehicle engine torque estimation via UI observer and adaptive parameter estimation[END_REF] have proposed an unknown input observer for online estimation of unknown effective engine torque with measured engine speed, load torque and air mass flow rate. Despite significant advances on vehicle estimation techniques, simultaneous estimation of both the sideslip angle and the steering angle as well as the effective engine torque has not been achieved. The significance of such an estimation solution is multi-fold. First, the estimated sideslip angle can be used for vehicle active safety control [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF], while the estimates of the steering angle and the effective engine torque are useful for fault diagnosis of intelligent automotive systems [START_REF] Shi | Fault diagnosis of an autonomous vehicle with an improved SVM algorithm subject to unbalanced datasets[END_REF]. Second, the proposed estimation solution is cost-effective since it only requires the online information from low-cost sensors. These issues motivate the new observer design method in this paper. The proposed unknown input (UI) observer design is based on a combined longitudinal-lateral vehicle model, whose steering angle and effective engine torque are considered as UIs. Two design challenges arise concerning: [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF] dealing with the unmeasured nonlinearities of vehicle dynamics, (2) guaranteeing an asymptotic convergence of both the vehicle state and the UIs. To meet similar challenges, Takagi-Sugeno (TS) fuzzy model-based techniques [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF], [START_REF] Nguyen | Fuzzy control systems: Past, present and future[END_REF] have been attempted to design nonlinear UI observers [START_REF] Zhang | A novel observer design for simultaneous estimation of vehicle steering angle and sideslip angle[END_REF], [START_REF] Jia | Fault reconstruction and faulttolerant control via learning observers in Takagi-Sugeno fuzzy descriptor systems with time delays[END_REF]. However, based on the Lipschitz property of the membership functions (MFs), the existing results generally lead to overconservative results or complex observer design frameworks [START_REF] Pan | A unified framework for asymptotic observer design of fuzzy systems with unmeasurable premise variables[END_REF]. Note also that due to technical challenges related to dealing with unmeasured MFs, fuzzy UI observer design and fuzzy fault detection/diagnosis have been mainly focused on systems with measurable nonlinearities [START_REF] Jia | Fault reconstruction and faulttolerant control via learning observers in Takagi-Sugeno fuzzy descriptor systems with time delays[END_REF], which is not the case of the considered nonlinear vehicle dynamics. The following contributions of this paper can overcome the abovementioned drawbacks.

• The nonlinear vehicle system is equivalently rewritten in a specific TS fuzzy form with both measured and unmeasured nonlinear consequents. Hence, the number of TS local subsystems, directly related to the realtime computational cost, can be significantly reduced [START_REF] Nguyen | Constrained output-feedback control for discrete-time fuzzy systems with local nonlinear models subject to state and input constraints[END_REF]. Moreover, this TS fuzzy form permits an effective application of the differential mean value theorem [START_REF] Jeon | Tracking of vehicle motion on highways and urban roads using a nonlinear observer[END_REF] to deal with unmeasured nonlinearities. • Using a generalized Luenberger observer structure, an effective UI decoupling can be achieved. Hence, an asymptotic estimation convergence of both the vehicle state and UIs can be guaranteed via Lyapunov stability without requiring any a priori UI information [START_REF] Jia | Fault reconstruction and faulttolerant control via learning observers in Takagi-Sugeno fuzzy descriptor systems with time delays[END_REF] or any specific choice of UI matrix [START_REF] Zhang | A novel observer design for simultaneous estimation of vehicle steering angle and sideslip angle[END_REF]. The UI observer design is recast as a convex optimization problem under linear matrix inequality (LMI) constraints, effectively solved with standard numerical solvers. • The effectiveness of the proposed fuzzy UI observer framework is practically verified with real-time hardwarein-the-loop (HiL) experiments. Notation. The set of nonnegative integers is denoted by Z + and I r = {1, 2, . . . , r} ⊂ Z + . For i ∈ I r , we denote ξ r (i) = [0, . . . , 0, ith 1 , 0, . . . , 0] ∈ R r a vector of the canonical basis of R r . For a vector x, x i denotes its ith entry. For two vectors x, y ∈ R n , the convex hull of these vectors is denoted as co(x, y) = {λx + (1 -λ)y : λ ∈ [0, 1]}. For a matrix X, X denotes its transpose, X 0 means X is positive definite, and HeX = X + X . When the existence is guaranteed, X † denotes the Moore-Penrose pseudo-inverse of matrix X, i.e., X † = X X -1 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. In block matrices, the symbol stands for the terms deduced by symmetry. Arguments are omitted when their meaning is clear.

II. VEHICLE MODELING AND PROBLEM FORMULATION

This section presents the nonlinear vehicle model used for UI observer design. Then, the related observer problem is formulated. The vehicle nomenclature is given in Table I. A. Nonlinear Vehicle Model Fig. 1 depicts a two degrees-of-freedom vehicle model. This model represents the vehicle motion in the horizontal plane, whose dynamics is described as follows [START_REF] Swaroop | The design of a controller for a following vehicle in an emergency lane change maneuver[END_REF]:

vx = T eng -C x v 2 x I e + v y r vy = F yf + F yr -C y v 2 y M -v x r ṙ = l f F yf -l r F yr I z (1)
where v x is the vehicle longitudinal speed, v y is the lateral speed, r is the vehicle yaw rate, and T eng represents the torque input for the vehicle longitudinal dynamics [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF]. The cornering forces at the front tires F yf and at the rear tires F yr are modeled using the magic formula [START_REF] Li | Integrated longitudinal and lateral tire/road friction modeling and monitoring for vehicle motion control[END_REF] as

F yi (α i ) = D i sin(∇ i ), i ∈ {f, r} ∇ i = C i arctan[(1 -E i )B i α i + E i arctan(B i α i ))], (2)
where the Pacejka parameters B i , C i , D i and E i depend on the characteristics of the tire, the road and the vehicle operating conditions. Note that the Pacejka tire model ( 2) is used for simulation validation purposes in Section IV. The wheel slip angles for the front and rear tires are modeled as follows [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF]:

α f = δ -arctan v y + l f r v x , α r = arctan l r r -v y v x ,
where δ is the front wheel steering angle.

B. Observer Problem Statement

For observer design, we consider the normal driving situation with small angle assumption [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF], [START_REF] Nguyen | Driver-automation cooperative approach for shared steering control under multiple system constraints: Design and experiments[END_REF], [START_REF] Piyabongkarn | Development and experimental evaluation of a slip angle estimator for vehicle stability control[END_REF]. Moreover, the lateral tire forces are proportional to the slip angles of each axle. Hence, the lateral tire forces (2) can be approximated by

F yf = 2C f δ - v y + l f r v x , F yr = 2C r l r r -v y v x , (3)
From ( 1) and ( 3), the nonlinear vehicle dynamics used for observer design can be obtained as follows:

ẋ = A v (x)x + D v d, (4)
where

x = v x v y r is the vehicle state vector, d = T eng δ
is the control input. The state-space matrices of the nonlinear vehicle model ( 4) are given by

A v (x) =   a 11 0 v y 0 a 22 a 23 0 a 32 a 33   , D v =   d 11 0 0 d 22 0 d 32   , with a 11 = - C x v x I e , d 11 = 1 I e a 22 = - 2(C f + C r ) M v x - C y v y M , a 23 = 2(C r l r -C f l f ) M v x -v x a 32 = 2(l r C r -C f l f ) I z v x , a 33 = - 2(C f l 2 f + C r l 2 r ) I z v x d 22 = 2C f M , d 32 = 2l f C f I z .
Taking into account the physical limitations during normal driving conditions [START_REF] Nguyen | Driver-automation cooperative approach for shared steering control under multiple system constraints: Design and experiments[END_REF], the compact set of the vehicle state is defined as

D x = v x ∈ [v x , v x ], v y ∈ [v y , v y ], r ∈ [r, r] . ( 5
)
where 4) is given by

v x = 5 [m/s], v x =
y = Cx, C = 1 0 0 0 0 1 . ( 6
)
In this work, the steering angle δ and the effective engine torque T eng are considered as unknown inputs to be estimated.

The vehicle system (4) has three nonlinearities (or premise variables), i.e., v x , 1 vx and v y . Using the sector nonlinearity approach [32, Chapter 2], a classical eight-rule TS fuzzy model of the nonlinear vehicle dynamics (4) can be easily derived as

ẋ = 8 i=1 h i (z)A i x + D v d, y = Cx, (7)
where

z = v x 1 vx
v y is the vector of premise variables. The local state-space matrices A i , and the corresponding MFs h i (z) of the TS fuzzy model [START_REF] Li | Integrated longitudinal and lateral tire/road friction modeling and monitoring for vehicle motion control[END_REF], for i ∈ I 8 , are not given here for brevity. Similar TS fuzzy representations as [START_REF] Li | Integrated longitudinal and lateral tire/road friction modeling and monitoring for vehicle motion control[END_REF] have been used for vehicle dynamics estimation, see for instance [START_REF] Zhang | A novel observer design for simultaneous estimation of vehicle steering angle and sideslip angle[END_REF]. However, this classical TS fuzzy form (7) leads to both theoretical and practical difficulties [START_REF] Pan | A unified framework for asymptotic observer design of fuzzy systems with unmeasurable premise variables[END_REF].

• Due to the practical unavailability of v y , the MFs h i (z), for i ∈ I 8 , are unmeasured. Designing TS fuzzy observers in this situation still remains challenging [START_REF] Pan | A unified framework for asymptotic observer design of fuzzy systems with unmeasurable premise variables[END_REF], especially in presence of unknown inputs [START_REF] Zhang | A novel observer design for simultaneous estimation of vehicle steering angle and sideslip angle[END_REF].

• The classical TS model ( 7) may yield a TS fuzzy observer with complex structure for real-time implementation. To overcome these drawbacks, we reformulate the vehicle system (4) in the following form:

ẋ = A(ξ)x + D v d + f v (ξ) + G v φ(x), y = Cx, (8)
where φ(x) = v 2 y , and

A(ξ) =    0 r 0 0 - 2(C f +Cr) M vx 0 0 2(lrCr-C f l f ) Izvx 0    , ξ = 1 vx r , f v (ξ) =     - Cxv 2 x Ie 2(Crlr-C f l f )r M vx -v x r - 2(C f l 2 f +Crl 2 r )r Izvx     , G v =   0 - Cy M 0   .
In this paper, we perform the observer design in the discretetime domain for real-time implementation. To this end, Euler's discretization method, with the sampling time T s = 0.01 [s], is used to obtain the discrete-time model of system (8) as

x k+1 = A(ξ k )x k + Dd k + f (ξ k ) + Gφ(x k ), y k = Cx k , (9)
where

A(ξ k ) = T s A v (ξ k ) + I, G = T s G v , f (ξ k ) = T s f v (ξ k ), D = T s D v .
For convenience of presentation, we explicitly denote

A(ξ) = A(V x , r) with V x = 1 vx ∈ [V x , V x ].
Using the sector nonlinearity approach [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] with the premise vector ξ ∈ R 2 , the following four-rule TS model of system (9) can be derived:

x k+1 = 4 i=1 h i (ξ k )A i x k + Dd k + f (ξ k ) + Gφ(x k ), y k = Cx k , (10)
where the local matrices A i , for i ∈ I 4 , are given by

A 1 = A (V x , r) , A 2 = A (V x , r) , A 3 = A V x , r , A 4 = A V x , r .
The corresponding membership functions h i (ξ k ), for i ∈ I 4 , of the TS fuzzy model [START_REF] Zhang | A novel observer design for simultaneous estimation of vehicle steering angle and sideslip angle[END_REF] are defined as

h 1 (ξ) = Ω v1 Ω r1 , h 2 (ξ) = Ω v1 Ω r2 , h 3 (ξ) = Ω v2 Ω r1 , h 4 (ξ) = Ω v1 Ω r1 , (11)
with

Ω v1 = V x -V x V x -V x , Ω r1 = r -r r -r , Ω v2 = V x -V x V x -V x , Ω r2 = r -r r -r .
Remark 1. The reformulated vehicle model ( 8) has two features deserving particular attention. First, the unmeasurable nonlinearity is isolated in φ(x k ). Then, the membership functions h i (ξ k ), for i ∈ I 4 , defined in [START_REF] Chen | Vehicle sideslip angle and road friction estimation using online gradient descent algorithm[END_REF], of the TS fuzzy model [START_REF] Zhang | A novel observer design for simultaneous estimation of vehicle steering angle and sideslip angle[END_REF] only depend on the measured premise vector ξ k .

Hence, these membership functions can be directly used to construct the UI observer structure as in [START_REF] Viehweger | Vehicle state and tyre force estimation: demonstrations and guidelines[END_REF]. Note that using the classical TS fuzzy modeling, the membership functions h i (z), for i ∈ I 8 , of the TS fuzzy model ( 7) cannot be directly incorporated into the UI observer structure due to their dependency on the unmeasured lateral speed v y . This induces major technical challenges in designing effective UI observers for systems with unmeasured nonlinearities [START_REF] Zhang | A novel observer design for simultaneous estimation of vehicle steering angle and sideslip angle[END_REF]. Second, a part of nonlinearities are retained in the measured consequent f (ξ k ) and unmeasured consequent φ(x k ). Hence, the resulting TS observer can be of much simpler structure, i.e., with only four fuzzy rules in place of eight-rule conservative TS representation [START_REF] Li | Integrated longitudinal and lateral tire/road friction modeling and monitoring for vehicle motion control[END_REF]. As shown in the next section, these features enable an effective framework for TS fuzzy UI observer design in terms of dealing with unmeasured nonlinearities and complexity reduction.

This paper provides an effective algorithm for the following vehicle dynamics estimation problem.

Problem 1. Consider the vehicle nonlinear model [START_REF] Zhang | A novel observer design for simultaneous estimation of vehicle steering angle and sideslip angle[END_REF] with the compact set D x defined in [START_REF] Nguyen | Sensor reduction for driver-automation shared steering control via an adaptive authority allocation strategy[END_REF]. Design an UI observer such that the lateral speed v y , the steering action δ and the engine torque T eng can be asymptotically and simultaneously estimated from the output information [START_REF] Wang | Online sensing of human steering intervention torque for autonomous driving actuation systems[END_REF].

III. UI OBSERVER DESIGN FOR TS FUZZY SYSTEMS WITH

UNMEASURED NONLINEAR CONSEQUENTS This section presents a new framework to design UI observers for a generalized class of TS fuzzy systems.

A. Observer Structure and Useful Lemmas

For generality, we consider the TS fuzzy system [START_REF] Zhang | A novel observer design for simultaneous estimation of vehicle steering angle and sideslip angle[END_REF] in a more general form

x k+1 = A(h)x k + Dd k + f (ξ k , u k ) + G(h)φ(x k , u k ), y k = Cx k , (12)
where

x k ∈ D x ⊆ R nx is the state vector, u k ∈ D u ⊆ R nu is the known input, d k ∈ R n d is the unknown input, y k ∈ R ny
is the output vector, and ξ k ∈ R n ξ is the vector measured premise variables. The nonlinear functions f : D x × D u → R nx and φ : D x × D u → R n φ are differentiable with respect to the state x k . The elements of function f (•) are measurable whereas those of φ(•) cannot be measured from the output. The state-space matrices of system (12) are given by

A(h) G(h) = N i=1 h i (ξ k ) A i G i .
Note that the MFs satisfy the following convex sum property:

N i=1 h i (ξ k ) = 1, 0 ≤ h i (ξ k ) ≤ 1, ∀i ∈ I N . ( 13
)
Let H be the set of membership functions satisfying (13), i.e., h = h 1 (ξ k ), h 2 (ξ k ), . . . , h r (ξ k ) ∈ H . The following assumptions are considered for the TS fuzzy system [START_REF] Selmanaj | Vehicle sideslip estimation: A kinematic based approach[END_REF].

Assumption 1. The differentiable function φ(x k , u k ) satisfies the following boundedness condition:

ρ ij ≤ ∂φ i ∂x j (x, u) ≤ ρ ij , (14)
with

ρ ij = min ρ∈Dx×Du ∂φi ∂xj (ρ) , ρ ij = max ρ∈Dx×Du ∂φi ∂xj (ρ) , for ∀(i, j) ∈ I n φ × I nx .
Note that the state x k and the input u k of engineering systems are always physically bounded, i.e., x k ∈ D x and u k ∈ D u . Then, the bounds ρ ij and ρ ij in ( 14) can be easily computed.

Assumption 2. We assume that rank(CD) = rank(D), (15a)

rank I D C 0 = n x + n d . (15b)
The rank conditions in [START_REF] Gadola | Development and validation of a Kalman filter-based model for vehicle slip angle estimation[END_REF] are standard in UIO design framework for unknown input decoupling [START_REF] Zhang | A novel observer design for simultaneous estimation of vehicle steering angle and sideslip angle[END_REF], [START_REF] Jia | Fault reconstruction and faulttolerant control via learning observers in Takagi-Sugeno fuzzy descriptor systems with time delays[END_REF], [START_REF] Wang | Observer design for discrete-time descriptor systems: An LMI approach[END_REF], [START_REF] Nguyen | Avoiding unmeasured premise variables in designing unknown input observers for Takagi-Sugeno fuzzy systems[END_REF]. Note that the vehicle nonlinear system (10) verifies these rank conditions.

For estimation purposes, we consider the following UI observer structure:

ζ k+1 = T Φk + L (h)(y k -ŷk ), (16a) xk = ζ k + N y k , (16b) dk = (CD) † (y k+1 -C Φk ), (16c)
with

Φk = A(h)x k + f (ξ k , u k ) + G(h)φ(x k , u k ).
The existence of the Moore-Penrose pseudo-inverse matrix (CD) † is guaranteed by condition (15a). The matrices T ∈ R nx×nx , N ∈ R nx×ny and L (h) ∈ R nx×ny are to be designed with

T + N C = I. ( 17
)
Remark 2. Note that selecting T = I and N = 0, the UI observer structure ( 16) reduces to the well-known Luenberger observer [START_REF] Besançon | Nonlinear Observers and Applications[END_REF], widely used in the literature.

From the matrix constraint [START_REF] Du | Sideslip angle estimation and stability control for a vehicle with a non-linear tyre model and a varying speed[END_REF], it follows that

x k+1 = T x k+1 + N Cx k+1 = T x k+1 + N y k+1 . (18)
Then, from ( 12) and ( 18), the dynamics of the TS fuzzy system can be rewritten as

x k+1 = T A(h)x k + T Dd k + T f (ξ k , u k ) + T G(h)φ(x k , u k ) + N y k+1 . (19)
Let us define the state estimation error as e k = x k -xk . To achieve an asymptotic state estimation, we impose

T D = 0. (20)
Then, the estimation error dynamics can be defined from (16a), (16b), ( 19) and ( 20) as

e k+1 = x k+1 -xk+1 = x k+1 -ζ k+1 -N y k+1 = T A(h)e k + T G(h)δ φ -L (h)Ce k , (21)
where

δ φ = φ(x k , u k )-φ(x k , u k ).
The mismatching nonlinear term δ φ caused by the unmeasured premise variables leads to technical difficulties in designing TS fuzzy observers [START_REF] Pan | A unified framework for asymptotic observer design of fuzzy systems with unmeasurable premise variables[END_REF]. To effectively deal with this term and achieve an asymptotic error convergence, the following lemma is useful to rewrite δ φ as a function of e k .

Lemma 1 (Differential Mean Value Theorem [START_REF] Jeon | Tracking of vehicle motion on highways and urban roads using a nonlinear observer[END_REF]). Let g(x) : R nx → R q and a, b ∈ R nx . If g(x) is differentiable on co(a, b), then there exist constant vectors c i ∈ co(a, b), c i = a, c i = b, for ∀i ∈ I q , such that

g(a) -g(b) =   q i=1 nx j=1 σ q (i)σ nx (j) ∂g i ∂x j (c i )   (a -b).
Applying Lemma 1 to function φ(x k , u k ), then there exist

ϑ i ∈ co(x k , xk ), for i ∈ I n φ , such that δ φ =   n φ i=1 nx j=1 σ n φ (i)σ nx (j) ∂φ i ∂x j (ϑ i , u)   (x -x). ( 22
)
We denote θ ij = ∂φi ∂xj (ϑ i , u), for ∀(i, j) ∈ I n φ × I nx , and θ = θ 11 , . . . , θ 1nx , . . . , θ n φ nx .

Due to the boundedness condition [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], the unknown parameter θ belongs to a bounded convex set S φ , whose set of 2 n φ nx vertices is given by

V φ = {θ = θ 11 , . . . , θ 1nx , . . . , θ n φ nx : θ ij ∈ {ρ ij , ρ ij }},
where the bounds ρ ij and ρ ij are given in [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]. From ( 21) and ( 22), the state estimation error dynamics can be rewritten as

e k+1 = (T A (h, θ) -L (h)C) e k , (23)
where

A (h, θ) = N i=1 h i (ξ k )A i (θ), A i (θ) = A i + n φ l=1 nx j=1 σ n φ (l)σ nx (j)θ lj G i . (24)
We are now ready to formulate the UIO design problem.

Problem 2. Consider the TS fuzzy system [START_REF] Selmanaj | Vehicle sideslip estimation: A kinematic based approach[END_REF]. Determine matrices of appropriate dimensions T , N and L (h) of the TS fuzzy UI observer [START_REF] Viehweger | Vehicle state and tyre force estimation: demonstrations and guidelines[END_REF] such that both the state estimate xk and the UI estimate dk asymptotically converge to the state x k and the UI d k , respectively.

The following technical lemmas are useful for the design of TS fuzzy UI observers.

Lemma 2 ([42]). Given matrices of appropriate dimensions A and B.

There exists a matrix X such that X A = B if and only if BA † A = B. Moreover, the general solution to X A = B is given by

X = BA † + Y(I -AA † ),
where Y is an arbitrary matrix of appropriate dimension.

Lemma 3 ([43]). Consider the MF-dependent inequality

Υ hhh+ = N i=1 N j=1 N l=1 h i (ξ k )h j (ξ k )h l (ξ k+1 )Υ ijl 0, ( 25
)
where h + = h 1 (ξ k+1 ), h 2 (ξ k+1 ), . . . , h N (ξ k+1 ) , and h, h + ∈ H . The symmetric matrices of appropriate dimensions Υ ijl , with i, j, l ∈ I N , are linearly dependent on the unknown decision variables. Inequality (25) holds if

Υ iil 0, i, l ∈ I N 2 N -1 Υ iil + Υ ijl + Υ jil 0, i, j, l ∈ I N , i = j. ( 26
)
Note that Lemma 3 allows to convert the infinite LMI-based condition [START_REF] Moon | Design, tuning, and evaluation of a fullrange adaptive cruise control system with collision avoidance[END_REF] in to a finite set of LMI constraints [START_REF] Franco | Real-time brake torque estimation for internal combustion engines[END_REF], which is numerically tractable.

B. LMI-Based Unknown Input Observer Design

The following theorem provides a numerical tractable solution for the UIO design in Problem 2.

Theorem 1. Consider the TS fuzzy system [START_REF] Selmanaj | Vehicle sideslip estimation: A kinematic based approach[END_REF], there is an asymptotic UI observer in the form ( 16) if there exist matrices T and N satisfying conditions [START_REF] Du | Sideslip angle estimation and stability control for a vehicle with a non-linear tyre model and a varying speed[END_REF] and [START_REF] Li | A reliable fusion methodology for simultaneous estimation of vehicle sideslip and yaw angles[END_REF], and if there exist positive definite matrices

P i ∈ R nx×nx , matrices M i ∈ R nx×nx , L i ∈ R nx×ny , for i ∈ I N , such that Φ iil (θ p ) 0 (27a) 2 N -1 Φ iil (θ p ) + Φ ijl (θ p ) + Φ jil (θ p ) 0 (27b)
for i, j, l ∈ I N , i = j, and θ p ∈ V φ , p ∈ I 2 n φ nx . The quantity Φ ijl (θ p ) is given by

Φ ijl (θ p ) = P j M j T A i (θ p ) -L j C M j + M j -P l ,
with A i (θ p ) defined in [START_REF] Yang | Front sensor and GPS-based lateral control of automated vehicles[END_REF]. Moreover, the matrix L (h) in (16a) is defined as

L (h) = M -1 (h)L(h) with M (h) L(h) = N i=1 h i (ξ k ) M i L i . (28)
Proof. Note that if matrices T and N satisfy conditions ( 17) and ( 20), then the TS fuzzy UI observer ( 16) leads to the state estimation error dynamics [START_REF] Si | High-precision estimation of steering angle of agricultural tractors using GPS and low-accuracy MEMS[END_REF]. Moreover, conditions ( 17) and ( 20) can be rewritten in the compact form

T N I D C 0 = I 0 . (29)
Due to the rank condition (15b), the solution of the algebraic matrix equation ( 29) exists. Applying Lemma 2 with

A = I D C 0 , B = I 0 , X = T N ,
we can compute matrices T and N as

T N = I 0 I D C 0 † + Y I - I D C 0 I D C 0 † , (30)
where Y is an arbitrary matrix of appropriate dimension.

For stability analysis, we consider the following MFdependent Lyapunov function candidate:

V(e k ) = e k P (h)e k , P (h) = N i=1 h i (ξ k )P i . (31)
By Lemma 3, it follows from (27a) and (27b) that

P (h) M (h)T A (h, θ) -L(h)C M (h, h + ) 0, (32)
for h, h + ∈ H , θ ∈ S φ , with

M (h, h + ) = M (h) + M (h) -P (h + ),
and

P (h + ) = N i=1 h i (ξ k+1 )P i . Since P (h + ) 0, condition (32) implies M (h) + M (h)
0. This guarantees the existence of M (h) -1 , thus the validity of the expression of L (h). [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] with I -A(h, θ) on the left and its transpose on the right, we obtain

Let us denote

A(h, θ) = T A (h, θ)-L (h)C = T A (h, θ)- M (h) -1 L(h)C. Premultiplying
A(h, θ) P (h + )A(h, θ) -P (h) ≺ 0. (33)
Note that inequality (33) guarantees a negative variation of the fuzzy Lyapunov function [START_REF] Na | Vehicle engine torque estimation via UI observer and adaptive parameter estimation[END_REF] along the trajectory of the error dynamics [START_REF] Si | High-precision estimation of steering angle of agricultural tractors using GPS and low-accuracy MEMS[END_REF], i.e.,

δV k = V(e k+1 ) -V(e k ) = e k+1 P (h + )e k+1 -e k P (h)e k < 0, ∀k ∈ Z + . (34)
Using Lyapunov-based argument, it is clear that condition [START_REF] Jia | Fault reconstruction and faulttolerant control via learning observers in Takagi-Sugeno fuzzy descriptor systems with time delays[END_REF] guarantees the asymptotic stability of the error dynamics [START_REF] Si | High-precision estimation of steering angle of agricultural tractors using GPS and low-accuracy MEMS[END_REF].

Hereafter, we show that the estimate dk defined in (16c) converges asymptotically to the of the unknown input d k . To this end, note from ( 12) that

d k = (CD) † (y k+1 -CΦ k ) , (35)
with

Φ k = A(h)x k + f (ξ k , u k ) + G(h)φ(x k , u k ).
The UI estimation error ε k = d k -dk can be computed from (16c) and [START_REF] Pan | A unified framework for asymptotic observer design of fuzzy systems with unmeasurable premise variables[END_REF] as

ε k = (CD) † C (A(h)e k + G(h)δ φ ) . (36)
Exploiting again expression [START_REF] Benloucif | Cooperative trajectory planning for haptic shared control between driver and automation in highway driving[END_REF], the UI estimation error ε k in ( 36) can be rewritten in the form

ε k = (CD) † CA (h, θ)e k . (37)
Since h and θ belong to bounded convex sets, i.e., h ∈ H and θ ∈ S φ , remark from the algebraic equation ( 37) that if e k → 0, then ε k → 0. This concludes the proof.

Remark 3. The observer design in Theorem 1 is recast as a convex optimization problem under strict LMI constraints [START_REF] Hedrick | Longitudinal vehicle controller design for IVHS systems[END_REF]. Hence, the decision matrices M i , L i , for i ∈ I N , constituting the observer gain L (h) as in [START_REF] Ahmed | Estimating SI engine efficiencies and parameters in second-order sliding modes[END_REF], can be efficiently solved with available numerical toolboxes, for instance YALMIP package with SDPT3 solver [START_REF] Löfberg | Yalmip: A toolbox for modeling and optimization in Matlab[END_REF].

The UI observer design is summarized in Algorithm 1. The proposed UI observer design can be now applied to the TS fuzzy model [START_REF] Zhang | A novel observer design for simultaneous estimation of vehicle steering angle and sideslip angle[END_REF] for the estimation of vehicle nonlinear dynamics as described in Problem 1. A. Hardware-in-the-Loop Simulation Platform Setup 1) Full Vehicle Model: The real-time verification of the proposed nonlinear UI observer is performed with a 15-DOF multibody vehicle model [START_REF] Anthonis | Recent Advances in Optimization and its Applications in Engineering[END_REF], developed in LMS Imagine.Lab AMESim environment. This full vehicle model consists of a powertrain-chassis subsystem, a tire-road subsystem, a vehicle dynamics sensing subsystem and a vehicle control unit as depicted in Fig. 2. The key vehicle parameters are given in Table I while others are initialized by AMESim [START_REF] Anthonis | Recent Advances in Optimization and its Applications in Engineering[END_REF] to create a challenging simulation environment. The vehicle control unit is constructed in Simulink and embedded in AMESim via a co-simulation interface. Without further precision on the test scenarios, it is assumed that the vehicle drives on a flat ground with an ideal or a time-varying road adherence. 2) Hardware-in-the-Loop Platform Setup: To evaluate the practical estimation performance, a HiL platform has been setup as shown in Fig. 3(a). This platform consists of two real-time (RT) machines: a MicroAutoboxII (MABII) and a Vector System (VTS), and a personal computer to monitor and download programmes into the RT machines via their application tools: ControlDesk and CANoe/Pro. For the RT code generation, the MABII has been selected as the platform to run the full vehicle model while the VTS has been selected as the platform to function as the vehicle sensing system as well as to implement the designed fuzzy UI observer. All the programs have been implemented in Simulink, allowing RT code generation with RT interfaces from the dSPACE and the VTS, see Figs.

C. Scenario 2: Driving with a Circular Vehicle Trajectory

For this driving scenario, the estimation performance is tested with a circular trajectory as depicted in Fig. 6(a). The corresponding vehicle speed is given in Fig. 6(b). We can see that the unmeasured lateral speed and unknown vehicle inputs can be accurately reconstructed with the proposed fuzzy UI observer as shown in Fig. 6(c) and Fig. 7, respectively. Moreover, as in the previous driving scenario, the estimation convergence is very quick for all estimated vehicle variables. This test is performed with a time-varying adherence condition to emphasize the robustness performance of the proposed UI observer. For this scenario, we assume that the autonomous vehicle performs a contour trajectory with radius varied from 45 [m] to 55 [m] and an increasing vehicle speed profile as shown in Figs. 8(a) and (b), respectively. To design a challenging driving situation, the interaction between the ground and the tires is now simulated by a road grip model as shown in Fig. 2, which is driven by the vehicle movement as

Road grip = µ grip 1 + sin π √ X 2 + Y 2 l r ,
where µ grip = 0.6 is the grip coefficient, X and Y are the vehicle positions. We can observe in Fig. 8 that the estimation of the vehicle states is also highly accurate in this situation. In particular, the proposed UI observer allows capturing precisely the high-frequency chattering behaviors of vehicle variables related to the lateral motion. The estimates of the steering angle and the effective engine torque also converges quickly to their respective measured signals as depicted in Fig. 9. For a quantitative performance analysis, the mean absolute errors (respectively root mean square deviations) of the unmeasured lateral speed v yM AE , steering angle δ M AE and effective engine torque T engM AE (respectively v yRM SD , δ RM SD and T engRM SD ) obtained with the proposed UI observer are computed. These performance indices are summarized in Table II for the three driving scenarios. The analysis results confirm that the proposed fuzzy UI observer can provide accurate V. CONCLUDING REMARKS A new nonlinear UI observer design method has been proposed to simultaneously estimate the vehicle state, the lateral speed as well as the effective engine torque of autonomous ground vehicles. TS fuzzy modeling with nonlinear consequents is exploited for observer design to deal with the unmeasured nonlinearities of the combined longitudinallateral vehicle dynamics. The proposed generalized Luenberger observer structure permits an effective UI decoupling to guarantee an asymptotic convergence of both the vehicle state and the UI estimation errors. LMI-based observer design conditions are derived using Lyapunov arguments. The practical performance of the new fuzzy UI observer is realtime tested with a high-fidelity AMESim vehicle model. The results of HiL experiments show that the proposed nonlinear UI observer can provide accurate estimates of both vehicle state variables and UIs. Future works focus on the extension of the proposed estimation method to deal with limit driving situations, e.g., by taking into account a nonlinear tire model or parametric uncertainties of the cornering stiffness parameters in the observer design. Moreover, exploiting the proposed UI observer structure for an effective fault-tolerant control scheme of autonomous vehicles is another promising research topic.

Fig. 1 .

 1 Fig. 1. Schematic of a two degrees-of-freedom vehicle model.

  30 [m/s], v y = -1.5 [m/s], v y = 1.5 [m/s], r = -0.55 [rad/s] and r = 0.55 [rad/s]. For system (4), we assume that the vehicle speed v x [m/s] and the yaw rate r [rad/s] can be directly measured whereas the measurement of the lateral speed v y [m/s] is not available. Hence, the output equation of system (

Algorithm 1 :

 1 Observer Design Procedure Input: Nonlinear system in the TS fuzzy form[START_REF] Selmanaj | Vehicle sideslip estimation: A kinematic based approach[END_REF] Output: Unknown input observer[START_REF] Viehweger | Vehicle state and tyre force estimation: demonstrations and guidelines[END_REF] such that xk → x k and dk → d k , when k → ∞ 1 Check the matrix rank conditions in Assumption 2 • If YES, then go to Step 2 • If NO, then unapplicable algorithm 2 Compute matrices T and N from (30) 3 Solve LMI conditions (27) to get M i , L i , for i ∈ I N 4 Construct matrices M (h) and L(h) from (28) 5 Construct TS fuzzy unknown input observer (16) IV. HARDWARE-IN-THE-LOOP EXPERIMENTS This section presents real-time results obtained with HiL experiments to demonstrate the effectiveness of the proposed fuzzy UI observer design. Three test scenarios, representing different normal driving situations, are performed on a highfidelity vehicle model to show the robustness of the new observer with respect to unmodeled vehicle dynamics.

Fig. 2 .

 2 Fig. 2. AMESim full vehicle model running on HiL platform.

  3(b) and (c), respectively. At each operation step, to mimic the practical automotive application, pulse-width modulation (PWM) output channels of the MABII are used to send the virtual sensor signals from the AMESim full vehicle model to the VTS. The VTS receives the MABII signals via its PWM input channels and converts them to physical signals.

  (a) Configuration of the HiL platform. (b) Simulink design of the MABII. (c) Simulink design of the VTS.

Fig. 3 .

 3 Fig. 3. Hardware-in-the-loop platform setup.

Fig. 4 .

 4 Fig. 4. Estimation performance in Scenario 1. (a) Vehicle trajectory. (b) Longitudinal speed vx. (c) Lateral speed vy. (d) Yaw rate r.

Fig. 5 .

 5 Fig. 5. Estimation performance in Scenario 1. (a) Steering angle δ. (b) Effective engine torque Teng.

Fig. 6 .

 6 Fig. 6. Estimation performance in Scenario 2. (a) Vehicle trajectory. (b) Longitudinal speed vx. (c) Lateral speed vy. (d) Yaw rate r.

Fig. 7 .

 7 Fig. 7. Estimation performance in Scenario 2. (a) Steering angle δ. (b) Effective engine torque Teng.

Fig. 8 .

 8 Fig. 8. Estimation performance in Scenario 3. (a) Vehicle trajectory. (b) Longitudinal speed vx. (c) Lateral speed vy. (d) Yaw rate r.

Fig. 9 .

 9 Fig. 9. Estimation performance in Scenario 3. (a) Steering angle δ. (b) Effective engine torque Teng.

TABLE I PARAMETER

 I VALUES OF VEHICLE MODEL.

	Parameter Description	Value
	M	Vehicle mass	1476 [kg]
	l f	Distance from gravity center to front axle	1.13 [m]
	lr	Distance from gravity center to rear axle	1.49 [m]
	Ie	Effective longitudinal inertia	442.8 [kgm 2 ]
	Iz	Vehicle yaw moment of inertia	1810 [kgm 2 ]
	C f	Front cornering stiffness	57000 [N/rad]
	Cr	Rear cornering stiffness	59000 [N/rad]
	Cx	Longitudinal aerodynamic drag coefficient 0.35 [-]
	Cy	Lateral aerodynamic drag coefficient	0.45 [-]

TABLE II QUANTITATIVE

 II ANALYSIS OF ESTIMATION PERFORMANCE.

	Error index	Scenario 1 Scenario 2 Scenario 3
	v yM AE [m/s]	0.0025	5.66e-4	3.81e-4
	δ M AE [rad]	0.0018	9.71e-5	0.0023
	T engM AE [Nm]	0.9002	0.7329	0.7553
	v yRM SD [m/s]	3.81e-4	2.19e-4	2.96e-6
	δ RM SD [rad]	2.53e-4	2.91e-6	2.14e-4
	T engRM SD [Nm]	5.164	5.212	5.319

He is currently an Associate Professor with North Minzu University, Yinchuan, China. His current research interests include fuzzy control, robust control, nonlinear control and their applications.