Anh-Tu Nguyen
email: tnguyen@uphf.fr

Víctor Campos

Thierry-Marie Guerra
email: guerra@uphf.fr

Juntao Pan
email: jtpan@aliyun.com

Wen-Bo Xie
email: xiewenbo@hrbust.edu.cn

A.-T Nguyen

T.-M Guerra

Takagi-Sugeno Fuzzy Observer Design for Nonlinear Descriptor Systems With Unmeasured Premise Variables and Unknown Inputs

Keywords: Takagi-Sugeno models, fuzzy-model-based observers, nonlinear descriptor systems, unmeasured premise variables, unknown inputs, Lyapunov method

This paper presents a new observer design framework for a class of nonlinear descriptor systems with unknown but bounded inputs. In the presence of unmeasured nonlinearities, i.e., premise variables, designing nonlinear observers is known as particularly challenging. To solve this problem, we rewrite the nonlinear descriptor system in the form of a Takagi-Sugeno (TS) fuzzy model with nonlinear consequents. This model reformulation enables an effective use of the differential mean value theorem to deal with the mismatching terms involved in the estimation error dynamics. These nonlinear terms, issued from the unmeasured nonlinearities of the descriptor system, cause a major technical difficulty for TS fuzzy-model-based observer design. The descriptor form is treated through a singular redundancy representation. For observer design, we introduce into the Luenberger-like observer structure a virtual variable aiming at estimating the one-step ahead state. This variable introduction allows for free-structure decision variables involved in the observer design to further reduce the conservatism. Using Lyapunov-based arguments, the observer design is reformulated as an optimization problem under linear matrix inequalities with a single line search parameter. The estimation error bounds of both the state and the unknown input can be minimized by means of a guaranteed ∞-gain performance level. The interests of the new ∞ TS fuzzy observer design are clearly illustrated with two physically motivated examples.

I. INTRODUCTION

O BTAINING the real-time state information of dynamical systems is crucial either for feedback control design or for decision-making and supervision purposes [START_REF] Gao | A survey of fault diagnosis and fault-tolerant techniques-Part I: Fault diagnosis with model-based and signal-based approaches[END_REF]. Hence, state estimation for dynamical systems is a fundamental problem in control theory and applications. To this end, numerous approaches have been proposed to design full-order or reduced-order observers, with or without unknown inputs (UIs), for both linear and nonlinear systems, see [START_REF] Besançon | Nonlinear Observers and Applications[END_REF]- [START_REF] Xie | A decoupling approach for observer-based controller design of T-S fuzzy system with unknown premise variables[END_REF] and references therein. Since the pioneer work on linear time-invariant systems [START_REF] Luenberger | Observers for multivariable systems[END_REF], numerous extensions have been developed for nonlinear cases [START_REF] Besançon | Nonlinear Observers and Applications[END_REF]. Although satisfactory solutions have been achieved for linear systems, finding a unified and systematic method for observer design of nonlinear systems is still widely open [START_REF] Pan | A unified framework for asymptotic observer design of fuzzy systems with unmeasurable premise variables[END_REF]. A promising idea to overcome this challenging issue is to represent highly nonlinear systems by Takagi-Sugeno (TS) fuzzy models [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. Note that TS fuzzy models can also be considered as polytopic quasi-linear parameter varying (quasi-LPV) models whose scheduling variables correspond to premise variables in TS fuzzy case. The analogies between TS and LPV systems can be found in [START_REF] Rotondo | Automated generation and comparison of Takagi-Sugeno and polytopic quasi-LPV models[END_REF]. Since TS fuzzy systems are constructed from local linear models blending with nonlinear membership functions (MFs) [START_REF] Nguyen | Fuzzy control systems: Past, present and future[END_REF], TS fuzzy observer design can be derived using Lyapunov method [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] and linear matrix inequality (LMI) formulation [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. Moreover, from the viewpoint of estimation problem formulation, unmodeled dynamics, uncertain disturbances and faults in engineering systems, or attack signals in cyberphysical systems can be considered as unknown inputs [START_REF] Chen | Disturbance-observer-based control and related methods-An overview[END_REF], [START_REF] Chakrabarty | State and unknown input observers for nonlinear systems with bounded exogenous inputs[END_REF]. As a result, designing TS fuzzy observers in the presence of UIs has been an active research topic in the areas of fault estimation and diagnosis [START_REF] Zhang | Sensor fault estimation of switched fuzzy systems with unknown input[END_REF]- [START_REF] Nguyen | Takagi-sugeno fuzzy unknown input observers to estimate nonlinear dynamics of autonomous ground vehicles: Theory and real-time verification[END_REF], fault-tolerant control [START_REF] Jia | Fault reconstruction and fault-tolerant 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].

TS fuzzy-model-based observer design can be classified into the two following categories [START_REF] Pan | A unified framework for asymptotic observer design of fuzzy systems with unmeasurable premise variables[END_REF]:

• Category 1: TS fuzzy observer design with measurable premise variables, i.e., system nonlinearities, and • Category 2: TS fuzzy observer design with unmeasurable premise variables. The first design category is a direct extension of the linear Luenberger-based method [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. Then, the goal focuses on reducing the design conservatism via the use of various Lyapunov candidate functions and/or relaxation techniques [START_REF] Nguyen | Simultaneous estimation of state and unknown input with ∞ guarantee on error-bounds for fuzzy descriptor systems[END_REF]. However, assuming that all the premise variables can be measured from sensors, the TS fuzzy observer design in Category 1 can only be applied to a restrictive class of nonlinear systems. To overcome this drawback, research attention has been paid to the TS fuzzy observer design in Category 2. The technical challenge here is to deal with the mismatching nonlinear term involved in the estimation error dynamics, which is caused by unmeasured premise variables. A solution for this issue is to simply consider the mismatching term as a disturbance of the error dynamics [START_REF] Pérez-Estrada | Generalized dynamic observers for quasi-LPV systems with unmeasurable scheduling functions[END_REF], which should be minimized through an L 2 -gain performance. Note that this solution cannot guarantee an asymptotic convergence of the estimation errors. Moreover, since the disturbance explicitly depends on the system state and control input, its effects on the error dynamics strongly depend on the behaviors of the nonlinear system. The most common approach, guaranteeing the asymptotic estimation performance, is based on the Lipschitz property of the mismatching nonlinear term, see [START_REF] Bergsten | Observers for Takagi-Sugeno fuzzy systems[END_REF], [START_REF] Dong | Output feedback fuzzy controller design with local nonlinear feedback laws for discrete-time nonlinear systems[END_REF] and related references. Despite its simplicity, this approach leads to conservative design results, especially for nonlinear systems with a large Lipschitz constant [START_REF] Ichalal | How to cope with unmeasurable premise variables in Takagi-Sugeno observer design: dynamic extension approach[END_REF]. By decomposing the space of premise variables into crisp regions, an H ∞ method to design piecewise TS fuzzy observers has been proposed in [START_REF] Wang | H∞ observer design for fuzzy system with immeasurable state variables via a new Lyapunov function[END_REF]. Note that the proposed piecewise region decomposition may induce difficulties for real-time implementation of complex nonlinear systems. An immersion-based method has been proposed in [START_REF] Ichalal | How to cope with unmeasurable premise variables in Takagi-Sugeno observer design: dynamic extension approach[END_REF] to avoid the issue of unmeasured premise variables. Unfortunately, this method can only be applied to a very restrictive class of nonlinear systems due to the non-convergence of the required dynamics-extension algorithm. Recently, using the differential mean value theorem (DMVT), the authors in [START_REF] Guerra | H∞ LMI-based observer design for nonlinear systems via Takagi-Sugeno models with unmeasured premise variables[END_REF] have proposed a new method to deal with the mismatch caused by unmeasurable MFs. The resulting TS fuzzy observer design requires a norm-bounded uncertainty approach to handle the time-varying terms stemmed from the use of DMVT. As a consequence, the proposed method leads to a complex design framework which may also induce conservative results. Note that most of the existing observer design results are usually based on the assumption that the premise variables are perfectly known. However, this is not the case in many practical situations, e.g., due to inaccurate sensors. Within the LPV framework, significant advances have been achieved in designing LPV observers/filters with imperfect information on the scheduling parameters [START_REF] Lacerda | A new approach to handle additive and multiplicative uncertainties in the measurement for LPV filtering[END_REF]- [START_REF] Sato | One-shot design of performance scaling matrices and observer-based gain-scheduled controllers depending on inexact scheduling parameters[END_REF]. However, this topic is still widely open for TS fuzzy-model-based observer design [START_REF] Gómez-Peñate | Sensor fault diagnosis based on a sliding mode and unknown input observer for Takagi-Sugeno systems with uncertain premise variables[END_REF], for which the premise variables directly depend on the state vector.

Descriptor models can naturally represent a large class of physical systems. The descriptor forms appear in various real-world applications including robotics [START_REF] Nguyen | Nonlinear tracking control with reduced complexity of serial robots: A robust fuzzy descriptor approach[END_REF], rehabilitation systems [START_REF] Guelton | An alternative to inverse dynamics joint torques estimation in human stance based on a Takagi-Sugeno unknown-inputs observer in the descriptor form[END_REF], [START_REF] Sun | Reduced adaptive fuzzy decoupling control for lower limb exoskeleton[END_REF], transportation and power systems [START_REF] Dai | Singular Control Systems[END_REF], etc. Over the years, observer designs for linear descriptor systems have been extensively studied with significant advances, see for instance [START_REF] Dai | Singular Control Systems[END_REF]- [START_REF] Mobayen | Robust global controller design for discrete-time descriptor systems with multiple time-varying delays[END_REF]. Various extensions have been proposed to LPV descriptor systems [START_REF] Shi | Fault estimation and active fault tolerant control for linear parameter varying descriptor systems[END_REF]- [START_REF] Estrada | Robust H -/H∞ fault detection observer design for descriptor-LPV systems with unmeasurable gain scheduling functions[END_REF], and TS fuzzy descriptor systems [START_REF] Jia | Fault reconstruction and fault-tolerant control via learning observers in Takagi-Sugeno fuzzy descriptor systems with time delays[END_REF], [START_REF] Nguyen | Simultaneous estimation of state and unknown input with ∞ guarantee on error-bounds for fuzzy descriptor systems[END_REF], [START_REF] Guelton | An alternative to inverse dynamics joint torques estimation in human stance based on a Takagi-Sugeno unknown-inputs observer in the descriptor form[END_REF], [START_REF] Marx | Design of observers for Takagi-Sugeno descriptor systems with unknown inputs and application to fault diagnosis[END_REF], [START_REF] Guerra | Observer design for Takagi-Sugeno descriptor models: An LMI approach[END_REF]. However, dealing with the descriptor forms in LPV and/or TS fuzzy observer designs still remains a major challenge. Indeed, most of existing results can only be applied to systems with a linear descriptor matrix [START_REF] Jia | Fault reconstruction and fault-tolerant control via learning observers in Takagi-Sugeno fuzzy descriptor systems with time delays[END_REF], [START_REF] Shi | Fault estimation and active fault tolerant control for linear parameter varying descriptor systems[END_REF]- [START_REF] Marx | Design of observers for Takagi-Sugeno descriptor systems with unknown inputs and application to fault diagnosis[END_REF]. In particular, up to now all results on TS fuzzy descriptor observers have mainly been concerned with the case of measured premise variables, which greatly restricts their real-world applicability.

Motivated by the above technical and practical issues, this paper provides a new framework to design discrete-time TS fuzzy observers for a large class of descriptor systems with unmeasurable nonlinearities and UIs. Note that existing TS fuzzy observer designs are exclusively based on the classical TS fuzzy modeling with linear consequent [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF], which causes major difficulties in dealing with unmeasured MFs. To avoid this drawback, we rewrite the nonlinear descriptor system as a TS fuzzy descriptor model with nonlinear consequents [START_REF] Dong | Output feedback fuzzy controller design with local nonlinear feedback laws for discrete-time nonlinear systems[END_REF], [START_REF] Moodi | Robust observer-based controller design for Takagi-Sugeno systems with nonlinear consequent parts[END_REF], called N-TS fuzzy model, whose MFs are measured. In particular, the unmeasured nonlinearities of the original descriptor system are isolated in local nonlinear consequents. These features of the proposed N-TS fuzzy reformulation allow for an effective use of the differential mean value theorem [START_REF] Mcleod | Mean value theorems for vector valued functions[END_REF] to deal with the mismatching nonlinear term involved in the estimation error dynamics. To deal with the descriptor form, the singular redundancy approach [START_REF] Tanaka | A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions[END_REF] is exploited to design Luenberger-like fuzzy observers. Moreover, a virtual variable, representing the estimate of the one-step ahead system state, is introduced into the descriptor observer structure [START_REF] Guerra | Observer design for Takagi-Sugeno descriptor models: An LMI approach[END_REF]. Together with special convexification techniques, this virtual variable allows for free-structure observer gains to reduce the design conservatism. Using Lyapunov stability arguments, the design conditions of N-TS fuzzy descriptor observers can be represented as LMI constraints, which can be effectively solved with available numerical solvers [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. The estimation errors of both the state and the UI can be minimized by optimizing a guaranteed ∞ -gain performance. Moreover, the proposed N-TS fuzzy observer design does not require any specific rank condition for UI decoupling as in [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] Jia | Fault reconstruction and fault-tolerant control via learning observers in Takagi-Sugeno fuzzy descriptor systems with time delays[END_REF], [START_REF] Marx | Design of observers for Takagi-Sugeno descriptor systems with unknown inputs and application to fault diagnosis[END_REF], [START_REF] Nguyen | Avoiding unmeasured premise variables in designing unknown input observers for Takagi-Sugeno fuzzy systems[END_REF]. Hence, our systematic LMI-based observer design framework can be applied to a large class of descriptor systems with UIs and unmeasured nonlinearities. This main contribution has not been yet observed in any previous work in the open literature.

The paper is organized as follows. Section II formulates the observer design problem with useful tools for theoretical developments. Section III provides sufficient LMI-based conditions to design N-TS fuzzy-model-based observers for the class of nonlinear descriptor systems. The reconstruction of the unknown input with a guarantee on ∞ -norm bounded error is also presented. In Section V, two physically motivated examples are given to show the interests of the proposed design framework. Section VI concludes the paper and discusses some future works. Notation. N denotes the set of non-negative integers, and I p = {1, 2, ..., p} ⊂ N. For a matrix X, X denotes its transpose, X 0 (respectively X ≺ 0) means that X is symmetric positive (respectively negative) definite, HeX = X + X , and X denotes its maximum singular value. For a vector x ∈ R n , we denote its 2-norm as x =

√

x x. For a sequence of vectors

{x k } k∈N (respectively matrices {X k } k∈N ), we denote x ∞ = sup k≥0 x k (respectively X ∞ = sup k≥0 X k ). Then, {x k } k∈N ∈ ∞ if x ∞ < ∞ and {X k } k∈N ∈ L ∞ if X ∞ < ∞. I
) in I p1 × • • • × I pn .
For simplicity, we can define a single index j in I r with r = n i=1 p i representing these multiple indices by defining a one-to-one relationship between j and (κ 1 , . . . , κ n ). Whenever this multi-index notation is used, we consider j{i} to represent the ith term of the n-tuple (κ 1 , . . . , κ n ) that corresponds to j. For example, consider an index j ∈ I 4 that represents the tuple (κ 1 , κ 2 ) ∈ I 2 × I 2 with j = 1 → (1, 1), j = 2 → (1, 2), j = 3 → (2, 1) and j = 4 → (2, 2). In this case, for j = 3, j{1} = 2 and j{2} = 1. Throughout the paper, the argument of a function is omitted when its meaning is clear.

II. PROBLEM FORMULATION

Based on a singular redundancy approach, this section formulates the ∞ observer design of nonlinear descriptor systems.

A. Description of N-TS Fuzzy Descriptor Systems

We consider the observer design problem of the following class of discrete-time nonlinear descriptor system:

E(x k )x k+1 = f(x k , u k , d k ), y k = g(x k , u k ), (1)
where Here, the descriptor matrix E(x k ) is regular for ∀x k ∈ D x to guarantee a unique solution for system (1).

x k ∈ D x ⊆ R nx is the state, u k ∈ D u ⊆ R nu is the known input, d k ∈ D d ⊆ R n d is
Remark 1. Various nonlinear systems in mechatronics and robotics engineering can be modeled in the form (1), for instance Lagrangian-Euler systems [START_REF] Nguyen | Nonlinear tracking control with reduced complexity of serial robots: A robust fuzzy descriptor approach[END_REF], [START_REF] Stevens | Aircraft Control and Simulation -Dynamics, Controls Design, and Autonomous Systems[END_REF], where E(x k ) represents the inertia matrix [START_REF] Guelton | An alternative to inverse dynamics joint torques estimation in human stance based on a Takagi-Sugeno unknown-inputs observer in the descriptor form[END_REF]. Due to the regularity of E(x k ), it is possible to transform the descriptor system (1) into the classical state-space representation, i.e.,

x k+1 = E(x k ) -1 f(x k , u k , d k ).
However, keeping the descriptor form for TS fuzzy-model-based designs allows for a significant reduction of both the number of fuzzy rules and that of LMI-based design conditions, see [START_REF] Guerra | Observer design for Takagi-Sugeno descriptor models: An LMI approach[END_REF], [START_REF] Tanaka | A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions[END_REF] and related references.

Let us denote xk = x k x k+1 , and z k = z(y k ) ∈ R nz a vector of measurable premise variables of system (1), which are bounded in a compact set D x of the state space. 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 multiple-parameterization approach for local stabilization of constrained Takagi-Sugeno fuzzy systems with nonlinear consequents[END_REF], we assume that system (1) can be reformulated in the form

E(z k )x k+1 + J(z k )λ(x k ) = A(z k )x k + f (z k , u k ) + F (z k )φ(x k , u k ) + B(z k )d k , y k = C(z k )x k + g(z k , u k ) + G(z k )ψ(x k , u k ), (2)
where the nonlinear vector-valued functions f :

D x ×D u → R nx , g : D x ×D u → R ny , φ : D x ×D u → R n φ , ψ : D x ×D u → R n ψ
are differentiable with respect to x k , and λ : D x × D x → R n λ is differentiable with respect to xk . Note that the elements of functions f (z k , u k ) and g(z k , u k ) are measurable, and those of φ(x k , u k ), ψ(x k , u k ) and λ(x k ) cannot be directly accessible from the system output. The state-space matrices

A(z k ), B(z k ), C(z k ), E(z k ), F (z k ), G(z k ) and J(z k ) depend on the measurable premise variables. Assume that B(z k ) is of full-column rank for ∀x k ∈ D x , and {d k } k∈N ∈ ∞ .
For observer design, we also consider the following assumption.

Assumption 1. The gradients of the continuously differentiable functions φ(x k , u k ), ψ(x k , u k ) and λ(x k ) respectively belong to the following known bounded convex sets:

∇ x k φ i ∈ co(ρ i1 , • • • , ρ in φ i ), ρ iκ ∈ R 1×nx , (i, κ) ∈ I n φ × I n φ i , ∇ x k ψ j ∈ co(ζ j1 , • • • , ζ jn ψ j ), ζ jκ ∈ R 1×nx , (j, κ) ∈ I n ψ × I n ψ j , ∇ xk λ ∈ co(υ 1 , • • • , υ n λ ), υ κ ∈ R 1×2nx , ( , κ) ∈ I n λ × I n λ . (3)
Remark 2. Note that the vertices constituting the convex hull of the bounded convex sets in (3) for the gradients can be determined in a similar way as the sector nonlinearity approach [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. For illustrations, we consider a system with three states x 1k , x 2k , x 3k and a nonlinearity φ(x k ) = sin(x 1k + x 2k ). The gradient of φ(x k ) is defined as

∇ x k φ = cos(x 1k + x 2k ) cos(x 1k + x 2k ) 0 . Since cos(•) ∈ [-1, 1], then it follows that ∇ x k φ ∈ co(ρ 1 , ρ 2 ), with ρ 1 = -1 -1 0 , ρ 2 = 1 1 0 .
Note that the proposed approach to form a convex bounded set from the whole gradient can lead to a smaller number of vertices than the element-by-element bounding approach in [START_REF] Zemouche | Observers for a class of Lipschitz systems with extension to H∞ performance analysis[END_REF]. Indeed, this latter leads to a convex bounded set with 4 vertices for the considered example because it does not take advantage of the fact that both non-zero terms of the gradient ∇ x k φ are always the same.

Applying the sector nonlinearity approach [10, Chapter 2], the nonlinear descriptor system (2) can be exactly expressed by r fuzzy IF-THEN rules in the compact set D x with local nonlinear consequents:

RULE R i : IF z 1k is M i 1 and . . . and z pk is M i p THEN E i x k+1 + J i λ(x k ) = A i x k + f (z k , u k ) + F i φ(x k , u k ) + B i d k y k = C i x k + g(z k , u k ) + G i ψ(x k , u k ) (4)
where

(A i , B i , C i , E i , F i , G i , J i )
are known constant matrices with appropriate dimensions, R i denotes the ith fuzzy inference rule. M i j , with i ∈ I r and j ∈ I p , is the fuzzy set. The fuzzy membership functions are given by

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

r i=1 h i (z) = 1, 0 ≤ h i (z) ≤ 1, ∀i ∈ I r . (5)
Let Ω be the set of MFs satisfying [START_REF] Lungu | Reduced-order multiple observer for Takagi-Sugeno systems with unknown inputs[END_REF]. Using the center-of-gravity method for defuzzification, the N-TS fuzzy descriptor system (4) can be represented in the compact form

E(h)x k+1 + J(h)λ(x k ) = A(h)x k + f (z k , u k ) + F (h)φ(x k , u k ) + B(h)d k , y k = C(h)x k + g(z k , u k ) + G(h)ψ(x k , u k ), (6)
where

h = h 1 (z), h 2 (z), . . . , h r (z) ∈ Ω, and
Π(h) = r i=1 h i (z)Π i , Π ∈ {A, B, C, E, F, G, J}.
Remark 3. Note that using the sector nonlinearity approach [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF], the number of TS fuzzy subsystems increases exponentially according to the number of premise variables. Due to the retained nonlinearities f (z k , u k ), g(z k , u k ), φ(x k , u k ) and ψ(x k , u k ) in the consequents of the N-TS fuzzy system (6), N-TS fuzzy modeling allows reducing significantly the number of fuzzy rules compared to the classical TS fuzzy modeling, especially for complex nonlinear systems [START_REF] Moodi | Robust observer-based controller design for Takagi-Sugeno systems with nonlinear consequent parts[END_REF], [START_REF] Nguyen | Constrained output feedback control for fuzzy systems with local nonlinear models subject to state and input constraints[END_REF]. Hence, N-TS fuzzymodel-based approaches can lead to less conservative results and less computational burden compared to those based on the classical 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 multiple-parameterization approach for local stabilization of constrained Takagi-Sugeno fuzzy systems with nonlinear consequents[END_REF].

For descriptor observer design, we rewrite system (6) in the following equivalent singular form:

Ē xk+1 = Ā(h)x k -J(h)λ(x k ) + f (z k , u k ) + F (h)φ(x k , u k ) + B(h)d k , y k = C(h)x k + g(z k , u k ) + G(h)ψ(x k , u k ), (7)
with

f (z k , u k ) = 0 f (z k , u k ) and Ā(h) = 0 I A(h) -E(h) ∈ R 2nx×2nx , B(h) = 0 B(h) ∈ R 2nx×n d , Ē = I 0 0 0 ∈ R 2nx×2nx , C(h) = C(h) 0 ∈ R ny×2nx , F (h) = 0 F (h) ∈ R 2nx×n φ , J(h) = 0 J(h) ∈ R 2nx×n λ .
Remark 4. The singular redundancy representation has been exploited for control and observer designs of TS fuzzy descriptor systems with linear consequents [START_REF] Nguyen | Simultaneous estimation of state and unknown input with ∞ guarantee on error-bounds for fuzzy descriptor systems[END_REF], [START_REF] Guelton | An alternative to inverse dynamics joint torques estimation in human stance based on a Takagi-Sugeno unknown-inputs observer in the descriptor form[END_REF], [START_REF] Tanaka | A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions[END_REF]. However, fuzzy observer design using the N-TS fuzzy form [START_REF] Xie | A decoupling approach for observer-based controller design of T-S fuzzy system with unknown premise variables[END_REF] has not yet been reported in the open literature. This singular model reformulation allows for an effective treatment of the nonlinear descriptor matrix E(x k ) in (1), especially in the context of TS fuzzy observer design with unmeasured premise variables.

B. N-TS Fuzzy Observer Structure and Useful Lemmas

For the state estimation of system (6), we consider a Luenberger-like observer of the N-TS fuzzy descriptor form

Ē xk+1 = Ā(h) xk -J(h)λ( xk ) + f (z k , u k ) + F (h)φ(x k , u k ) + L (h)(y k -ŷk ), ŷk = C(h) xk + g(z k , u k ) + G(h)ψ(x k , u k ), (8)
where xk = x k χ k , xk is an estimate of x k and x0 = 0. The observer gain L (h) ∈ R 2nx×ny is to be determined.

Let us define the state estimation errors as

e k = x k -xk , ēk = xk -xk = e k x k+1 -χ k .
From ( 7) and ( 8), the estimation error dynamics is given by

Ēē k+1 = A(h)ē k -J(h)δ λ + F (h)δ φ -L (h)G(h)δ ψ + B(h)d k , (9)
where

A(h) = Ā(h) -L (h) C(h), δ φ = φ(x k , u k ) -φ(x k , u k ), δ ψ = ψ(x k , u k ) -ψ(x k , u k ), δ λ = λ(x k ) -λ( xk ).
Remark 5. Note that if d k = 0, for ∀k ∈ N, and the estimation error ēk converges to the origin, then xk → x k and χ k → xk+1 when k → ∞. Hence, the variable χ k can be viewed as an estimate of x k+1 . The key idea to introduce the virtual variable χ k into the observer structure ( 8) is to guarantee the consistency between the proposed observer form and the singular redundancy representation [START_REF] Xie | A decoupling approach for observer-based controller design of T-S fuzzy system with unknown premise variables[END_REF], which allows avoiding any special structure of the observer gain L (h). Hence, a convex observer design framework with a reduced degree of conservatism can be achieved [START_REF] Guerra | Observer design for Takagi-Sugeno descriptor models: An LMI approach[END_REF].

The nonlinear terms δ φ , δ ψ and δ λ in (9) raise the technical challenges for TS fuzzy-model-based observer design with unmeasured premise variables. To deal with this major difficulty, we reformulate these mismatching terms as functions of the estimation error e k through the following lemma.

Lemma 1 (Differential Mean Value Theorem [START_REF] Mcleod | Mean value theorems for vector valued functions[END_REF]). Let f (x) :

R n → R and a, b ∈ R n . If f (x) is continuously differentiable on co(a, b), then ∃c ∈ co(a, b), such that f (a) -f (b) = ∇ x f (c)(a -b).
Applying Lemma 1 to each element of δ φ , δ ψ and δ λ , these mismatching terms can be written as

δ φ =    ∇ x k φ 1 (t 1 )
. . .

∇ x k φ n φ (t n φ )    (x k -xk ), δ ψ =    ∇ x k ψ 1 (s 1 ) . . . ∇ x k ψ n ψ (t n ψ )    (x k -xk ), δ λ =    ∇ xk λ 1 (w 1 ) . . . ∇ xk λ n λ (w n λ )    (x k -xk ). (10)
Moreover, since the gradients belong to convex bounded sets defined in (3), it follows that

∇ x k φ i (t i ) = n φ i κ=1 β iκ (t i )ρ iκ , β iκ (t i ) ≥ 0, n φ i κ=1 β iκ (t i ) = 1, ∇ x k ψ i (s i ) = n ψ i κ=1 θ iκ (s i )ζ iκ , θ iκ (s i ) ≥ 0, n ψ i κ=1 θ iκ (s i ) = 1, ∇ xk λ i (w i ) = n λ i κ=1 ν iκ (w i )υ iκ , ν iκ (w i ) ≥ 0, n λ i κ=1 ν iκ (w i ) = 1. (11)
Let us define the following matrices:

K( β) = n φ 1 κ1=1 • • • n φn φ κn φ =1 n φ times n φ i=1 β iκi (t i )    ρ 1κ1 . . . ρ n φ κn φ    ∈ R n φ ×nx , (12a)
M ( θ) = n ψ 1 κ1=1 • • • n ψn ψ κn ψ =1 n ψ times n ψ i=1 θ iκi (s i )    ζ 1κ1 . . . ζ n ψ κn ψ    ∈ R n ψ ×nx , (12b)
N (ν) = n λ 1 κ1=1 • • • n λn λ κn λ =1 n λ times n λ i=1 ν iκi (w i )    υ 1κ1 . . . υ n λ κn λ    ∈ R n λ ×2nx , (12c)
with

r φ = n φ i=1 n φi , r ψ = n ψ
i=1 n ψi and r λ = n λ i=1 n λi . For observer design, we make use of the multi-index notation to simplify the representation of matrices K( β), M ( θ) and N (ν). To this end, let j ∈ I r φ represent the multiple indices

(κ 1 , . . . , κ n φ ) ∈ I n φ 1 × • • • × I n φn φ in (12a), ∈ I r ψ represent the multiple indices (κ 1 , . . . , κ n ψ ) ∈ I n ψ 1 × • • • × I n ψn ψ in (12b), and m ∈ I r λ represent the multiple indices (κ 1 , . . . , κ n λ ) ∈ I n λ 1 × • • • × I n λn λ
in (12c). We also denote j{q}, {q}, m{q} the κ q subindex in j, and m, respectively. Then, the parameter-dependent matrices in [START_REF] Nguyen | Fuzzy control systems: Past, present and future[END_REF] can be compactly represented in the following single-index form:

K( β) = r φ j=1 βj (t)K j , M ( θ) = r ψ =1 θ (s)M , N (ν) = r λ m=1 νm (w)N m , (13)
with β = β1 (t) β2 (t) . . . βr φ (t) , θ = θ1 (s) θ2 (s) . . . θr ψ (s) , ν = ν1 (w) ν2 (w) . . . νr λ (w) , and

K j =    ρ 1j{1} . . . ρ n φ j{n φ }    , βj (t) = n φ i=1 β ij{i} t j{i} , M =    ζ 1 {1} . . . ζ n ψ {n ψ }    , θ (s) = n ψ =1 θ i {i} s j{i} , N m =    υ 1m{1} . . . υ n λ m{n λ }    , νm (w) = n λ i=1 ν im{i} w j{i} . (14)
Moreover, it follows from ( 11) and ( 14) that

βj (t) ≥ 0, r φ j=1 βj (t) = 1, (15a)
θ (s) ≥ 0, r ψ =1 θ (s) = 1, (15b)
νm (w) ≥ 0,

r λ m=1 νm (w) = 1. (15c)
Let Ω β , Ω θ and Ω ν be the sets of unknown time-varying parameters β, θ and ν verifying the convex sum properties in (15a), (15b) and (15c), respectively. From the matrix definition (13), the mismatching nonlinear terms in [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] can be expressed by

δ φ = K( β)(x k -xk ), δ ψ = M ( θ)(x k -xk ), δ λ = N (ν)(x k -xk ). (16)
Then, from ( 9) and ( 16), the error dynamics is rewritten as

Ēē k+1 = r i=1 h i (z) Āi ( β, ν) -L (h) Ci ( θ) ēk + B(h)d k , (17)
with

Āi ( β, ν) = Āi -Ji N (ν) + Fi K( β) I 0 and Ci ( θ) = Ci + Ḡi M ( θ) I 0 .
This paper provides an LMI-based procedure for the following N-TS fuzzy observer design problem.

Problem 1. Consider a nonlinear descriptor system (1), which can be reformulated in the form [START_REF] Besançon | Nonlinear Observers and Applications[END_REF]. Design a N-TS fuzzy observer [START_REF] Luenberger | Observers for multivariable systems[END_REF] with the MF-dependent gain L (h) such that the error dynamics (17) verifies the following properties. (P1) For d k = 0, ∀k ∈ N, the error dynamics ( 17) is globally uniformly exponentially stable with respect to the origin. (P2) For zero initial error e 0 = 0 and any unknown input sequence {d k } k∈N ∈ ∞ , we have e k < γ d ∞ , ∀k ∈ N, for a predefined scalar γ > 0. (P3) The estimation error e k is uniformly bounded for any initial condition e 0 and any input sequence {d k } k∈N ∈ ∞ . That is, there exists a bound ϕ(e 0 , d ∞ ) such that e k ≤ ϕ(e 0 , d ∞ ), for ∀k ≥ 0. Moreover, we have

lim k→∞ sup e k < γ d ∞ . (18)
System [START_REF] Nguyen | Unknown input observers for simultaneous estimation of vehicle dynamics and driver torque: Theoretical design and hardware experiments[END_REF] verifying properties (P1)-(P3) is said to be globally uniformly ∞ -stable with a performance level γ. More details on the global uniform ∞ -stablity can be found in [START_REF] Zak | Systems and Control[END_REF]Chapter 4].

The following relaxation result is useful to convert a MF-dependent inequality into a finite set of LMI constraints.

Lemma 2 ([55]

). Consider the MF-dependent inequality

Υ hhh+ = r i=1 r j=1 r l=1 h i (z k )h j (z k )h l (z k+1 )Υ ijl ≺ 0, (19)
where h + = h 1 (z k+1 ), h 2 (z k+1 ), . . . , h r (z k+1 ) , and h, h + ∈ Ω. The symmetric matrices of appropriate dimensions Υ ijl , with i, j, l ∈ I r , are linearly dependent on the unknown decision variables. Inequality [START_REF] Zhao | A novel approach to state and unknown input estimation for Takagi-Sugeno fuzzy models with applications to fault detection[END_REF] holds if

Υ iil ≺ 0, i, l ∈ I r , 2 r -1 Υ iil + Υ ijl + Υ jil ≺ 0, i, j, l ∈ I r , i = j.
Lemma 2 provides a good tradeoff between numerical complexity and design conservatism [START_REF] Guerra | Observer design for Takagi-Sugeno descriptor models: An LMI approach[END_REF]. Other relaxation results with different degrees of complexity and/or conservatism can be found in [START_REF] Sala | Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya's theorem[END_REF].

III. FUZZY DESCRIPTOR OBSERVER WITH UNMEASURED PREMISE VARIABLES AND UNKNOWN INPUT

This section presents an LMI-based solution to design an N-TS fuzzy descriptor observer for system [START_REF] Besançon | Nonlinear Observers and Applications[END_REF]. We provide sufficient conditions to guarantee the global uniform ∞ -stability of the error dynamics (9) in the following theorem.

Theorem 1. Consider the estimation error dynamics [START_REF] Nguyen | Unknown input observers for simultaneous estimation of vehicle dynamics and driver torque: Theoretical design and hardware experiments[END_REF]. If there exist MF-dependent symmetric matrices

P (h) ∈ R nx×nx , R(h) ∈ R nx×nx , a regular MF-dependent matrix H(h) ∈ R 2nx×2nx , matrices Q(h) ∈ R nx×nx , L(h) ∈ R 2nx×ny
, and positive scalars τ ∈ (0, 1), , γ such that

Σ(h, h + ) + Φ(h, β, ν, θ) + Φ(h, β, ν, θ) ≺ 0, (20)
P (h) I γI 0, (21)
for h, h + ∈ Ω, β ∈ Ω β , ν ∈ Ω ν , θ ∈ Ω θ ,
where

P (h) = P (h) Q(h) R(h) , Σ(h, h + ) =   (τ -1) Ē P (h) Ē 0 P (h + ) 0 0 -τ γI   , Φ(h, β, ν, θ) =   Γ(h, β, ν, θ) -H(h) H(h) B(h) Γ(h, β, ν, θ) -H(h) H(h) B(h) 0 0 0   , Γ(h, β, ν, θ) = H(h) Ā (h, β, ν) -L(h) C (h, θ), Ā (h, β, ν) = r i=1 h i (z) Āi ( β, ν), C (h, θ) = r i=1 h i (z) Ci ( θ).
Then, system ( 17) is globally uniformly ∞ -stable with performance level γ. Moreover, the N-TS fuzzy observer corresponding to the nonlinear descriptor system (2) is given by

E(h)χ k + J(h)λ(x k , χ k ) = A(h)x k + f (z k , u k ) + F (h)φ(x k , u k ) + L 2 (h)(y k -ŷk ), xk+1 = χ k + L 1 (h)(y k -ŷk ), ŷk = C(h)x k + g(z k , u k ) + G(h)ψ(x k , u k ), (22)
with

L (h) . = L 1 (h) L 2 (h) = H(h) -1 L(h).
Proof. For the observer design, we consider the following MF-dependent Lyapunov function:

V k = ē k Ē P (h) Ēē k . (23)
Note that Ē P (h) Ē = I 0 0 0

P (h) Q(h) R(h) I 0 0 0 = P (h) 0 0 0 .
Then, the Lyapunov function V k in ( 23) can be rewritten as V k = e k P (h)e k , which is positive definite for ∀k ∈ N since P (h) 0. Inequality (20) can be represented in the form

Σ(h, h + ) + Π(h, β, ν, θ) + Π(h, β, ν, θ) ≺ 0, (24)
where

Π(h, β, ν, θ) = H (h) Λ(h, β, ν, θ) -I B(h) , H (h) = H(h) H(h) 0 , Λ(h, β, ν, θ) = Ā (h, β, ν) -L (h) C (h, θ).
From the error dynamics equation ( 17), it follows that

2E (h) Λ(h, β, ν, θ)ē k -Ēē k+1 + B(h)d k = 0, ∀k ∈ N, (25)
with

E (h) = ē k H(h)+ ē k+1 Ē H(h).
Pre-and postmultiplying [START_REF] Pérez-Estrada | Generalized dynamic observers for quasi-LPV systems with unmeasurable scheduling functions[END_REF] with ē k ē k+1 Ē d k and its transpose while taking into account equality [START_REF] Bergsten | Observers for Takagi-Sugeno fuzzy systems[END_REF], it follows that

ē k+1 Ē P (h + ) Ēē k+1 + (τ -1)ē k Ē P (h) Ēē k -τ γd k d k ≺ 0, ∀k ∈ N. ( 26
)
Condition ( 26) can be rewritten in the form

∆V k + τ (V k -γd k d k ) < 0, ∀k ∈ N, (27)
where V k is defined in [START_REF] Nguyen | Simultaneous estimation of state and unknown input with ∞ guarantee on error-bounds for fuzzy descriptor systems[END_REF], and ∆V k = V k+1 -V k is its difference along the trajectory of the error dynamics [START_REF] Nguyen | Unknown input observers for simultaneous estimation of vehicle dynamics and driver torque: Theoretical design and hardware experiments[END_REF]. It follows from ( 27) that

V k < (1 -τ )V k-1 + τ γ d k-1 2 , ∀k ≥ 1. ( 28
)
Since τ ∈ (0, 1), by recursively it follows from ( 28) that

V k < (1 -τ ) k V 0 + τ γ k-1 i=0 (1 -τ ) i d k-1-i 2 < (1 -τ ) k V 0 + γ d 2 ∞ , ∀k ≥ 1. ( 29
)
If

d k = 0, ∀k ∈ N, it follows from (29) that V k < (1 -τ ) k V 0 ,

which guarantees property (P1)

. By Schur complement lemma [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF], it follows from ( 21) and ( 29) that

e k 2 < γV k < γ(1 -τ ) k V 0 + γ 2 d 2 ∞ , ∀k ≥ 1. (30)
Note that inequality (30) implies [START_REF] Rotondo | Actuator multiplicative fault estimation in discrete-time LPV systems using switched observers[END_REF], and V 0 = 0, for e 0 = 0. Then, it follows from ( 30) that

e k 2 < γ 2 d 2 ∞ , ∀k ≥ 1. (31)
Properties (P2)-(P3) follow directly from [START_REF] Chandra | Fault detection in uncertain LPV systems with imperfect scheduling parameter using sliding mode observers[END_REF]. Moreover, the N-TS fuzzy observer structure ( 22) is straightforwardly recovered from [START_REF] Luenberger | Observers for multivariable systems[END_REF]. This concludes the proof.

Remark 6. Note from (18) that the estimation error e k can be minimized by minimizing the ∞ -gain performance γ.

Remark 7. The positive scalar and the MF-dependent matrix H(h) are introduced in the descriptor observer design through equality [START_REF] Bergsten | Observers for Takagi-Sugeno fuzzy systems[END_REF] to reduce the conservatism. This null-term relaxation approach has been effectively exploited for TS fuzzy-modelbased control and observer design, see for instance [START_REF] Nguyen | Simultaneous estimation of state and unknown input with ∞ guarantee on error-bounds for fuzzy descriptor systems[END_REF], [START_REF] Mozelli | A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems[END_REF].

Remark 8. Theorem 1 can directly be applied to the classical TS fuzzy systems with unmeasured premise variables and unknown input by setting E(h) = I and λ(x k ) = 0. Moreover, if φ(x k , u k ) = 0, ψ(x k , u k ) = 0 and λ(x k ) = 0, then we recover the class of nonlinear descriptor systems considered in [START_REF] Nguyen | Simultaneous estimation of state and unknown input with ∞ guarantee on error-bounds for fuzzy descriptor systems[END_REF]. Hence, the proposed N-TS fuzzy observer design is more general than the existing results in the open literature.

Theorem 1 cannot be directly applied to design ∞ N-TS fuzzy observer (8) due to the MF-dependent nature of conditions ( 20)- [START_REF] Jia | Fault reconstruction and fault-tolerant control via learning observers in Takagi-Sugeno fuzzy descriptor systems with time delays[END_REF]. On the basis of Theorem 1, the following theorem provides a finite set of tractable design conditions while minimizing the state estimation error. Theorem 2. Consider the estimation error dynamics [START_REF] Nguyen | Unknown input observers for simultaneous estimation of vehicle dynamics and driver torque: Theoretical design and hardware experiments[END_REF] and a positive scalar τ ∈ (0, 1). If there exist symmetric matrices

P i ∈ R nx×nx , R i ∈ R nx×nx , regular matrices H i ∈ R 2nx×2nx , matrices Q i ∈ R nx×nx , L i ∈ R 2nx×ny , i ∈ I r ,
subject to

Ξ iilκ m ≺ 0, ( 33
) 2 r -1 Ξ iilκ m + Ξ ijlκ m + Ξ jilκ m ≺ 0, (34)
P i I γI 0, (35)
for i, j, l ∈ I r , i = j, κ ∈ I r φ , ∈ I r ψ , m ∈ I r λ . The quantity Ξ ijlκ m in ( 34) is given by

Ξ ijlκ m = Σ il + Φ ijκ m + Φ ijκ m ,
where

Σ il =   (τ -1) Ē Pi Ē 0 Pl 0 0 -τ γI   , Φ ijκ m =   Γ ijκ m -H i H i Bj Γ ijκ m -H i H i Bj 0 0 0   , Γ ijκ m = H j Āi -Ji N m + Fi K κ I 0 -L j Ci + Ḡi M .
Then, the estimation error dynamics ( 17) is globally uniformly ∞ -stable with performance level γ. Moreover, the N-TS fuzzy observer structure corresponding to the initial descriptor system ( 6) is given in [START_REF] Lan | Integrated design of fault-tolerant control for nonlinear systems based on fault estimation and T-S fuzzy modeling[END_REF], with L (h) = H(h) -1 L(h) and

H(h) L(h) = r i=1 h i (z k ) H i L i .
Proof. The proof is based on the convex combination form of the state-space matrices A(h), B(h), C(h), E(h), F (h), G(h), J(h) and the decision matrices P (h), R(h), Q(h), L(h), H(h). Indeed, using Lemma 2 and the convexity property of the bounded sets for the gradients, inequalities ( 33)-( 34) imply clearly [START_REF] Nguyen | Takagi-sugeno fuzzy unknown input observers to estimate nonlinear dynamics of autonomous ground vehicles: Theory and real-time verification[END_REF] in Theorem 1. Moreover, condition (21) can also be rewritten as

r i=1 h i (z k ) P i I γI 0. (36)
Remark that condition [START_REF] Nguyen | Nonlinear tracking control with reduced complexity of serial robots: A robust fuzzy descriptor approach[END_REF] guarantees [START_REF] Guelton | An alternative to inverse dynamics joint torques estimation in human stance based on a Takagi-Sugeno unknown-inputs observer in the descriptor form[END_REF]. Then, the proof can be concluded by following the results of Theorem 1.

Remark 9. The design conditions in Theorem 2 are expressed in terms of LMIs with a line search over the scalar . A gridding method can be performed to search for . Then, the optimization problem (32) under LMI constraints ( 33)-( 35) can be effectively solved with available numerical solvers [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF].

IV. UNKNOWN INPUT ESTIMATION WITH ∞ NORM-BOUNDED GUARANTEE

This section presents a method to reconstruct the unknown input of system (2). Especially, the ultimate norm upper-bound of the UI estimation error is also characterized. To this end, we derive from (6) that

B(h)d k = E(h)x k+1 + J(h)λ(x k ) -A(h)x k -F (x k , z k , u k ), (37)
with

F (x k , z k , u k ) = f (z k , u k ) + F (h)φ(x k , u k ). Since B(h) is of full-column rank for ∀h ∈ Ω, there exists B(h) † such that B(h) † B(h) = I. Premultiplying (37) with B(h) † yields d k = B(h) † (E(h)x k+1 -A(h)x k -F (x k , z k , u k )) . (38)
We propose the estimate dk of d k of the form

dk = B(h) † E(h)x k+1 + J(h)λ( xk ) -A(h)x k -F (x k , z k , u k ) . (39)
Let us define the UI estimation error as ε k = d k -dk . It follows from ( 38) and ( 39) that

ε k = B(h) † (E(h)e k+1 + J(h)δ λ -A(h)e k -F (h)δ φ ) , (40)
where δ φ and δ λ are given in [START_REF] Zhang | Sensor fault estimation of switched fuzzy systems with unknown input[END_REF]. We decompose N (ν), defined in (13), as N (ν) = N 1 (ν) N 2 (ν) . Expression (40) can be compactly rewritten as

ε k = B(h) † E d (h, ν)e k+1 -A d (h, β, ν)e k , (41)
with

E d (h, ν) = E(h) + J(h)N 2 (ν) and A d (h, β, ν) = A(h) + F (h)K( β) -J(h)N 1 (ν). It follows from (41) that ε k ≤ B(h) † E d (h, ν) e k+1 + B(h) † A d (h, β, ν) e k ≤ B(h) † ( E d (h, ν) e k+1 + A d (h, β, ν) e k . (42)
If the optimization problem ( 32) is feasible, it follows that

lim k→∞ sup e k < γ d ∞ , lim k→∞ sup e k+1 < γ d ∞ . (43)
From ( 42) and ( 43), the ultimate norm upper-bound of the estimation error ε k is given by

lim k→∞ sup ε k ≤ γη d ∞ , (44)
with η = B(h) † ∞ E d (h, ν) ∞ + A d (h, β, ν ∞ .
The following result summarizes the ∞ norm-bounded UI estimation.

Corollary 1. Consider the error dynamics [START_REF] Pan | A unified framework for asymptotic observer design of fuzzy systems with unmeasurable premise variables[END_REF]. If the optimization problem (32) in Theorem 2 is feasible, then there exists a N-TS fuzzy descriptor observer [START_REF] Lan | Integrated design of fault-tolerant control for nonlinear systems based on fault estimation and T-S fuzzy modeling[END_REF] such that the following properties are verified.

• The error dynamics ( 9) is globally uniformly ∞ -stable with a guaranteed performance level γ.

• With the UI estimate dk defined in [START_REF] Wang | Robust fault estimation based on zonotopic Kalman filter for discrete-time descriptor systems[END_REF], the UI estimation error ε k is ultimately norm-bounded as in [START_REF] Marx | Design of observers for Takagi-Sugeno descriptor systems with unknown inputs and application to fault diagnosis[END_REF].

Remark 10. Assume that the optimization problem ( 32) is feasible with an arbitrarily small ∞ -gain γ. Then, for a bounded η, both the state and the UI can be estimated with an arbitrary degree of accuracy as respectively shown in [START_REF] Estrada | Robust H -/H∞ fault detection observer design for descriptor-LPV systems with unmeasurable gain scheduling functions[END_REF] and [START_REF] Marx | Design of observers for Takagi-Sugeno descriptor systems with unknown inputs and application to fault diagnosis[END_REF].

The UI observer design is summarized in Algorithm 1.

Algorithm 1: Observer Design Procedure Input: Nonlinear descriptor system (1) and its equivalent N-TS fuzzy form [START_REF] Zhang | Singular system full-order and reduced-order fixed-time observer design[END_REF].

Output: N-TS fuzzy observer [START_REF] Lan | Integrated design of fault-tolerant control for nonlinear systems based on fault estimation and T-S fuzzy modeling[END_REF] and UI estimator [START_REF] Wang | Robust fault estimation based on zonotopic Kalman filter for discrete-time descriptor systems[END_REF] such that

lim k→∞ sup x k -xk < γ d ∞ , lim k→∞ sup d k -dk < γη d ∞ ,
with γ and η respectively specified in Theorem 2 and inequality [START_REF] Marx | Design of observers for Takagi-Sugeno descriptor systems with unknown inputs and application to fault diagnosis[END_REF]. 1 Check the regularity of matrix E(x k ) and the boundedness condition (3) in Assumption 1.

• If YES, then go to Step 2.

• If NO, then the proposed method is unapplicable to the considered system. 2 Solve the LMI-based optimization problem [START_REF] Tan | ZKF-based optimal robust fault estimation of descriptor LPV systems with measurement error-affected scheduling variables[END_REF] to get H i and L i , for i ∈ I r .

3 Construct the N-TS fuzzy observer [START_REF] Lan | Integrated design of fault-tolerant control for nonlinear systems based on fault estimation and T-S fuzzy modeling[END_REF] to estimate x k . 4 Construct the UI estimator [START_REF] Wang | Robust fault estimation based on zonotopic Kalman filter for discrete-time descriptor systems[END_REF] to estimate d k .

The proposed fuzzy UI observer design is illustrated in the following section.

V. ILLUSTRATIVE EXAMPLES

This section presents two real-world examples to illustrate the interests of the proposed N-TS fuzzy observer design. The first example is concerned with the estimation problem of a chaotic Lorenz system with an external input. For this example, the unmeasured nonlinearities are involved in the descriptor matrix E(x k ). In the second example, we focus on estimating the angle of attack and aerodynamical forces and moments of a simulated aircraft, for which unmeasurable nonlinearities are involved in the right-hand side of the system state dynamics and in the system output. It is important to note that existing TS fuzzy-model-based observer designs in the literature cannot be applied to these two examples.

All LMI-based optimizations were solved using YALMIP toolbox [START_REF] Löfberg | Yalmip: A toolbox for modeling and optimization in Matlab[END_REF] and MOSEK 8.1.0.67 solver within MATLAB R2018b environment. For the gridding line search, we consider a logarithmically spaced grid as ∈ [10 -1 , 10 3 ].

Example 1 (Chaotic dynamical system). We revisit the following Lorenz system [START_REF] Ichalal | How to cope with unmeasurable premise variables in Takagi-Sugeno observer design: dynamic extension approach[END_REF], widely used in secured communications, whose dynamics is given by

ẋ1 = -px 1 + px 2 , ẋ2 = qx 1 -x 1 x 3 -x 2 , ẋ3 = x 1 x 2 -sx 3 + d, (45)
where

x 1 ∈ [-21, 21], x 2 ∈ [-25, π 12
], x 3 ∈ [0, 50] are the system states, d is the external input, and p = 10, q = 8 3 , s = 28 are the system parameters. Assume that signals x 2 and x 3 can be measured and the goal is to estimate x 1 and d. For this example, we consider a backward-Euler-like discretization with a time step T = 0.01s to transform system (45) into its discrete-time representation of the following form:

E(z k )x k+1 + Jλ(x k ) = Ax k + Bd k , y k = Cx k , (46)
with

x k = x 1k x 2k x 3k , y k = x 2k x 3k , z k = x 2k , A = I, and
E(z k ) =   1 + pT -pT 0 -qT 1 + T 0 -T x 2k 0 1 + sT   , J =   0 0 T 0 0 T   , λ(x k ) = x 1k x 3(k+1) x 1k x 2(k+1) , B =   0 0 T   , C = 0 1 0 0 0 1 .
Note that λ(x k ) is continuously differentiable with respect to xk , and its gradients are given by

∇ x λ 1 = x 3(k+1) 0 0 0 0 x 1k , ∇ x λ 2 = x 2(k+1) 0 0 0 x 1k 0 . (47)
Using the sector nonlinearity approach [10, Chapter 2], an equivalent N-TS fuzzy descriptor representation [START_REF] Zhang | Singular system full-order and reduced-order fixed-time observer design[END_REF] with two fuzzy rules (r = 2) can be obtained for the nonlinear descriptor system [START_REF] Moodi | Robust observer-based controller design for Takagi-Sugeno systems with nonlinear consequent parts[END_REF]. Moreover, with the gradient definitions (47), a convex bounded set representation is obtained for N (ν) in ( 13) with 16 vertices. The state-space matrices and the nonlinear MFs of the corresponding N-TS fuzzy representation, as well as the vertex matrices for the convex bounded set are omitted for brevity. Applying Theorem 2 to system [START_REF] Moodi | Robust observer-based controller design for Takagi-Sugeno systems with nonlinear consequent parts[END_REF], the suboptimal ∞ -gains found for different values of the decay rate τ are depicted in Fig. 1. Note that γ is infinite when the optimization problem [START_REF] Tan | ZKF-based optimal robust fault estimation of descriptor LPV systems with measurement error-affected scheduling variables[END_REF] is not feasible. Remark that a larger value of τ can improve the estimation performance, however too large value of τ may lead to infeasibility. To highlight the interest of introducing the virtual variable χ k into the observer structure [START_REF] Luenberger | Observers for multivariable systems[END_REF], we compare the feasibility regions obtained with the conditions in Theorem 2 and those derived without the introduction of χ k , i.e., similar design conditions as in Theorem 2 obtained by setting L 1 (h) = 0. The comparison result is shown in Fig. 2. Observe that Theorem 2 leads to less conservative results in this case because the corresponding feasibility region is clearly larger than that obtained without using the virtual variable χ k . For illustrations, the estimation results obtained from Theorem 2 with τ = 0.17 and = 2 are summarized in Figs. 3 and4. We can see that the proposed N-TS fuzzy descriptor observer provides an accurate estimation performance for both the state x 1k and the unknown input d k . Example 2 (Aircraft Longitudinal Dynamics). The longitudinal movement dynamics of an aircraft in wind axis can be expressed as follows [50, eq. (2.5-32)]:

m v = F t cos(α) -D -mg sin(θ -α), mv α = -F t sin(α) -L + mg cos(θ -α) + mvq, J y q = M y , θ = q, ( 48
)
where v is the speed relative to the wind, α is the angle of attack of the aircraft, q is the pitch rate, θ is the pitch angle, m is the mass of the aircraft, J y is the pitch moment of inertia, g is the acceleration of gravity, F t is the thrust force, D is the drag force, L is the lift force and M y the total pitch moment (torque). We consider that m = 9357.61kg, J y = 75673.62kgm 2 and g = 9.81m/s 2 , which correspond to the simulation of an F16 aircraft longitudinal model taken from the FlightGear open-source flight simulator [START_REF]Flightgear flight simulator[END_REF]. The signals v, q, θ, F t and v h = v sin(θ -α), with v h the rate at which the aircraft is changing its altitude, can be measured by sensors. The goal is to estimate the remaining state α and the unknown inputs D, L and M y .

Using the forward-Euler discretization with a time step T = 0.1s, the discrete-time counterpart of system ( 48) can be obtained as

E(z k )x k+1 = A(z k )x k + F (z k )φ(x k , u k ) + Bd k , y k = Cx k + G(z k )ψ(x k ), (49)
with

x k = v k α k q k θ k , u k = F tk , d k = D L M y , z k = v k sin(θ k ) cos(θ k ) , y k = v k q k θ k v h k , φ(x k , u k ) = F tk cos(α k ) F tk sin(α k ) cos(α k ) sin(α k ) , ψ(x k ) = cos(α k ) sin(α k ) and E(z k ) =     m 0 0 0 0 mz 1k 0 0 0 0 J y 0 0 0 0 1     , B =     -T 0 0 0 -T 0 0 0 T 0 0 0     , A(z k ) =     m 0 0 0 0 mz 1k T mz 1k 0 0 0 J y 0 0 0 T 1     , F (z k ) =     T 0 -T mgz 2k T mgz 3k 0 -T T mgz 3k T mgz 2k 0 0 0 0 0 0 0 0     , G(z k ) =     0 0 0 0 0 0 z 1k z 2k -z 1k z 3k     , C =     1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0     .
The value ranges of the variables of system (49) are given by [START_REF] Stevens | Aircraft Control and Simulation -Dynamics, Controls Design, and Autonomous Systems[END_REF]

v k ∈ [90, 150], α k ∈ [- π 18 , π 12 ], θ k ∈ [- π 18 , π 12
], F tk ∈ [0, 2.66 × 10 4 ].

Since both nonlinear functions φ(x k , u k ) and ψ(x k ) are continuously differentiable with respect to x k , whose gradients are respectively defined as ∇ x φ 1 = 0 -F tk sin(α k ) 0 0 , ∇ x φ 2 = 0 F tk cos(α k ) 0 0 , ∇ x φ 3 = 0 -sin(α k ) 0 0 , ∇ x φ 4 = 0 cos(α k ) 0 0 , ∇ x ψ 1 = 0 -sin(α k ) 0 0 , ∇ x ψ 2 = 0 cos(α k ) 0 0 . [START_REF] Besançon | Nonlinear Observers and Applications[END_REF], an equivalent N-TS fuzzy descriptor representation [START_REF] Zhang | Singular system full-order and reduced-order fixed-time observer design[END_REF] with eight fuzzy rules (r = 8) can be obtained for the nonlinear descriptor system [START_REF] Nguyen | Avoiding unmeasured premise variables in designing unknown input observers for Takagi-Sugeno fuzzy systems[END_REF]. A convex bounded set representation can be determined for K( β) and M ( θ) in ( 13) with 16 and 4 vertices, respectively. The state-space matrices and the nonlinear MFs corresponding to this N-TS fuzzy representation, as well as the vertex matrices for the convex bounded sets are omitted for brevity. By Theorem 2, an observer solution can be obtained with τ = 0.27, = 117.6812, and a suboptimal ∞ -gain level of γ = 2.432 × 10 -4 . The corresponding estimation results are summarized in Figs. 5 and6. We can see that the designed N-TS fuzzy observer can provide a satisfactory estimation performance for this real-world application, except for a small bias on the estimation of α k and D k .

Moreover

VI. CONCLUDING REMARKS

A new ∞ observer design has been proposed for a large class of nonlinear descriptor systems with UIs. To deal with the challenging issue on unmeasured premise variables, the nonlinear system is rewritten in a specific N-TS fuzzy form which permits isolating unmeasurable nonlinearities. Together with Lyapunov-based arguments and a judicious use of the DMVT, this allows for an effective N-TS fuzzy observer design framework guaranteeing the global uniform ∞ -stablity of the estimation error dynamics. Moreover, for observer design, a special singular redundancy representation is exploited for the treatment of the nonlinear descriptor form. As a result, all involved decision variables for the observer design can be of free structure to further reduce the design conservatism. The estimation accuracy of both the state and the UI can be improved by minimizing a suboptimal ∞ -gain level. The interests of the proposed N-TS fuzzy observer design are highlighted with two real-world applications. Future works focus on exploiting the proposed results for N-TS fuzzy observer-based control of descriptor systems with unmeasured nonlinearities. Another interesting future direction is to extend the proposed observer design method to the case where the descriptor matrix E(x k ) in ( 1) is singular and/or the premise variables cannot be precisely obtained from sensors.

  and positive scalars , γ such that the following optimization problem is feasible: minimize ζi=( ,γ,Pi,Ri,Qi,Li,Hi), i∈Ir γ

Fig. 1 .

 1 Fig. 1. Effect of the decay rate parameter τ on the ∞-gain bound γ for the Lorenz dynamical system in Example 1.

Fig. 2 .

 2 Fig. 2. Feasibility regions for τ ∈ [0.01, 0.17] and ∈ [2, 50] obtained the Lorenz dynamical system in Example 2. The colored areas indicate points where LMI-based design conditions were feasible, whereas the white areas represent the infeasibility areas. The left plot represents the conditions in Theorem 2, while the right plot represents the results with L 1 (h) = 0.

Fig. 3 .Fig. 4 .

 34 Fig. 3. System trajectory of Lorenz system in Example 1. The blue solid line indicates the trajectory of the nonlinear system (46), whereas the red line indicates the trajectory given by the proposed N-TS fuzzy observer.

Fig. 5 .Fig. 6 .

 56 Fig. 5. State estimation for the aircraft in Example 2. The blue solid lines indicate the original state, whereas the red dash-dotted lines indicate their estimates.

  is the identity matrix of appropriate dimension. " " stands for the terms deduced by symmetry. ∇ z f = ∂f ∂z is the gradient of function f with relation to the variables in vector

z ∈ R κ and represents the row vector ∂f ∂z1

  the unknown input, and y k ∈ R ny is the output. The nonlinear functions f : D x × D u × D d → R nx and g : D x × D u → R ny are differentiable with respect to the state x k .

  , it follows from[START_REF] Stevens | Aircraft Control and Simulation -Dynamics, Controls Design, and Autonomous Systems[END_REF] that sin(θ k ) ∈ [-0.1736, 0.2588], cos(θ k ) ∈ [0.9659, 1], sin(α k ) ∈ [-0.1736, 0.2588], cos(α k ) ∈ [0.9659, 1], F tk sin(α k ) ∈ [-6884, 4618] and F tk cos(α k ) ∈ [0, 2.66×10 4 ].Using the sector nonlinearity approach[START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] Chapter

ACKNOWLEDGEMENT This work was supported in part by the French Ministry of Higher Education and Research; in part by the National Center for Scientific Research (CNRS); in part by the Nord-Pas-de-Calais Region under Project ELSAT 2020; in part by Key Research Project of North Minzu University under Grant 2019KJ39; in part by the Natural Science Foundation of Ningxia Hui Autonomous Region under Grant 2019AAC03118; in part by the National Natural Science Foundation of China under Grant 61463001; in part by the Major Special Project of North Minzu University under Grant ZDZX201902; in part by the Third Batch of Ningxia Youth Talents Supporting Program under Grant TJGC2018097; in part by the Advanced Intelligent Perception and Control Technology Innovative Team of Ningxia.

DATA AVAILABILITY STATEMENT

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.