A.-T Nguyen

Reduced-Complexity LMI Conditions for Admissibility Analysis and Control Design of

Singular Nonlinear Systems Amine Dehak, Anh-Tu Nguyen * , Senior Member, IEEE, Antoine Dequidt, Laurent Vermeiren, Michel Dambrine

Abstract-We present a reduced-complexity control approach for a class of descriptor nonlinear systems with a nonlinear derivative matrix, possibly singular. To this end, a systematic approach is proposed to obtain an equivalent polytopic representation of a given nonlinear system within a compact set of the state space. This modeling approach has two particular features compared to the related Takagi-Sugeno (TS) fuzzy model-based framework. First, the model complexity only grows proportionally, rather than exponentially, with the number of premise variables. Second, the vertices of the proposed polytopic models can admit an infinite number of representations for the same predefined set of premise variables. This non-uniqueness feature allows introducing some specific slack variables at the modeling step to reduce the control design conservatism. Based on the proposed polytopic representation and Lyapunov stability theory, we derive reduced-complexity admissibility analysis and design conditions, expressed in terms of linear matrix inequalities, for the considered class of descriptor systems. In particular, a new nonlinear control law is proposed for regular descriptor systems to avoid using the extended redundancy form, which may yield numerically complex and conservative results due to the imposed special control structure. Both numerical and physically motivated examples are given to demonstrate the interests of the new control approach with respect to existing TS fuzzy modelbased control results.

Index Terms-Admissibility, Takagi-Sugeno fuzzy descriptor systems, linear matrix inequality (LMI), complexity reduction.

I. INTRODUCTION

Differential-algebraic systems, also known as descriptor systems or generalized state-space systems, provide a natural framework to represent and analyze a large class of engineering applications, including robotics [START_REF] Lewis | Robot Manipulator Control: Theory and Practice[END_REF], rehabilitation systems [START_REF] Gao | Adaptive fuzzyregion-based control of Euler-Lagrange systems with kinematically singular configurations[END_REF], chemical processes [START_REF] Kumar | Control of Nonlinear Differential Algebraic Equation Systems: with Applications to Chemical Processes[END_REF], transportation and power systems [START_REF] Dai | Singular Control Systems[END_REF], etc. However, the analysis of such systems requires more involved techniques when compared to the classical statespace systems since not only stability but also regularity and admissibility have to be addressed [START_REF] Wang | H∞ control for nonlinear descriptor systems[END_REF], [START_REF] Arceo | Convex stability analysis of nonlinear singular systems via linear matrix inequalities[END_REF]. Stability analysis of descriptor systems has been classically studied based on the system index or the coordinates reduction techniques [START_REF] Dai | Singular Control Systems[END_REF]. However, such methods require extensive algebraic manipulations, which can be unsuitable for a large class of engineering problems [START_REF] Di Francor | Stability of nonlinear differential-algebraic systems via additive identity[END_REF]. To avoid this drawback, Lyapunov methods directly based on descriptor system formulation have been proposed [START_REF] Gao | Actuator fault robust estimation and fault-tolerant control for a class of nonlinear descriptor systems[END_REF]- [START_REF] He | Admissibility analysis and robust H∞ control for T-S fuzzy descriptor systems with structured parametric uncertainties[END_REF]. Despite significant advances in numerical analysis and simulation, the problem of stability analysis and control for general descriptor systems remains challenging, especially for systems with a large number of nonlinearities.

Takagi-Sugeno (TS) fuzzy paradigm has become one of the most popular techniques for nonlinear control systems [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], which is due to several factors. First, TS fuzzy models can be used to approximate any smooth nonlinear system with any degree of accuracy. In particular, an exact TS fuzzy model of a given nonlinear system can be obtained within a state-space compact set via the sector nonlinearity approach [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. Second, using TS fuzzy modeling and Lyapunov stability theory, stability analysis and control design conditions can be derived for nonlinear systems in the form of linear matrix inequalities (LMIs), which can be efficiently solved using convex optimization techniques [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. Third, TS fuzzy modelbased technique has been successfully applied to various engineering applications [START_REF] Li | Adaptive sliding mode control for Takagi-Sugeno fuzzy systems and its applications[END_REF]- [START_REF] Precup | A survey on industrial applications of fuzzy control[END_REF]. From a theoretical viewpoint, it is possible to derive asymptotically necessary and sufficient stability conditions for TS fuzzy systems [START_REF] Sala | Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya's theorem[END_REF]. Nevertheless, in practice these stability conditions are conceptual rather than implementable since the computational burden swiftly increases such that most numerical solvers crash [START_REF] Nguyen | Fuzzy control systems: Past, present and future[END_REF]. This leads to another challenge of TS fuzzy approaches in deriving less conservative sufficient conditions for stability analysis of nonlinear systems with a reasonable numerical burden. To reduce the design conservatism, various fuzzy Lyapunov functions have been effectively exploited for TS fuzzy systems [START_REF] Zhang | Dissipativity analysis and synthesis for a class of T-S 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]- [START_REF] Xie | Membership functiondependent local controller design for T-S fuzzy systems[END_REF]. In contrast to quadratic stability, exploiting the information on the membership functions and their timederivatives plays a key role for fuzzy Lyapunov-based stability of TS fuzzy systems [START_REF] Zhang | Dissipativity analysis and synthesis for a class of T-S fuzzy descriptor systems[END_REF], [START_REF] Xie | Membership functiondependent local controller design for T-S fuzzy systems[END_REF]. However, the information on the time-derivative of premise variables is generally not available for control design, which implies much more numerical and theoretical challenges when using fuzzy Lyapunov functions for TS fuzzy stability analysis and control design. Note that within local TS fuzzy control framework, it is still not possible to theoretically demonstrate that nonquadratic results include those derived from quadratic Lyapunov functions [START_REF] Nguyen | Fuzzy control systems: Past, present and future[END_REF].

Concerning descriptor nonlinear systems, most of TS fuzzy based approaches have been devoted to the case with regular derivative matrices, see for instance [START_REF] Bouarar | Robust fuzzy Lyapunov stabilization for uncertain and disturbed Takagi-Sugeno descriptors[END_REF]- [START_REF] Guerra | Observer design for Takagi-Sugeno descriptor models: An LMI approach[END_REF] and related references, for which the regularity and admissibility analysis is not required. However, such a class of descriptor systems cannot take into account the physical differential-algebraic equations into the model description. TS fuzzy model-based admissibility analysis conditions, in the form of LMI constraints, have been proposed in [START_REF] Chadli | On stability and stabilization of singular uncertain Takagi-Sugeno fuzzy systems[END_REF]- [START_REF] Chen | New admissibility and admissibilization criteria for nonlinear discrete-time singular systems by switched fuzzy models[END_REF]. However, these results can only be applied to singular TS fuzzy systems with a constant derivative matrix, which may be restrictive for many engineering problems. To avoid this drawback, LMI-based design conditions have been developed for singular TS fuzzy systems with a nonlinear derivative matrix, which is more challenging [START_REF] Zhang | Dissipativity analysis and synthesis for a class of T-S fuzzy descriptor systems[END_REF], [START_REF] He | Admissibility analysis and robust H∞ control for T-S fuzzy descriptor systems with structured parametric uncertainties[END_REF], [START_REF] Huang | Admissibility and design issues for T-S fuzzy descriptor systems with perturbed derivative matrices in the rules[END_REF]. For existing TS fuzzy approaches, the numerical complexity of stability analysis, observation and control design conditions exponentially grows with respect to the number of premise variables [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. This limits the applicability of these results to systems with only few nonlinearities. To overcome this major limitation, several approaches have been proposed to reduce the numerical complexity of TS fuzzy systems. A singular value decomposition was proposed in [START_REF] Yam | Reduction of fuzzy rule base via singular value decomposition[END_REF] to reduce the fuzzy rules, leading to approximate TS fuzzy models. The authors in [START_REF] Taniguchi | Model construction, rule reduction, and robust compensation for generalized form of Takagi-Sugeno fuzzy systems[END_REF] proposed to transform some nonlinearities into system uncertainties to reduce the number of vertices of TS fuzzy models. Based on the linear dependencies between the TS fuzzy local sub-models obtained with the sector nonlinearity approach, a reduced-complexity model can be directly derived from the initial TS fuzzy model in [START_REF] Dehak | Reduced-complexity affine representation for Takagi-Sugeno fuzzy systems[END_REF]. However, the reduction methods in [START_REF] Taniguchi | Model construction, rule reduction, and robust compensation for generalized form of Takagi-Sugeno fuzzy systems[END_REF], [START_REF] Dehak | Reduced-complexity affine representation for Takagi-Sugeno fuzzy systems[END_REF] can lead to over-conservative design results. Data-based methods have been proposed for dimensionality reduction of the premise variables, e.g., principal component analysis [START_REF] Kwiatkowski | PCA-based parameter set mappings for LPV models with fewer parameters and less overbounding[END_REF], deep neural network [START_REF] Koelewijn | Scheduling dimension reduction of LPV models -A deep neural network approach[END_REF]. However, these modeling methods suffer some major drawbacks. First, we can only deal with nonlinear systems in the sense of approximation, which could be unsuitable for fast dynamical systems. Second, these methods fundamentally rely on experimental data obtained from typical trajectories of the premise variables, which require not only additional optimization steps but also extensive simulations to collect data. Third, due to the "data-based" feature, the stability analysis and control performance obtained with the resulting TS fuzzy models highly depend on the collected data.

Motivated by the above technical and practical issues, this paper aims at finding an effective solution to reduce the numerical complexity of LMI-based stability analysis and design conditions with respect to classical TS fuzzy approaches. A special focus is paid on a class of descriptor systems with a nonlinear derivative matrix. We propose a new approach to derive an equivalent polytopic representation for a given nonlinear system within a compact set. Although all powerful tools of TS fuzzy framework can be directly applied to the proposed approach, the model complexity only grows proportionally with the number of premise variables, rather than exponentially when compared to the conventional TS fuzzy modeling. Moreover, for the same predefined set of premise variables, the vertices of the proposed polytopic models can admit an infinite number of representations. This non-uniqueness feature allows introducing specific slack variables at the modeling step, which are useful to reduce the control design conservatism. Using the proposed modeling and the descriptor redundancy approach [START_REF] Bouarar | Robust fuzzy Lyapunov stabilization for uncertain and disturbed Takagi-Sugeno descriptors[END_REF], [START_REF] Tanaka | A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions[END_REF], reducedcomplexity admissibility analysis and control design condi-tions for singular nonlinear systems are derived in terms of LMIs. Then, we propose a new nonlinear control law for the nonsingular case to avoid using the extended redundancy form, which is unnecessarily complex from a numerical viewpoint and may yield conservative results due to the imposed special control structure as similarly discussed in [START_REF] Nguyen | Takagi-Sugeno fuzzy observer design for nonlinear descriptor systems with unmeasured premise variables and unknown inputs[END_REF], [START_REF] Guerra | Observer design for Takagi-Sugeno descriptor models: An LMI approach[END_REF] for fuzzy descriptor observer design. The main contributions can be summarized as follows.

• A reduced-complexity polytopic representation of nonlinear systems, which offers a possibility to introduce specific slack variables for control design relaxation. • LMI-based admissibility analysis and control design conditions for descriptor nonlinear systems with less complex and less conservative results compared to related TS fuzzy model-based results.

This paper is organized as follows. Section II formalizes the control problem of singular nonlinear systems. Section III presents the LMI-based control design of descriptor nonlinear systems for both singular and regular cases. Numerical and physically motivated examples are given in Section IV to demonstrated the proposed control approach. Section V concludes the paper and discusses future works. Notations. I n denotes the set of natural numbers {1, 2, . . . , n}. For a vector z, we denote by z i its ith entry. R m×n denotes the set of m × n matrices with real elements. I is an identity matrix of appropriate dimensions. For a matrix X, X is the transpose of X, X -1 is its inverse, det(X) is its determinant, and He[X] = X + X . The expression X Y (respectively X ≺ Y ) means that X-Y (respectively Y -X) is a symmetric positive (respectively negative) definite matrix. The symbol stands for matrix blocks that can be deduced by symmetry. Arguments are omitted when their meaning is straightforward.

II. PROBLEM FORMULATION

This section first recalls preliminary results on admissibility analysis and TS fuzzy modeling of singular nonlinear systems. Then, we propose a new polytopic representation to reduce the numerical complexity compared to the conventional TS fuzzy technique. Finally, the control problem is formalized.

A. Preliminaries

We consider the nonlinear differential-algebraic system

E 1 (z) ẋd (t) = A 11 (z)x d (t) + A 12 (z)x a (t) + B 1 (z)u(t),(1a) 0 = A 21 (z)x d (t) + A 22 (z)x a (t) + B 2 (z)u(t),(1b)
where x d (t) ∈ R q is the differential state vector, x a (t) ∈ R s is the vector of algebraic variables, and u(t) ∈ R m is the control input. The vector of premise variables z(t) ∈ R r continuously depends on the system state and

z(t) ∈ D z = {z ∈ R r : z i min ≤ z i ≤ z i max , ∀i ∈ I r } , (2)
where z i min and z i max are given bounds. The system matrices

E 1 (z) ∈ R q×q , A 11 (z) ∈ R q×q , A 12 (z) ∈ R q×s , A 21 (z) ∈ R s×q , A 22 (z) ∈ R s×s , B 1 (z) ∈ R q×m ,
and B 2 (z) ∈ R s×m affinely depend on z(t). Moreover, we consider the case where E 1 (z) is regular for all z ∈ D z . For compactness, system (1) can be rewritten in the singular form

E(z) ẋ(t) = A(z)x(t) + B(z)u(t), (3)
where x = x d x a ∈ R n , with n = q + s, E(z) = diag(E 1 (z), 0) and

A(z) = A 11 (z) A 12 (z) A 21 (z) A 22 (z) , B(z) = B 1 (z) B 2 (z) .
Remark 1. Various engineering systems can be represented in the form (1), e.g., constrained robot systems [START_REF] Krishnan | Tracking in nonlinear differentialalgebraic control systems with applications to constrained robot systems[END_REF], chemical processes [START_REF] Kumar | Control of Nonlinear Differential Algebraic Equation Systems: with Applications to Chemical Processes[END_REF], etc. Moreover, if x = x d and x a ∈ ∅, then system (1a) becomes regular, which has been widely studied in the literature [START_REF] Lewis | Robot Manipulator Control: Theory and Practice[END_REF], [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF], [START_REF] Bouarar | Robust fuzzy Lyapunov stabilization for uncertain and disturbed Takagi-Sugeno descriptors[END_REF], [START_REF] Taniguchi | Model construction, rule reduction, and robust compensation for generalized form of Takagi-Sugeno fuzzy systems[END_REF].

The following definition is important to analyze the regularity and impulse-free proprieties of system (3).

Definition 1 ([45]

). The unforced singular system (3), i.e., u = 0, is of index-one if there exists a unique solution x a = f (x d ) in some neighborhood of the equilibrium x(0) = 0 satisfying f (0) = 0 and

A 21 (z)x d (t) + A 22 (z)f (x d (t)) = 0. (4)
We consider the following assumption for system (1).

Assumption 1. The premise variables only depend on x d , i.e., z(x) = g(x d ), where g : R q → R r is a differentiable function with respect to x d .

Under Assumption 1, the unforced system (3) is of indexone if det(A 22 (z)) = 0, ∀z(∈ D z . Moreover, in this case the unique solution of equation ( 4) is given by

x a = f (x d ) = -A 22 (z) -1 A 21 (z)x d . (5)
Note from (5) that if x d → 0, then x a → 0. The following definition is useful for the admissibility analysis of system (1).

Definition 2. The unforced singular system (3), i.e., u = 0, is admissible if it is of index-one and its solution x = 0 is asymptotically stable.

The nonlinear singular system (3) can be described by the following TS fuzzy model with IF-THEN fuzzy rules:

RULE R i : IF z 1 (t) is M i 1 and . . . and z r (t) is M i r , THEN Ẽi ẋ(t) = Ãi x(t) + Bi u(t), (6)
where ( Ẽi , Ãi , Bi ) are known constant matrices with appropriate dimensions, R i denotes the ith fuzzy inference rule, and M i j is the fuzzy set, for i ∈ I 2 r and j ∈ I r . The fuzzy membership functions are defined as

h i (z) = r j=1 µ i j (zj ) 2 r i=1 r j=1 µ i j (zj ) , ∀i ∈ I 2 r
, where µ i j (z j ) represents the membership grade of z j in the respective fuzzy set M i j . Using the fuzzy method in [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF], system (6) can be inferred as

2 r i=1 h i (z) Ẽi ẋ(t) = 2 r i=1 h i (z)( Ãi x(t) + Bi u(t)). (7)
The membership functions h i (z), for i ∈ I 2 r , satisfy the property

2 r i=1 h i (z) = 1 and 0 ≤ h i (z) ≤ 1, ∀i ∈ I 2 r .
TS fuzzy model-based technique has been shown as one of the most promising approaches for LMI-based admissibility analysis and control design of the singular system [START_REF] Kumar | Control of Nonlinear Differential Algebraic Equation Systems: with Applications to Chemical Processes[END_REF], see e.g., [START_REF] He | Admissibility analysis and robust H∞ control for T-S fuzzy descriptor systems with structured parametric uncertainties[END_REF], [START_REF] Qiao | Admissibility analysis and control synthesis for T-S fuzzy descriptor systems[END_REF], [START_REF] Li | Robust stabilization of T-S fuzzy stochastic descriptor systems via integral sliding modes[END_REF], [START_REF] Chen | New admissibility and admissibilization criteria for nonlinear discrete-time singular systems by switched fuzzy models[END_REF], [START_REF] Huang | Admissibility and design issues for T-S fuzzy descriptor systems with perturbed derivative matrices in the rules[END_REF], [START_REF] Yang | Lyapunov stability and strong passivity analysis for nonlinear descriptor systems[END_REF]. However, the numerical complexity of model [START_REF] Di Francor | Stability of nonlinear differential-algebraic systems via additive identity[END_REF], specifically the number of vertices 2 r and by conjunction the number of related LMI conditions, exponentially grows with the number of premise variables r. This may prevent the use of the TS fuzzy model [START_REF] Di Francor | Stability of nonlinear differential-algebraic systems via additive identity[END_REF] to deal with complex engineering systems, i.e., r 1. To avoid this major drawback, we propose hereafter a reduced-complexity polytopic representation of the nonlinear singular system [START_REF] Kumar | Control of Nonlinear Differential Algebraic Equation Systems: with Applications to Chemical Processes[END_REF], where the number of vertices only grows proportionally with r rather than exponentially as for TS fuzzy model [START_REF] Di Francor | Stability of nonlinear differential-algebraic systems via additive identity[END_REF].

B. Reduced-Complexity Polytopic Representation

Since the matrices E(z), A(z) and B(z) of the nonlinear system (3) affinely depend on z, then they can be represented in the form

X(z) = X0 + r p=1 z p Xp , (8)
where X ∈ {E, A, B} and Xi , for i ∈ I r ∪ {0}, are constant matrices of appropriate dimensions, which can be directly obtained from the affine structure of X(z). Moreover, for z ∈ D z , the term z p can be convexly rewritten as

z p = ω 0 p (z)z p min + ω 1 p (z)z p max , (9)
with ω 0 p (z) = zp max-zp zp max-zp min and ω 1 p (z) = 1 -ω 0 p (z), ∀p ∈ I r . Then, it follows from (9) that expression (8) can be equivalently rewritten as

X(z) = X0 + r p=1 ω 1 p (z)z p max + ω 0 p (z)z p min Xp (10) = r p=1 (w 2p-1 (z)X 2p-1 + w 2p (z)X 2p ) = 2r i=1 w i (z)X i , with w 2p-1 (z) = 1 r ω 1 p (z), X 2p-1 = X0 + rz p max Xp , w 2p (z) = 1 r ω 0 p (z), X 2p = X0 + rz p min Xp . (11)
Using the matrix decomposition [START_REF] Wang | Fuzzy-modelbased sliding mode control of nonlinear descriptor systems[END_REF], the nonlinear singular system (3) can be rewritten in the polytopic form

2r i=1 w i (z)E i ẋ(t) = 2r i=1 w i (z)(A i x(t) + B i u(t)), (12)
where the nonlinear functions w i (z), ∀i ∈ I 2r , satisfy the following properties:

2r i=1 w i (z) = 1, 0 ≤ w i (z) ≤ 1, w 2p-1 (z) + w 2p (z) = 1 r , ∀p ∈ I r . (13)
Let Ω be the set of nonlinear functions w i (z), ∀i ∈ I 2r , satisfying (13), i.e., w(z

) = w 1 (z), w 2 (z), . . . , w 2r (z) ∈ Ω.
The vertices X i can be directly obtained from X(z), for X ∈ {E, A, B} and i ∈ I 2r . To this end, we define the vectors ζ 1 , . . . , ζ r forming the canonical basis of R r as

ζ i = [0, . . . , 0, ith 1 , 0, . . . , 0] , i ∈ I r .
Then, it follows from the matrix decomposition ( 10)-( 11) that

X i = X(r • z p max • ζ p ) if i = 2p -1, X(r • z p min • ζ p ) if i = 2p, for i ∈ I 2r . (14)
It is important to note that the polytopic model ( 12) is an algebraic rewriting of the nonlinear model (3), i.e., there is no approximation error between these two models. For model ( 12), the nonlinear functions w i (z), ∀i ∈ I 2r , precisely defined in [START_REF] He | Admissibility analysis and robust H∞ control for T-S fuzzy descriptor systems with structured parametric uncertainties[END_REF], capture the nonlinearities in model (3).

Remark 2. Note that the nonlinear functions w j (z), j ∈ I 2r , in [START_REF] He | Admissibility analysis and robust H∞ control for T-S fuzzy descriptor systems with structured parametric uncertainties[END_REF], of the new polytopic model ( 12) only depend on the jth component z j of z. Indeed, their construction is fundamentally different compared to that of the membership functions h i (z), for i ∈ I 2 r , of the TS fuzzy model ( 7), for which a combination product is required [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF], i.e.,

h i (z) = r p=1 ω ip p (z p ), i ∈ I 2 r , i p ∈ {0, 1},
with ω ip p (z p ) defined in [START_REF] Zhang | Dissipativity analysis and synthesis for a class of T-S fuzzy descriptor systems[END_REF]. Hence, model ( 12) only has 2r vertices, which can significantly reduce the numerical complexity of LMI-based design conditions of system (3) compared to the conventional TS fuzzy model [START_REF] Di Francor | Stability of nonlinear differential-algebraic systems via additive identity[END_REF] with 2 r vertices. This feature is particularly interesting when dealing with nonlinear complex systems with a large number r of premise variables. However, as shown in [START_REF] Shen | Nonfragile H∞ control for fuzzy Markovian jump systems under fast sampling singular perturbation[END_REF], the bounds z p min and z p max , p ∈ I r , are multiplied by r when constructing the vertices of model [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. This may lead to a modeling overbounding [START_REF] Kwiatkowski | PCA-based parameter set mappings for LPV models with fewer parameters and less overbounding[END_REF], thus the design conservatism. Remark 3. Let {T p } p∈I2r be a family of matrices with appropriate dimensions such that T 2p = T 2p-1 , for p ∈ I r , and T = 1 r r p=1 T 2p . We define

X * i = X i + T i -T, X ∈ {E, A, B}, i ∈ I 2r . (15)
Then, for X ∈ {E, A, B}, it follows from ( 13) and [START_REF] Nguyen | Fuzzy control systems: Past, present and future[END_REF]

that 2r i=1 w i (z)X * i = 2r i=1 w i (z)(X i + T i -T ) = 2r i=1 w i (z)X i + 2r i=1 w i (z)T i -T = X(z) + r p=1 (w 2p-1 (z)T 2p-1 + w 2p (z)T 2p ) -T = X(z) + 1 r r p=1 T 2p -T = X(z). (16)
Note from ( 16) that X(z) can admit an infinite number of representations of the form [START_REF] Wang | Fuzzy-modelbased sliding mode control of nonlinear descriptor systems[END_REF] with the same predefined set of nonlinear functions w(z) ∈ Ω. This non-uniqueness feature of the vertices of model ( 12) offers a flexibility to introduce r slack variables, i.e., similar to matrices T p in [START_REF] Nguyen | Fuzzy control systems: Past, present and future[END_REF], into LMIbased design conditions to further reduce the conservatism that may be induced by the overbounding discussed in Remark 2. This will be numerically illustrated in Section IV.

This paper investigates the following control problem.

Problem 1. Consider the singular nonlinear system (3) under Assumption 1. Determine a control law u(t) such that the closed-loop system (3) is admissible and the corresponding trajectory x(t) exponentially converges towards the origin with a decay rate less than a predefined positive scalar α. Moreover, for any admissible initial state x d (0), the corresponding closed-loop trajectory x d (t) of system ( 3) is required to remain inside the validity domain D x d ⊆ R q , described by

D x d = x d ∈ R q : H (l) x d ≤ 1, l ∈ I ns , (17)
where H (l) denotes the lth row of the matrix H ∈ R ns×q , which characterizes the domain D x d .

Remark 4. The system states are generally bounded in engineering applications due to physical and/or safety reasons [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF], [START_REF] Nguyen | Constrained output-feedback control for discrete-time fuzzy systems with local nonlinear models subject to state and input constraints[END_REF]. Then, the characterizing matrix H can be directly derived from the ultimate bounds x di min and x di max of the state x d , i.e., x di min ≤ x di ≤ x di max , for i ∈ I q , as illustrated in Example 3 in Section IV. Note that due to Assumption 1 and the admissibility property, any limitation bounds on the algebraic variables x a of system (3) can be also transformed into those on the differential state x d . Moreover, since z(x) = g(x d ), from [START_REF] Li | Adaptive sliding mode control for Takagi-Sugeno fuzzy systems and its applications[END_REF] we can directly obtain the ultimate bounds z i min and z i max , for i ∈ I r , defining the admissible set D z in (2) of the premise variables. Hence, if x d (t) ∈ D x d , then the bounds of the premise variables in (2) are always valid for the proposed polytopic representation.

The following lemma is useful for the control design.

Lemma 1 ([47]

). Let Υ ij be symmetric matrices of appropriate dimensions where i, j ∈ I r . Then, the inequality

r i=1 r j=1 w i (z)w j (z)Υ ij ≺ 0, (18)
holds if

Υ ii ≺ 0, i ∈ I r 2 r -1 Υ ii + Υ ij + Υ ji ≺ 0, i, j ∈ I r , i = j.
Note that other relaxation results to convert (18) into a finite set of LMI constraints with different degrees of complexity and/or conservatism can be found in [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF], [START_REF] Sala | Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya's theorem[END_REF].

III. LMI-BASED CONTROL DESIGN FOR NONLINEAR DESCRIPTOR SYSTEMS

This section first presents the control design for system (1) using the proposed model [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. Then, we pay a special attention to the control design of regular systems, i.e., there is no algebraic equation (1b), which has been widely treated in TS fuzzy control framework.

A. Descriptor-Redundancy Based Control Design

We consider the nonlinear singular system (3), which can be rewritten in the following extended form:

Ē ẋe (t) = Ā(z)x e (t) + B(z)u(t), (19)
with

x e =   x d ẋd x a   , Ā(z) =   0 I 0 A 11 (z) -E 1 (z) A 12 (z) A 21 (z) 0 A 22 (z)   , B(z) =   0 B 1 (z) B 2 (z)   , Ē =   I 0 0 0 0 0 0 0 0   .
Applying the reduced-complexity modeling in [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF], system [START_REF] Precup | A survey on industrial applications of fuzzy control[END_REF] can be equivalently represented as

Ē ẋe (t) = 2r i=1 w i (z)( Āi x e (t) + Bi u(t)). (20)
For the admissibilization of system (20), we consider the following control law:

u(t) = 2r i=1 w i (z) Ki x e (t) . = K(z)x e (t), (21)
where the feedback gains Ki = K i 0 , with K i ∈ R m×q , i ∈ I 2r , are to be determined such that the closed-loop system (3) is admissible. The following theorem provides LMI conditions to design controller [START_REF] Mozelli | A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems[END_REF].

Theorem 1. Consider system (3) with Assumption 1. If there exist a positive definite matrix V ∈ R q×q , symmetric matrices

T i , S j ∈ R 2(n+q)×2(n+q) , matrices L j , R j ∈ R (n+q)×(n+q) , F j ∈ R m×q , W ij ∈ R n×q , Z ij ∈ R n×n
, for i, j ∈ I 2r , and a positive scalar α, such that

Ξ ii ≺ 0, i ∈ I 2r , ( 22
) 2 2r -1 Ξ ii + Ξ ij + Ξ ji ≺ 0, i, j ∈ I 2r , i = j, (23)
V H (l) V 1 0, l ∈ I ns , (24)
with

T 2p = T 2p-1 , S 2p = S 2p-1 , ∀p ∈ I r , and
Ξ ij = Ψ ij + T i + S j - 1 r (T + S), S = r p=1 S 2p , T = r p=1 T 2p , Ψ ij = He ( Āi + α Ē)L j + Bi Fj Āi R j P ij -L j -R j , Fj = F j 0 , P ij = V 0 W ij Z ij .
Then, the closed-loop system (20) is admissible with a decay rate less than α, and for any x d (0) belonging to the set E V , specified in [START_REF] Huang | Admissibility and design issues for T-S fuzzy descriptor systems with perturbed derivative matrices in the rules[END_REF], the corresponding trajectory x d (t) remains inside D x d . Moreover, the feedback gains of the control law [START_REF] Mozelli | A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems[END_REF] are given by

K i = F i V -1 , i ∈ I 2r . ( 25
)
Proof. For brevity, we denote

Θ(z) = 2r j=1 w j (z)Θ j , Θ ∈ {L, R, F }, Λ(z) = 2r i=1 2r j=1 w i (z)w j (z)Λ ij , Λ ∈ {Ξ, Ψ} , P(z) = V 0 W (z) Z(z) = 2r i=1 2r j=1 w i (z)w j (z)P ij .
It follows from the definition of Ψ ij that

Ψ(z) = He ( Ā(z) + α Ē)L(z) + B(z) F (z) Ā(z)R(z) P(z) -L(z) -R(z) ,
with F (z) = F (z) 0 . Moreover, it follows from (13) that

2r i=1 2r j=1 w i (z)w j (z)(T i + S j ) = 2r i=1 w i (z)T i + 2r j=1 w j (z)S j = r k=1 (w 2k-1 (z)T 2k + w 2k (z)T 2k ) + r =1 (w 2 -1 (z)S 2 + w 2 (z)S 2 ) (26) = 1 r r p=1 (T 2p + S 2p ) = 1 r (T + S).
Then, it follows from the definition of Ψ(z) and relation [START_REF] Nguyen | Takagi-Sugeno fuzzy observer design for nonlinear descriptor systems with unmeasured premise variables and unknown inputs[END_REF] that Ξ(z) = Ψ(z). Applying Lemma 1, LMI conditions ( 22) and [START_REF] Zhao | Novel stability criteria for TS systems[END_REF] imply that Ξ(z) ≺ 0, thus

He ( Ā(z) + α Ē)L(z) + B(z) F (z) Ā(z)R(z) P(z) -L(z) -R(z) ≺ 0.( 27
)
Pre-and post-multiplying [START_REF] Tanaka | A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions[END_REF] with I ( Ā(z) + α Ē) and its transpose, it follows that

∆(z) = He ( Ā(z) + α Ē)P(z) + B(z) F (z) ≺ 0. ( 28
)
Note that

∆(z) = • • • -He[ Ā22 (z)Z(z)] , (29)
where "•" denotes the matrix terms irrelevant to the theoretical developments, and

Ā22 (z) = -E 1 (z) A 12 (z) 0 A 22 (z) ∈ R n×n .
It follows from ( 28) and (29) that

Ā22 (z)Z(z) + Z(z) Ā22 (z) 0,
which ensures that det( Ā22 (z)Z(z)) = 0, thus det(Z(z)) = 0, ∀z ∈ D z . Combining with the fact that V 0, it follows that P(z) is invertible on D z with

P(z) -1 = V -1 0 -Z -1 (z)W (z)V -1 Z -1 (z) . ( 30
)
Note from (30) that

Ē P -1 (z) = P -1 (z) Ē = diag(V -1 , 0) 0. (31)
From ( 25) and [START_REF] Chen | Regularity and controllability robustness of TS fuzzy descriptor systems with structured parametric uncertainties[END_REF], it follows that

K(z) = F (z)P(z) -1 . (32)
Pre-and post-multiplying [START_REF] Guerra | Observer design for Takagi-Sugeno descriptor models: An LMI approach[END_REF] with P(z) -and P(z) -1 while considering expression [START_REF] Qiao | Admissibility analysis and control synthesis for T-S fuzzy descriptor systems[END_REF], it follows that

He P(z) -( Ā(z) + B(z) K(z)) + 2α Ē P(z) -1 ≺ 0.( 33
)
We consider the Lyapunov function candidate

V(z, x e ) = x e Ē P(z) -1 x e . (34)
It follows from ( 31) and ( 34) that V(z, x e ) = x d V -1 x d > 0, ∀x e = 0. The time derivative of V(z, x e ) in [START_REF] Li | Robust stabilization of T-S fuzzy stochastic descriptor systems via integral sliding modes[END_REF] along the solution of system ( 19) is given by

V(z, x e ) = He x e P(z) -( Ā(z) + B(z) K(z))x e . ( 35
)
It is clear from [START_REF] Feng | Admissibilization of singular interval-valued fuzzy systems[END_REF] and ( 35) that V(z, x e ) + 2αV(z, x e ) < 0, ∀x e = 0. Hence, the solution x e = 0 of system ( 19) is asymptotically stable with a decay rate less than α. Moreover, since E(z) is regular, the expression of V(z, x e ) can be also computed using (1) as

V(z, x e ) = He ẋ d V -1 x d = He ẋ d E 1 (z) E 1 (z) -V -1 x d = He (A c11 (z)x d + A 12 (z)x a ) Y (z) -1 x d , (36)
with 36) can be rewritten as follows:

A c11 (z) = A 11 (z) + B 1 (z)K(z) and Y (z) = V E 1 (z) . Note that A c21 (z)x d + A 22 (z)x a = 0, with A c21 (z) = A 21 (z) + B 2 (z)K(z). Then, V(z, x e ) in (
V(z, x e ) = He (A c11 (z)x d + A 12 (z)x a ) Y (z) -1 x d + He (A c21 (z)x d + A 22 (z)x a ) Y (z) -1 x a = x d x a • • • He A 22 (z) Y (z) -1 x d x a . (37)
Since V(z, x e ) < 0, ∀x = 0, it follows from (37) that He A 22 (z) Y (z) -1 ≺ 0, or det(A 22 (z)Y (z) -1 ) = 0. This latter ensures that det(A 22 (z)) = 0. Hence, the nonlinear singular system (3) under Assumption 1 is of index-one.

Following the same line as in [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF], [START_REF] Nguyen | Constrained output-feedback control for discrete-time fuzzy systems with local nonlinear models subject to state and input constraints[END_REF], we can prove that condition [START_REF] Xie | Membership functiondependent local controller design for T-S fuzzy systems[END_REF] guarantees the inclusion E V ⊆ D x d , with

E V = {x d ∈ R q : x d V -1 x d ≤ 1}. ( 38
)
Since V(z, x e ) < 0, ∀x = 0, then by the set invariance property [START_REF] Khalil | Nonlinear Systems[END_REF], we deduce that ∀x d (0) ∈ E V , the corresponding trajectory x d (t) converges to the origin while remaining inside this set, thus x d (t) ∈ D x d . This concludes the proof.

The following corollary provides LMI-based design conditions without using slack variables T i and S i , ∀i ∈ I 2r , specifically offered by the proposed polytopic model [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF].

Corollary 1. Consider system (3) with Assumption 1. If there exist a positive definite matrix V ∈ R q×q , matrices

L j , R j ∈ R (n+q)×(n+q) , F j ∈ R m×q , W ij ∈ R (n+q)×q , Z ij ∈ R (n+q)×(n+q)
, for i, j ∈ I 2r , and a positive scalar α, such that condition [START_REF] Xie | Membership functiondependent local controller design for T-S fuzzy systems[END_REF] is verified, and

Φ ii ≺ 0, i ∈ I 2r , (39) 2 2r -1 Φ ii + Φ ij + Φ ji ≺ 0, i, j ∈ I 2r , i = j, (40)
with

Φ ij = He ( Āi + α Ē)L j + Bi Fj Āi R j P ij -L j -R j .
Then, the closed-loop system ( 20) is admissible with a decay rate less than α, and for any x d (0) ∈ E V , the corresponding trajectory x d (t) remains inside the set D x d . Moreover, the feedback gains of the control law [START_REF] Mozelli | A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems[END_REF] are given in [START_REF] Bouarar | Robust fuzzy Lyapunov stabilization for uncertain and disturbed Takagi-Sugeno descriptors[END_REF].

Proof. In Theorem 1, setting T i = S j = 0, ∀i, j ∈ I 2r , it follows that Φ ij = Ξ ij . Then, conditions ( 22) and ( 23) become ( 39) and ( 40), respectively. The proof is concluded following the result of Theorem 1.

Remark 5. Following the same lines of fuzzy Lyapunov-based control approaches for TS fuzzy systems, see [START_REF] Zhang | Dissipativity analysis and synthesis for a class of T-S 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]- [START_REF] Xie | Membership functiondependent local controller design for T-S fuzzy systems[END_REF] and related references, the Lyapunov function candidate

V (z, x e ) = x e Ē P(z) -1 x e , (41)
with

P(z) = V (z) 0 W (z) Z(z) = 2r i=1 2r j=1 w i (z)w j (z) V ij 0 W ij Z ij ,
can be directly applied to reduce the design conservatism. However, such control results require a fundamental assumption that

x ∈ R = {x ∈ R n : | ẇi (z)| ≤ φ i , ∀i ∈ I 2r } ,
for some predefined positive scalars φ i . Note that such an assumption can be only verified a posteriori by extensive simulations with the designed controllers. Hence, as for conventional TS fuzzy control approaches [START_REF] Nguyen | Fuzzy control systems: Past, present and future[END_REF], it is still not possible to theoretically demonstrate that control results based on V (z, x e ) in ( 41) include those derived from the quadratic Lyapunov function V(z, x e ) in [START_REF] Li | Robust stabilization of T-S fuzzy stochastic descriptor systems via integral sliding modes[END_REF]. By numerical experiments in Section IV, we show that in many cases, the proposed quadratic control approach with specific slack variables in Theorem 1, i.e., S i and T i , for i ∈ I 2r , offered by the new representation ( 12) can provide a better performance, in terms of conservatism reduction, compared to many existing fuzzy Lyapunov based control results in TS fuzzy control framework.

B. Control Design without Descriptor-Redundancy Approach

In the following, a special attention is paid to the control design of regular systems, for which the use of the well-known descriptor-redundancy approach [START_REF] Tanaka | A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions[END_REF], i.e., the extended form [START_REF] Precup | A survey on industrial applications of fuzzy control[END_REF], is still possible but can be unnecessarily complex and conservative from a numerical viewpoint. To this end, let us consider the following descriptor nonlinear system:

E(z) ẋ(t) = A(z)x(t) + B(z)u(t), (42)
where x ∈ R n , u ∈ R m , the matrix E(z) is non-singular, i.e., rank(E(z)) = n, and

E(z) A(z) B(z) = 2r i=1 w i (z) E i A i B i .
In this case, as in [START_REF] Li | Adaptive sliding mode control for Takagi-Sugeno fuzzy systems and its applications[END_REF] we consider the validity domain

D x = x ∈ R n : H (l) x ≤ 1, l ∈ I ns , (43)
where the matrix H ∈ R ns×n characterizes the domain D x .

Note that system (42) is admissible as long as it admits an asymptotically stable solution x = 0. We consider the nonlinear control law

u(t) = K(z)x(t). (44)
The following theorem provides LMI conditions to determine a feedback gain K(z) ∈ R m×n such that the origin of system ( 42) is asymptotically stable, and its trajectory x(t) remains inside D x defined in ( 43) under an admissible initial condition.

Theorem 2. If there exist a positive definite matrix P ∈ R n×n , symmetric matrices T i , S j ∈ R 2n×2n , matrices F i ∈ R m×n , L i , R j ∈ R n×n , for i, j ∈ I 2r , and a positive scalar α, such that

Γ ii ≺ 0, i ∈ I 2r , (45) 2 2r
-1 Γ ii + Γ ij + Γ ji ≺ 0, i, j ∈ I 2r , i = j, (46)
P H (l) P 1 0, l ∈ I n , (47)
with T 2p = T 2p-1 , S 2p = S 2p-1 , ∀p ∈ I r , and

Γ ij = Σ ij + T i + S j - 1 r (T + S), S = r p=1 S 2p , T = r p=1 T 2p , Σ ij = He (A i + αE i )L j + B i F j A i R j P E i -L j -R j .
Then, the solution x = 0 of the non-singular system (42) is asymptotically stable with a decay rate less than α, and ∀x(0) belonging to the set E P , specified in (51), the corresponding trajectory x(t) remains inside D x . Moreover, the feedback gain of the control law ( 44) is given by

K(z) = F (z)E(z) -P -1 , F (z) = 2r j=1 w j (z)F j . ( 48
)
Proof. Following the same steps in the proof of Theorem 1, we can prove that conditions (45)-( 46) guarantee that

He (A(z) + αE(z))P E(z) + B(z)F (z) ≺ 0. ( 49
)
Since P 0 and E(z) is non-singular, pre-and postmultiplying [START_REF] Löfberg | YALMIP: A toolbox for modeling and optimization in MATLAB[END_REF] with P -1 E(z) -1 and E(z) -P -1 , it follows that

Π = He P -1 E(z) -1 (A(z) + αE(z) + B(z)K(z)) ≺ 0 (50)
Condition [START_REF] Sakthivel | Fault estimation for mode-dependent IT2 fuzzy systems with quantized output signals[END_REF] ensures that V (z, x) + 2αV (z, x) = x Πx < 0, ∀x = 0, where V (x, z) is the time derivative of the Lyapunov function candidate V (z, x) = x P -1 x along the solution of system [START_REF] Kwiatkowski | PCA-based parameter set mappings for LPV models with fewer parameters and less overbounding[END_REF].

As in the proof of Theorem 1, we can prove that condition (47) guarantees the inclusion E P ⊆ D x , with

E P = {x ∈ R n : x P -1 x ≤ 1}. ( 51
)
Since V (z, x) < 0, ∀x = 0, using again the set invariance property [START_REF] Khalil | Nonlinear Systems[END_REF], we deduce that ∀x(0) ∈ E P , the corresponding trajectory x(t) converges to the origin while remaining inside this set, thus x(t) ∈ D x . This concludes the proof.

Remark 6. The new nonlinear control law [START_REF] Krishnan | Tracking in nonlinear differentialalgebraic control systems with applications to constrained robot systems[END_REF] with the feedback gain ( 48) exploits the invertibility of E(z) to avoid using the extended form as in [START_REF] Precup | A survey on industrial applications of fuzzy control[END_REF]. Then, no specific matrix structure is required for the feedback gain as K(z) in ( 21) or the Lyapunov matrix as Ē P -1 (z) in [START_REF] Han | Nonfragile control with guaranteed cost of TS fuzzy singular systems based on parallel distributed compensation[END_REF]. Compared to the conventional TS fuzzy descriptor approaches, see for instance [START_REF] Zhang | Dissipativity analysis and synthesis for a class of T-S fuzzy descriptor systems[END_REF], [START_REF] He | Admissibility analysis and robust H∞ control for T-S fuzzy descriptor systems with structured parametric uncertainties[END_REF], [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF], [START_REF] Bouarar | Robust fuzzy Lyapunov stabilization for uncertain and disturbed Takagi-Sugeno descriptors[END_REF], these features can further reduce not only the numerical complexity but also the conservatism of the control results for complex non-singular nonlinear systems as illustrated in Example 3.

Remark 7. The computational complexity of LMI-based optimization problems can be evaluated with the number of scalar decision variables N var and the number of LMI constraints N row . To illustrate the complexity reduction of the proposed approach, Table I shows these characteristics numbers corresponding to the LMI constraints of different control results for non-singular descriptor systems with n = q states, m control inputs and r = r e + r a premise variables, where r e and r a respectively represent the number of the premise variables in the matrix E(z) and in both matrices A(z) and B(z). For illustrations, the evolution of N var and N row with respect to r for the case of stability analysis with n = 5 and m = 0 is depicted in Fig. 1. Note that for a relatively low number of premise variables, i.e., r ≤ 6, all considered methods have a similar level of numerical complexity. We can also observe an exponential growth of N var and N row with respect to r for TS fuzzy model-based results, which is not the case of the proposed approach. Further numerical studies are performed in the next section to demonstrate the interests of the new method in reducing both computational complexity and design conservatism compared to existing TS fuzzy model-based results.

Remark 8. The decay rate α is related to the closed-loop time performance, i.e., a larger value of α leads to a faster convergence time [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF].

Remark 9. When there are no required state constraints, i.e., D x d ≡ R q for system (3) or D x ≡ R n for system [START_REF] Kwiatkowski | PCA-based parameter set mappings for LPV models with fewer parameters and less overbounding[END_REF], the control design for these systems can be performed by simply removing condition [START_REF] Xie | Membership functiondependent local controller design for T-S fuzzy systems[END_REF] in Theorem 1 and Corollary 1, or condition [START_REF] Tuan | Parameterized linear matrix inequality techniques in fuzzy control system design[END_REF] in Theorem 2, respectively. Moreover, it is possible to maximize the size of the invariant sets E V and E P using convex optimization techniques. This issue, omitted here, has been well addressed in [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] and related references.

n(2 r+ra+2 + 2 r + 2 r+1 ) n(2 r+ra+1 + 2 r + 2 r+1 ) + m2 r+ra+1 Nvar n 2 (8r 2 + 32r + 1 2 ) n 2 (12r + 1 2 ) n 2 ( 5 2 2 r + 2 re-1 + 2 ra-1 + 6)+ n 2 (1 + 2 2ra+1 + 2 r-1 )+ +n( 1 2 + 4r) + 2rmn + n 2 + 2rmn n(2 r-1 + 2 re-1 + 2 ra-1 ) + nm2 r n(1 + 2 r-1 + 3m2 r + m2 2ra ) + m 2 2 r

IV. NUMERICAL EXAMPLES

This section presents three examples with different degrees of numerical complexity to illustrate the effectiveness of the proposed approach. All the involved LMI-based conditions are solved using YALMIP toolbox with SeDuMi solver [START_REF] Löfberg | YALMIP: A toolbox for modeling and optimization in MATLAB[END_REF]. Note that without an explicit statement, all the design conditions are solved with the decay rate α = 0 for fair comparisons with related existing results. Moreover, for Examples 1 and 2 where D x d ≡ R q , the LMI condition [START_REF] Xie | Membership functiondependent local controller design for T-S fuzzy systems[END_REF] is not considered in the admissibility analysis and control design, see Remark 9. Example 1. This example is used for two purposes: i) to illustrate the proposed polytopic modeling procedure, ii) to put in evidence the interest of the specific slack variables offered by the proposed modeling method. To this end, we consider a simple unforced singular nonlinear system of the form

1 0 0 0 ẋ1 (t) ẋ2 (t) = -1 z 1 (t) + 5z 2 (t) az 2 (t) b x 1 (t) x 2 (t) , (52)
where a ∈ [-10, 10] and b ∈ [-10, 10] are the parameters. The two premise variables are given by z 1 = cos(x 1 ) and z 2 = sin(x 1 ), i.e., z = z 1 z 2 , which both depend on the differential state x 1 . We consider the validity domain D x d ≡ R. The matrices of system (52) are given by

E = 1 0 0 0 , A(z) = -1 z 1 + 5z 2 az 2 b . (53)
Applying the proposed modeling procedure in [START_REF] Wang | Fuzzy-modelbased sliding mode control of nonlinear descriptor systems[END_REF], ( 11) and ( 14) to matrix A(z) with -1 ≤ z 1 ≤ 1 and -1 ≤ z 2 ≤ 1, ∀x 1 ∈ R, we obtain 2 × r = 4 following vertices:

A 1 = -1 -2 0 b , A 2 = -1 2 0 b , A 3 = -1 -10 -2a b , A 4 = -1 10 2a b . (54)
The nonlinear functions w i (z), for i ∈ I 4 , corresponding to the four above vertices are given by

w 1 (z) = 1 -cos(x 1 ) 4 , w 2 (z) = cos(x 1 ) + 1 4 , w 3 (z) = 1 -sin(x 1 ) 4 , w 4 (z) = sin(x 1 ) + 1 4 . (55)
Let us define x = x 1 x 2 . From (53), ( 54) and (55), the singular nonlinear system (52) can be equivalently rewritten in the polytopic form E ẋ = 4 i=1 w i (z)A i x. We examine the conservatism of the admissibility analysis for system (52) using the following approaches:

• Theorem 1 with u(t) = 0, i.e, F i = 0, for i ∈ I 4 ,

• Corollary 1 with u(t) = 0, i.e, F i = 0, for i ∈ I 4 ,

• Affine TS fuzzy-model-based result in [START_REF] Dehak | Reduced-complexity affine representation for Takagi-Sugeno fuzzy systems[END_REF]Theorem 2] adapted to singular systems, • TS fuzzy-model-based result in [START_REF] He | Admissibility analysis and robust H∞ control for T-S fuzzy descriptor systems with structured parametric uncertainties[END_REF]Theorem 3.3] with u(t) = 0, i.e, N jk = 0, for j, k ∈ I 4 .

Fig. 2 shows the feasibility regions obtained with the four above control results. Observe that using the reducedcomplexity affine representation for TS fuzzy systems, the result in [START_REF] Dehak | Reduced-complexity affine representation for Takagi-Sugeno fuzzy systems[END_REF] leads to a conservative admissibility analysis compared to the proposed modeling method in this paper.

Theorem 1 provides a larger feasibility region compared to that obtained with Corollary 1. This means that using the slack variables, specifically offered by the proposed modeling, can contribute to reduce the conservatism induced by overbounding, see Remarks 2 and 3. For this simple singular system (52), the proposed approach and the recent result in [START_REF] He | Admissibility analysis and robust H∞ control for T-S fuzzy descriptor systems with structured parametric uncertainties[END_REF] lead to the same feasibility region.

- Example 2. This example aims at studying the design conservatism of the proposed approach with respect to recent related TS fuzzy control results. For this purpose, we consider the singular system (3), whose matrices are taken from [START_REF] He | Admissibility analysis and robust H∞ control for T-S fuzzy descriptor systems with structured parametric uncertainties[END_REF] as

E(z) =   1 0 0 z 1 1 0 0 0 0   , B(z) =   0 0.16z 3 + a 0.12z 3 + b   , A(z) = A 0 z 3 +   -z 2 -5 1 z 2 z 2 bz 2 -b 1 (a + 2)z 2 -2 0 1   , A 0 =  
0.05 -0.05 0.15 0.04 -0.04 0.12 0.06 -0.06 0.18

  , (56)
where a ∈ [-13, -8] and b ∈ [-27, -20] are system parameters. The three premise variables are given by

z 1 = e -2x1 1 + e -2x1 , z 2 = 1 + sin 2 (x 2 )
2 , z 3 = sin(0.1x 1 ).

For this example, we have

x d = x 1 x 2 , x a = x 3 and D x d ≡ R 2 . Note that 0 ≤ z 1 ≤ 1, 0 ≤ z 2 ≤ 1 and -1 ≤ z 3 ≤ 1, ∀x d ∈ R 2 .
Applying the proposed modeling approach, the considered nonlinear system can be exactly represented in the polytopic form [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] with 2 × 3 = 6 vertices, whose six nonlinear functions w i (z), ∀i ∈ I 6 , can be derived using ( 9) and ( 11) as

w 1 (z) = 1 3 + 3e -x1 , w 2 (z) = e -x1 3 + 3e -x1 , w 3 (z) = 1 -sin 2 (x 2 ) 6 , w 4 (z) = 1 + sin 2 (x 2 ) 6 , w 5 (z) = 1 -sin(0.1x 1 ) 6
, w 6 (z) = 1 + sin(0.1x 1 ) 6 .

The details on the corresponding system matrices of the six vertices are omitted here for brevity. We examine the design conservatism between the following control results: i) LMI conditions in Corollary 1; ii) LMI conditions in Theorem 1; iii) TS fuzzy-model-based results in [START_REF] He | Admissibility analysis and robust H∞ control for T-S fuzzy descriptor systems with structured parametric uncertainties[END_REF]Theorem 3.3] and [9, Theorem 3]; iv) TS fuzzy-model-based result in [25, Theorem 2]; v) affine TS fuzzy-model-based result in [41, Theorem 2]. Fig. 3 depicts the feasibility regions obtained with these control results. Remark that Corollary 1 cannot provide any feasible control solution for the considered parameter space, which confirms the negative overbounding effect of the proposed modeling approach on the control design conservatism. However, taking into account the slack variables specifically introduced by this modeling, Theorem 1 outperforms other recent TS fuzzy control results. The characteristics numbers N row and N var , representing the numerical burden of LMIbased optimization problems, for the considered control results are given in Table II. Remark that as a price for conservatism reduction, the numerical complexity of design conditions in Theorem 1 is slightly higher than that of other control results.

For illustrations, we consider system (56) with a = -11.37 and b = -20 to evaluate the control performance. Note that only Theorem 1; [START_REF] He | Admissibility analysis and robust H∞ control for T-S fuzzy descriptor systems with structured parametric uncertainties[END_REF]Theorem 3.3] and [START_REF] Zhang | Dissipativity analysis and synthesis for a class of T-S fuzzy descriptor systems[END_REF]Theorem 3] can provide a feasible solution for this system. Table III shows the comparison of the decay rate performance obtained with these control results. Observe that the proposed control approach provides a larger value of the maximal decay rate α max . Fig. 4 illustrates the closed-loop behaviors obtained with the corresponding controllers. As expected, we can see that compared to the recent related results, the proposed control approach yields a faster closed-loop time response under the same initial condition, see Remark 8. Example 3. To demonstrate the interests of the proposed control approach for complex engineering systems with a large number of nonlinearities, we consider a three-degreeof-freedom (3DoF) serial robot depicted in Fig. 5, whose dynamics can be described as [START_REF] Dehak | Reduced-complexity affine representation for Takagi-Sugeno fuzzy systems[END_REF] M(θ)

θ(t) + N(θ, θ) θ(t) + G(θ) = u(t), (57)
where θ(t) ∈ R 3 is the vector of generalized coordinates in

x(t) = θ(t) θ(t) ∈ R 6 , B(z) = 0 I , E(z) = I 0 0 M(z) , A(z) = 0 I -H(z) -N(z) . (58)
The matrices M(z), N(z) and H(z) in ( 58 57) can be directly defined as

D x = x ∈ R 6 : H (l) x ≤ 1, l ∈ I 12 , (59)
where the matrix H, characterizing the physical limitations of the robot states, is given by

H =                      1 θmax 0 0 0 0 0 -1 θmax 0 0 0 0 0 0 1 θmax 0 0 0 0 0 -1 θmax 0 0 0 0 0 0 1 θmax 0 0 0 0 0 -1 θmax 0 0 0 0 0 0 1 50 0 0 0 0 0 -1 50 0 0 0 0 0 0 1 50 0 0 0 0 0 -1 50 0 0 0 0 0 0 1 50 0 0 0 0 0 -1 50                      .
Given the above bounds of θ i and θi , for i ∈ I 3 , we can straightforwardly derive the bounds of the twelve premise variables from their mathematical expressions shown in Table IV, i.e., z p min ≤ z p ≤ z p max , for p ∈ I 12 , which are used to construct the TS fuzzy model ( 7) and the polytopic model [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF] of the 3DoF robot model (57). We consider the case without the viscous friction coefficient, i.e., f v = 0, and the values of the constant parameters c i , for i ∈ I 9 , are given in Table V. With r = 12 premise variables z i , for i ∈ I 12 , the TS fuzzy modeling (7), using the classical sector nonlinearity approach [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF], requires 2 12 = 4096 vertices to exactly represent system (57)-(58), which is computationally excessive for control design. Moreover, the corresponding TS fuzzy controller is impractical for real-time control purposes. However, the proposed modeling (12) leads to only 2 × 12 = 24 vertices, allowing for a practically feasible control solution. For comparison purposes, we examine the following quadratic control results: i) LMI conditions in Theorems 1 and 2 with α = 0, [rad] of the robot positions. Fig. 6 shows the feasibility regions obtained with the five considered control results. Remark that without including the acceleration information θ(t) (which is generally unavailable in practice) in the controller structure, the control result in [41, Theorem 2] cannot provide any feasible solution. The existing TS fuzzy control results in [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF], [START_REF] Bouarar | Robust fuzzy Lyapunov stabilization for uncertain and disturbed Takagi-Sugeno descriptors[END_REF] and Theorem 1 lead to the same feasibility region with a very restrictive joint range, i.e., θ max ≤ π 5 [rad]. We can observe that without using the descriptor-redundancy approach, Theorem 2 allows finding feasible control solutions for the whole robot workspace. This clearly confirms the interests of the new control approach for complex nonlinear systems in reducing the numerical complexity and design conservatism. The complexity characteristics numbers N row and N var of the considered control approaches are given in Table VI. Remark that despite the introduction of slack variables, the computational complexity of the proposed control results are much lower compared to the existing TS fuzzy control results. The reduced-complexity affine model-based approach in [41, Theorem 2] leads to numerically simple design conditions at the price of over-conservativeness. For illustrations, we consider system (57) with m 3 = 1 [kg] and θ max = π 10 [rad]. Fig. 7 depicts the closed-loop response of the corresponding robot system (57) obtained from the nonlinear controller (21), designed with Theorem 1 and α = 3. Note that the initial condition x(0) = 0.7 θ max -θ max -θ max 50 -50 -50 is selected such that x(0) ∈ E V . We observe that the robot states converge to the origin while respecting their physical limitations defined by the validity domain D x in (59). Hence, the computed bounds of the premise variables given in Table IV are always valid for the proposed polytopic representation during the simulation as discussed in Remark 4. However, without taking into account condition [START_REF] Xie | Membership functiondependent local controller design for T-S fuzzy systems[END_REF] in the control design to ensure the invariance property, see Remark 9, the closed-loop trajectories can go outside the validity domain D x under the same simulation conditions as shown in Fig 8. Note that in this case, the bounds of the premise variables are not respected anymore for the proposed polytopic representation as well as the TS fuzzy modeling.

L j = L, R j = R, W ij = W and Z ij = Z, ∀i,
To evaluate numerically the control performance, Table VII compares the decay rates obtained with different related control results for the robot system (57) with m 3 = 1 [kg] and θ max = π 10 [rad]. We note that the proposed conditions in Theorems 1 and 2 yield much larger values of α max compared to the control approach in [START_REF] Bouarar | Robust fuzzy Lyapunov stabilization for uncertain and disturbed Takagi-Sugeno descriptors[END_REF]. In particular, the largest value of α max obtained with Theorem 2 confirms the great potential of this control result, in terms of design conservatism and control performance, for non-singular nonlinear systems.

V. CONCLUDING REMARKS

A new LMI-based control approach has been proposed for a class of descriptor nonlinear systems. Compared to the classical TS fuzzy-model-based control approaches, the numerical complexity of the new approach grows proportionally, rather than exponentially, with respect to the number of premise variables. This is particularly interesting when dealing with complex descriptor systems with a large number of nonlinearities. Moreover, the system vertices obtained from the proposed polytopic representation are not unique even with the same predefined set of premise variables. This nonuniqueness representation feature allows introducing specific slack variables into the control design to reduce the conservatism of the proposed approach. Strict LMI conditions are derived via Lyapunov stability theory for control design and admissibility analysis. A special attention is paid to the control design of non-singular nonlinear systems, for which the descriptor-redundancy approach is not required to further reduce the design complexity and conservatism. The interests of the new control results are clearly demonstrated with both numerical and physically motivated examples. From numerical experiments, it is observed that for nonlinear systems with a low number of premise variables the proposed approach leads to admissibility analysis and control design results with a similar level of conservatism and numerical complexity as related existing TS fuzzy model-based approaches. However, when the number of premise variables becomes sufficiently Fig. 8: Closed-loop response of the robot system (57) with m 3 = 1 [kg] and θ max = π 10 [rad] obtained with controller (21) designed from Theorem 1 without condition [START_REF] Xie | Membership functiondependent local controller design for T-S fuzzy systems[END_REF], α = 3 under the same initial condition as in Fig. 7. large, as illustrated in Example 3, the strong interest of the new method related to numerical complexity and design conservatism reduction over TS fuzzy model-based approaches is put in evidence. Future works focus on extending the proposed approach to output feedback tracking control of singular nonlinear systems. Considering the proposed polytopic modeling method to deal with the fault detection issue of singular nonlinear systems with unmeasured premise variables and delayed/quantized output signals [START_REF] Sakthivel | Fault estimation for mode-dependent IT2 fuzzy systems with quantized output signals[END_REF] is another promising research direction.

Fig. 1 :

 1 Fig. 1: Characteristics numbers of numerical complexity with respect to the number of premise variables r for the case n = 5 and m = 0: Theorem 1 ; Theorem 2 ; [11, Theorem 3.3] ; [25, Theorem 2] .

Fig. 2 :

 2 Fig. 2: Feasibility regions obtained with [41, Theorem 2] ( ); Corollary 1 (×, ); Theorem 1 and [11, Theorem 3.3] (•,×, ).

20 Fig. 3 :

 203 Fig. 3: Feasibility regions obtained with [41, Theorem 2] ( ); [11, Theorem 3.3] and [9, Theorem 3] (•,×, ); [25, Theorem 2] ( ,×, ); and Theorem 1 (+, ,•,×, ).

Fig. 4 :

 4 Fig. 4: Closed-loop responses of system (56) with a = -11.37 and b = -20 obtained under the same initial condition from Theorem 1 with α max = 2.25 ( ); [11, Theorem 3.3] and [9, Theorem 3] with α max = 0.2 ().

n 12 n 13 n 21 n 22 n 23 n 31 n 32 n 33 1 z

 331 2 + c 8 c 6 z 2 + c 9 10 + c 2 z 11 + c 3 z 12 c 2 z 11 + c 3 z 12 c 3 z 12 c 2 z 11 + c 3 z 12 c 3 z 12 c 3 z 12   , with m 11 = 2(c 4 z 1 + c 6 z 2 + c 5 z 3 + c 7 ), m 12 = c 4 z 1 + 2c 6 z 2 + c 5 z 3 + c 8 , m 13 = c 6 z 2 + c 5 z 3 + c 9 , n 11 = c 5 (z 6 -z 9 ) + c 6 (z 7 -z 8 ) + f v , n 12 = -c 4 z 4 + c 6 (z 7 -z 8 ) -c 5 z 9 , n 13 = -c 6 z 8 -c 5 z 9 , n 21 = c 4 z 5 + c 5 z 6 + c 6 (z 7 -z 8 ), n 22 = c 6 (z 7 -z 8 ) + 2f v , n 23 = -c 6 z 8 , n 31 = c 5 z 6 + c 6 z 7 , n 32 = c 6 z 7 , n 33 = 2f v . The expressions of the premise variables z i , for i ∈ I 12 , are given in Table IV. Note that {z 1 , z 2 , z 3 } are the nonlinear terms involved in the inertia matrix M(θ); {z 4 , z 5 , . . . , z 9 } are the nonlinear terms involved in the Coriolis matrix N(θ, θ); and {z 10 , z 11 , z 12 } are the nonlinear terms involved in the gravity-related matrix H(θ). For the robot model (57), we consider the joint mechanical limits and the maximum motor velocities as |θ i | ≤ θ max [rad], and | θi | ≤ 50 [rad/s], for i ∈ I 3 . The parameter θ max ∈ [0, π] is used in the following to study the control design conservatism. As a result, the validity domain D x of the robot model (

Fig. 5 :

 5 Fig. 5: Schematic of the 3DoF robot in the vertical plane.

Fig. 6 :

 6 Fig. 6: Feasibility regions obtained with TS fuzzy approaches [12, Chapter 10] (•); [25, adapted Theorem 2] (•); Theorem 1 (•); and Theorem 2 (•,×).

Fig. 7 :

 7 Fig. 7: Closed-loop response of the robot system (57) with m 3 = 1 [kg] and θ max = π 10 [rad] obtained with controller (21) designed from Theorem 1, α = 3 and x(0) ∈ E V .

TABLE I :

 I Complexity Characteristics Numbers of Different Control Results.

	Control Design	Theorem 1	Theorem 2	[11, Theorem 3.3]	[25, Theorem 2]
	Number of vertices	2r	2r	2 r	2 r
	Nrow	n(16r 2 + 1)	n(8r 2 + 1)

TABLE II :

 II Numerical Complexity of Different Control Results.

	Control Design	Theorem 1 Corollary 1 [41, Theorem 2] [25, Theorem 2] [11, Theorem 3.3]
	Number of vertices	6	6	4	8	8
	Nrow	362	362	146	224	408
	Nvar	1185	855	119	334	306

TABLE III :

 III Comparison of Decay Rate for Example 2. ∈ R 3 is the vector of generalized control forces, M(θ) ∈ R 3×3 is the inertia matrix, N(θ, θ) ∈ R 3×3 is the Coriolis/centripetal matrix plus the viscous friction coefficients of the joints, and G(θ) ∈ R n represents the gravity matrix. Since the vector-valued function G(θ) is smooth and G(0) = 0, we can then parameterize G(θ) = H(θ)θ. System (57) can be rewritten in the nonlinear descriptor form (3) with θ(t) = θ 1 (t) θ 2 (t) θ 3 (t) and

	Control Design Theorem 1 [11, Theorem 3.3] [9, Theorem 3]
	αmax	2.25	0.20	0.20
	joint space, u(t)

TABLE IV :

 IV Premise Variables of the 3DoF Robot.

	Premise variables Expression
	z 1	cos(θ 2 )
	z 2	cos(θ 3 )
	z 3	cos(θ 2 + θ 3 )
	z 4	(2 θ1 + θ2 ) sin(θ 2 )
	z 5	θ1 sin(θ 2 )
	z 6	θ1 sin(θ 2 + θ 3 )
	z 7	( θ1 + θ2 ) sin(θ 3 )
	z 8	( θ1 + θ2 + θ3 ) sin(θ 3 )
	z 9	( θ1 + θ2 + θ3 ) sin(θ 2 + θ 3 )
	z 10	sinc(θ 1 )
	z 11	sinc(θ 1 + θ 2 )
	z 12	sinc(θ 1 + θ 2 + θ 3 )

TABLE V :

 V Mechanical Parameters of the 3DoF Robot.

	Symbol Expression	Value
	c 1	g(L 1 m 2 + L 1 m 3 + m 1 r 1 )	121.64
	c 2	g(L 2 m 3 + m 2 r 2 )	67.69
	c 3	gm 3 r 3	23.54
	c 4	L 1 (L 2 m 3 + m 2 r 2 )	6.9
	c 5	L 1 m 3 r 3	2.4
	c 6	L 2 m 3 r 3	2.16
	c 7 c 8 c 9	(m 2 + m 3 )L 2 1 + L 2 2 m 3 + 3 i=1 I i L 2 2 m 3 + I 2 + I 3 I 3	19.75 7.85 1.9

TABLE VI :

 VI Numerical Complexity of Different Control Results.

	Control Design	Theorem 1 Theorem 2 [41, Theorem 2] [12, Chapter 10] [25, Theorem 2]
	Number of vertices	24	24	13	4096	4096
	Nrow	582	294	162	49158	61446
	Nvar	813	525	255	73821	73821

TABLE VII :

 VII Comparison of Decay Rate for Example 3.

	Control Design	Theorem 1	Theorem 2 [25, adapted Theorem 2]
	αmax	10.6 × 10 4	25 × 10 4	750

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

The authors are with the LAMIH laboratory, UMR CNRS 8201,