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

Tadanari Taniguchi

Luka Eciolaza

Víctor Campos

Reinaldo Palhares

Michio Sugeno

Fuzzy Control Systems: Past, Present and Future

More than 40 years after fuzzy logic control appeared as an effective tool to deal with complex processes, the research on fuzzy control systems has constantly evolved. Mamdani fuzzy control was originally introduced as a model-free control approach based on expert's experience and knowledge. Due to the lack of a systematic framework to study Mamdani fuzzy systems, we have witnessed growing interest in fuzzy model-based approaches with Takagi-Sugeno fuzzy systems and singleton-type fuzzy systems (also called piecewise multiaffine systems) over the past decades. This paper reviews the key features of the three above types of fuzzy systems. Through these features, we point out the historical rationale for each type of fuzzy systems and its current research mainstreams. However, the focus is put on fuzzy model-based approaches developed via Lyapunov stability theorem and linear matrix inequality (LMI) formulations. Finally, our personal viewpoint on the perspectives and challenges of the future fuzzy control research is discussed.

I. INTRODUCTION

Fuzzy control was initiated by Mamdani [START_REF] Mamdani | Application of fuzzy algorithms for control of simple dynamic plant[END_REF] in 1974 stimulated by the Zadeh's two seminal papers on fuzzy algorithms [START_REF] Zadeh | Fuzzy algorithms[END_REF] in 1968 and linguistic analysis [START_REF] Zadeh | Outline of a new approach to the analysis of complex systems and decision processes[END_REF] in 1973. In these papers, Zadeh presented a method of system modeling based on fuzzy IF-THEN rules with linguistic variables. The first application of fuzzy logic control was performed by Mamdani and Assilian on a laboratory steam engine [START_REF] Mamdani | An experiment in linguistic synthesis with a fuzzy logic controller[END_REF] which led to a great impact on the fuzzy control research. Indeed, many fuzzy control systems have been proposed since the publication of the original paper in 1975. Generally speaking, there are three types of fuzzy systems as classified by Sugeno in [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF] according to the consequent parts of IF-THEN rules. First, Mamdani-type fuzzy systems [START_REF] Mamdani | An experiment in linguistic synthesis with a fuzzy logic controller[END_REF] are defined by IF-THEN rules associated with linguistic variables as Rule R i :

IF x 1 is G i 1 , x 2 is G i 2 , . . . , x m is G i
where H i and G i j , j = 1, 2, . . . , m, are fuzzy sets, n is the number of fuzzy rules. Second, Takagi-Sugeno (T-S) fuzzy systems [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF] are with functional consequents Rule R i :

IF x 1 is G i 1 , x 2 is G i 2 , . . . , x m is G i m THEN y is f i (x 1 , x 2 , . . . , x m ), i = 1, 2, . . . , n
The function f i (•) is usually linear as follows:

f i (x 1 , x 2 , . . . , x m ) = b i + a i 1 x 1 + a i 2 x 2 + • • • + a i m x m ,
where the coefficients b i , a i j , with j = 1, . . . , m, are constants. Third, singleton-type fuzzy systems [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF] are with fuzzy rules of the following form:

Rule R i :

IF x 1 is G i 1 , x 2 is G i 2 , . . . , x m is G i m THEN y is b i , i = 1, 2, . . . , n
where b i is a singleton, i.e., a real number. A singleton-type fuzzy system is recently called piecewise multiaffine (PMA) system in [START_REF] Nguyen | LMI-based stability analysis for piecewise multi-affine systems[END_REF] since its input-output relation with respect to an affine-in-control system is found to be a multiaffine function. This type of fuzzy systems is used to be called PB (piecewise bilinear, more precisely biaffine) systems because for a two-dimensional case, the output can be expressed as y = ax 1 + bx 2 + cx 1 x 2 + d, for some scalars a, b, c and d, see [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF], [START_REF] Kim | A new computational approach to stability analysis and synthesis of linguistic fuzzy control system[END_REF].

Mamdani-type fuzzy systems are linguistically understandable since fuzzy variables are used in both the premises and the consequents. However, T-S fuzzy systems do not have linguistic variables since only functional membership functions (MFs) are used without labels. PMA systems are a simplified special case of both previous types of fuzzy systems. Indeed, PMA systems are Mamdani fuzzy systems with singleton consequents and they can be obtained from T-S fuzzy systems when only the constant terms in the consequents are present. While Mamdani fuzzy systems are deeply concerned with fuzzy set and logic, T-S fuzzy systems are only concerned with fuzzy set in their premises. PMA systems stand independently of fuzzy set and logic since there is no need to use neither linguistic variables nor MFs. In the case of PMA systems, these functions in the premises are of a triangular shape and only play roles as parameters for interpolation which are not necessarily interpreted as membership functions in the conventional "fuzzy" sense. However, PMA systems derived from Mamdani may keep their status in fuzzy control and linguistic labels could be assigned to singletons if necessary. Note that all three types of fuzzy systems are known to have general approximation capability for any nonlinear functions [START_REF] Kosko | Fuzzy systems as universal approximators[END_REF]. However, compared to two other types, T-S fuzzy modeling can drastically reduce the number of fuzzy rules, especially for high dimensional complex systems [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF].

During the first ten years since Mamdani's successful application of fuzzy logic control, researchers were faced with a lot of criticisms from the conventional control theorists as reported in [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF], [START_REF] Sala | Perspectives of fuzzy systems and control[END_REF], [START_REF] Guerra | Fuzzy control turns 50: 10 years later[END_REF]. The main reason is that there was no stability analysis available for fuzzy control at that time [START_REF] Sugeno | An introductory survey of fuzzy control[END_REF]. To answer these criticisms, T-S fuzzy systems were newly introduced in 1985. Since T-S fuzzy systems are associated with linear models in their consequents, model-based stability analysis and control design can be performed for fuzzy systems using conventional Lyapunov-based approaches as firstly shown in [START_REF] Tanaka | Stability analysis and design of fuzzy control systems[END_REF].

T-S fuzzy modeling can be used to represent exactly a nonlinear system in a compact set of the state space, where the nonlinearities are embedded in membership functions [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. Therefore, it is generally impossible to make a non-conservative stability analysis of a given nonlinear system with T-S fuzzy model-based approaches. More clearly, T-S fuzzy systems belong to a class of polytopic uncertain systems whose uncertainties are caused by the nonlinearity of the MFs. As such, T-S fuzzy model-based stability analysis always remains conservative [START_REF] Sala | On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems[END_REF]. PMA models were then presented to overcome this drawback of T-S fuzzy models. As shown in [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF], PMA models are fully parametric like linear systems, though they are approximate models of nonlinear systems in their nature. Because of this advantage, it is theoretically possible to derive necessary and sufficient stability conditions for PMA systems just as in the case of linear systems [START_REF] Sugeno | On improvement of stability conditions for continuous Mamdani-like fuzzy systems[END_REF]. This cannot be the case of other types of fuzzy systems. Since T-S fuzzy systems and PMA systems include linear systems as a special case, it is expected that any theoretical framework developed for these both types of fuzzy systems would be more general than its linear counterpart.

The intention of this paper is to provide a concise overview on the fuzzy control research since the pioneering works performed by Mamdani's group at Queen Mary College. Here, no attempt is made to comprehensively review the literature which includes several excellent books [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF], [START_REF] Pedrycz | Fuzzy Control and Fuzzy Systems[END_REF]- [START_REF] Babuška | Fuzzy Modeling for Control[END_REF] and thousands of technical articles, see for instance [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF], [START_REF] Sala | Perspectives of fuzzy systems and control[END_REF], [START_REF] Sugeno | An introductory survey of fuzzy control[END_REF],

[21]- [START_REF] Precup | A survey on industrial applications of fuzzy control[END_REF] and references therein. Instead, only a selective and exemplary list of some breakthrough results is given to tracing back the evolution of fuzzy control systems. Our primary aim is to provide a historical rationale for each type of fuzzy systems and its current research status without excessive mathematical complexity. The emphasis is put more on the fuzzy model-based stability analysis than model-free fuzzy one. In particular, for the sake of simplicity and illustration, only approaches based on Lyapunov stability theory for continuous-time dynamical systems without any specific performance issue are focused in the paper. However, it is stressed that theoretical results on stability analysis and control design with or without various performance specifications, for instance H ∞ and H 2 with respect to external disturbances, robustness with respect to time delay or modeling uncertainty, etc., and their discrete-time counterparts have also been widely reported in the fuzzy control literature [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF], [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF].

Finally, we present our viewpoint on the perspectives and challenges of the future fuzzy control research. Notation: Ω N denotes the set {1, 2, . . . , N }. R is the field of real numbers. For a vector x ∈ R n and i ∈ Ω n , x i denotes the ith entry of x. I denotes the identity matrix of appropriate dimension. For a matrix X, X indicates its transpose and

X (ij)
denotes its element of the ith row and jth column. For any square matrix X, X > 0 indicates a symmetric positive definite matrix, and HeX = X + X . The symbol stands for matrix blocks that can be deduced by symmetry. The time dependency of the variables is omitted when convenient.

II. MAMDANI FUZZY SYSTEMS

The basic idea of Mamdani fuzzy control (MFC) is to represent the process states by means of linguistic variables and to exploit these variables as inputs to control rules [START_REF] Yager | Essentials of Fuzzy Modeling and Control[END_REF]. Hence, MFC enables to incorporate the expert's skills and experience through a set of fuzzy IF-THEN rules [START_REF] Pedrycz | Fuzzy Control and Fuzzy Systems[END_REF], [START_REF] Passino | Fuzzy Control[END_REF]. Thanks to this particular feature, the effectiveness of MFC has been clearly proved in the following situations [START_REF] Sugeno | An introductory survey of fuzzy control[END_REF]. First, no acceptable mathematical model is available for the controlled plant. Second, human operators play a crucial role in the control process and can provide qualitative control rules in terms of fuzzy logic sentences. Moreover, as pointed out by Zadeh, MFC is task-oriented control which is in contrast to set-point-oriented feature of conventional control approaches [START_REF] Zadeh | The evolution of systems analysis and control: A personal perspective[END_REF]. This allows MFC to achieve easily multi-objective goals by simply setting some fuzzy control rules under one criterion and others under a different performance criterion. The coordination between different control objectives can be performed by fuzzy reasoning. For these reasons, until now MFC has been successfully applied to a large number of industrial processes, see [START_REF] Sugeno | An introductory survey of fuzzy control[END_REF], [START_REF] Precup | A survey on industrial applications of fuzzy control[END_REF], [START_REF] Bonissone | Industrial applications of fuzzy logic at General Electric[END_REF], [START_REF] Van Der Wal | Application of fuzzy logic control in industry[END_REF] for constructive surveys on prominent applications of MFC.

Unfortunately, except for a few exceptions, the design of MFC remains model-free and essentially heuristic. As a direct consequence, we still lack at present a systematic framework as well as analytical tools to study rigorously the stability of Mamdani fuzzy systems [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF]. Establishing such a theoretical stability framework for Mamdani fuzzy systems is expected to be particularly challenging due to the novelty of the fuzzy mathematics and language [START_REF] Nguyen | LMI-based stability analysis for piecewise multi-affine systems[END_REF]. The mainstream idea for stability analysis has been to consider the Mamdani fuzzy controller as a nonlinear controller, then the fuzzy control design is recast as a nonlinear control approach using absolute stability theory, sliding mode control, adaptive fuzzy control, etc. Excellent reviews on the stability issues of MFC systems can be found, for instance, in [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF], [START_REF] Sugeno | An introductory survey of fuzzy control[END_REF], [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF].

To overcome the "model-free" major drawback of MFC, fuzzy model-based control approaches were originally proposed by Sugeno's research group at Tokyo Institute of Technology [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF], [START_REF] Tanaka | Stability analysis and design of fuzzy control systems[END_REF]. Such control approaches enable a systematic framework to deal with the stability analysis and control design of nonlinear dynamical systems in the following general form:

ẋ(t) = f (x(t)) + g(x(t))u(t) (1)
where x(t) ∈ R n is the state vector and u(t) ∈ R m is the control input. Without loss of generality, the nonlinear system [START_REF] Mamdani | Application of fuzzy algorithms for control of simple dynamic plant[END_REF] satisfies the following assumption.

Assumption 1. The origin x 0 = 0 ∈ R n is an equilibrium of system (1) such that f (0) = 0. Moreover, the vector fields f (•)

and g(•) are sufficiently smooth, i.e., f ∈ C 2 and g ∈ C 2 .

Fuzzy model-based approches to study theoretically the nonlinear system (1) are discussed in the subsequent sections.

III. TAKAGI-SUGENO FUZZY SYSTEMS

After a brief description of T-S fuzzy models, this section reviews some key points on the stability analysis and control design of this type of fuzzy systems.

A. System Description

The T-S fuzzy model of the continuous-time nonlinear system (1) is described in the following form [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]:

Rule R i : IF z 1 (t) is M i 1 and • • • and z p (t) is M i p THEN ẋ(t) = A i x(t) + B i u(t) (2)
where R i denotes the ith fuzzy inference rule, r is the number of inference rules, M i j , with i ∈ Ω r and j ∈ Ω p , are the fuzzy sets, and (A i , B i ) the state-space matrices of appropriate dimensions of the ith local model. The vector of premise variables is defined as

z(t) = [z 1 (t) • • • z p (t)].
Using the center-of-gravity method for defuzzification, the T-S fuzzy model (2) can be represented in the following compact form:

ẋ(t) = r i=1 h i (z)(A i x(t) + B i u(t)) (3)
where the normalized MF h i (z) is defined as

h i (z) = ω i (z) r i=1 ω i (z) , ω i (z) = p j=1 µ i j (z j ), i ∈ Ω r .
The grades of membership of the premise variables in the respective fuzzy sets M i j are given as µ i j (z j ). Note that the normalized MFs satisfy the following convex sum property:

0 ≤ h i (z) ≤ 1, r i=1 h i (z) = 1, r i=1 ḣi (z) = 0 (4)
Remark 1. Although the premise variables can represent any type of variables in nonlinear systems, most of the works in the T-S fuzzy literature deal with the case for which the premise variables are composed of a subset of the system state x(t). This assumption is also adopted along this paper.

Remark 2. Using the sector nonlinearity approach in [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF], the T-S fuzzy model (3) can be directly derived from [START_REF] Mamdani | Application of fuzzy algorithms for control of simple dynamic plant[END_REF]. In this case, both representations of the affine nonlinear system are strictly equivalent in a compact set of the state space.

B. Lyapunov-Based Stability Analysis

Consider the T-S fuzzy system (3) with u(t) ≡ 0 as follows:

ẋ(t) = r i=1 h i (z)A i x(t) (5)
The first results on the stability analysis were proposed in [START_REF] Tanaka | Stability analysis of fuzzy systems using Lyapunov's direct method[END_REF] for the T-S fuzzy system (5), and in [START_REF] Tanaka | Stability analysis and design of fuzzy control systems[END_REF] for its discrete-time counterpart. Consider a quadratic Lyapunov function (QLF) of the form

V (x) = x P x, P > 0 (6)
The following result is readily obtained.

Theorem 1. [START_REF] Tanaka | Stability analysis of fuzzy systems using Lyapunov's direct method[END_REF] The equilibrium of the T-S fuzzy system (5) is globally asymptotically stable if there exists a common positive definite matrix P such that

A i P + P A i < 0, i ∈ Ω r (7)
The three following remarks deserve particular attention.

• Conditions [START_REF] Nguyen | LMI-based stability analysis for piecewise multi-affine systems[END_REF] are expressed in terms of linear matrix inequalities (LMIs). Hence, the stability analysis can be easily checked with available numerical solvers [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF].

• A common Lyapunov matrix variable P has to exist for all local linear subsystems.

• The MFs of the T-S fuzzy system (5), i.e., its nonlinearities, are considered as uncertainty. Then, the stability analysis is embedded in the conventional robust control theory.

Concerning the first remark, the possibility to reformulate the stability analysis of T-S fuzzy systems as a convex optimization problem has undeniably sparked the growing interest in this type of fuzzy systems [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. This enables systematic and effective frameworks for stability analysis and control design of general nonlinear systems, see detailed discussions on prominent results

in the 90s and in the early of 2010s in [START_REF] Guerra | Fuzzy control turns 50: 10 years later[END_REF], [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF]. Another very recent survey on this research topic can be found in [START_REF] Lam | A review on stability analysis of continuous-time fuzzy-model-based control systems: From membership-function-independent to membership-function-dependent analysis[END_REF].

The two other remarks are concerned with the conservativeness issue of T-S fuzzy control theory [START_REF] Sala | On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems[END_REF]. Currently, most of the research effort has been focused on this issue. There are two mainstreams to reduce the conservatism of T-S fuzzy model-based approaches, which are briefly discussed hereafter. The first one is based on the choice of different families of Lyapunov function candidates eventually combined with slack variables introduced via robust control tools such as Finsler lemma, S-procedure, etc. [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. The second mainstream consists in finding ways to exploit more efficiently the information on the MFs for stability analysis.

1) Relaxations with Different Choices of Lyapunov Functions:

The conservatism of the results due to the use of a single quadratic Lyapunov function ( 6) was emphasized in [START_REF] Johansson | Piecewise quadratic stability of fuzzy systems[END_REF], [START_REF] Tanaka | A multiple Lyapunov function approach to stabilization of fuzzy control systems[END_REF]. To overcome this drawback, more general classes of Lyapunov functions have been suggested in the T-S fuzzy literature, including piecewise Lyapunov functions (PLFs) [START_REF] Johansson | Piecewise quadratic stability of fuzzy systems[END_REF], [START_REF] Feng | Stability analysis of discrete-time fuzzy dynamic systems based on piecewise Lyapunov functions[END_REF], [START_REF] Campos | New stability conditions based on piecewise fuzzy Lyapunov functions and tensor product transformations[END_REF],

fuzzy Lyapunov functions (FLFs) depending on the MFs [START_REF] Tanaka | A multiple Lyapunov function approach to stabilization of fuzzy control 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] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF], line integral Lyapunov functions (LILFs) [START_REF] Mozelli | A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems[END_REF], [START_REF] Rhee | A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design[END_REF], [START_REF] González | A generalised integral polynomial Lyapunov function for nonlinear systems[END_REF], polynomial Lyapunov functions depending on the MFs with arbitrary degree [START_REF] Tognetti | Selective H 2 and H∞ stabilization of Takagi-Sugeno fuzzy systems[END_REF]- [START_REF] Chen | Stability analysis and region-of-attraction estimation using piecewise polynomial Lyapunov functions: Polynomial fuzzy model approach[END_REF], multidimensional fuzzy Lyapunov functions [START_REF] Lee | Relaxed LMI conditions for local stability and local stabilization of continuous-time Takagi-Sugeno fuzzy systems[END_REF], [START_REF] Lee | Local stability analysis of continuous-time Takagi-Sugeno fuzzy systems: a fuzzy Lyapunov function approach[END_REF], and so on.

An interesting remark when using FLFs for stability analysis of the continuous-time T-S fuzzy system ( 5) is that the stability conditions depend on the time-derivatives of the MFs, thus the derivatives of the system state. This implies more numerical and theoretical complexities and requires a much larger effort to obtain convex formulations than a quadratic Lyapunov based framework. To overcome this difficulty, different alternatives have been proposed to consider the upper bounds of the timederivatives of the MFs. However, this usually leads to a local analysis setting [START_REF] Lee | Relaxed LMI conditions for local stability and local stabilization of continuous-time Takagi-Sugeno fuzzy systems[END_REF]- [START_REF] Campos | A comparison of different upper-bound inequalities for the membership functions derivative[END_REF]. Another alternative is to make use of PLFs or LILFs to avoid the presence of the time-derivatives of the MFs in the stability conditions. However, in these cases some special structures should be imposed on the Lyapunov matrices which introduce some conservatism. In addition, using PLFs is only suitable for T-S fuzzy systems with triangular or trapezoidal MFs inducing particular state-space partitions [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF], [START_REF] Johansson | Piecewise quadratic stability of fuzzy systems[END_REF], [START_REF] Feng | Stability analysis of discrete-time fuzzy dynamic systems based on piecewise Lyapunov functions[END_REF]. Moreover, LILFs-based approaches usually require that the premise variables are the system state, i.e., z ≡ x.

Nevertheless, this assumption can be recently avoided in [START_REF] González | A generalised integral polynomial Lyapunov function for nonlinear systems[END_REF]. Applying PLFs for the stability analysis of T-S fuzzy systems was already discussed in detail in [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF]. Hence, only the two other cases, namely FLFs and LILFs, are given below to illustrate the above discussion.

Consider a simple example of FLFs depending explicitly on MFs as follows [START_REF] Tanaka | A multiple Lyapunov function approach to stabilization of fuzzy control systems[END_REF]:

V (x) = x r i=1 h i (z)P i x, P i > 0 (8)
whose time-derivative is given by

V (x) = x ẋ r i=1 ḣi (z)P i r i=1 h i (z)P i 0 x ẋ .
Remark 3. When the positive definite matrices are imposed as P 1 = • • • = P r = P , then the QLF ( 6) is straightforwardly recovered from [START_REF] Kim | A new computational approach to stability analysis and synthesis of linguistic fuzzy control system[END_REF]. Therefore, the QLFs are only a special case of the FLFs. Observe also that the terms ḣi (z), i ∈ Ω r , appear explicitly in the expression of V (x). This makes the stability analysis much more involved with this choice. It should be stressed that although the fuzzy Lyapunov function ( 8) belongs to a more general class of Lyapunov function candidates compared to the quadratic one, there is no guarantee that FLFs-based results are less conservative than those derived from QLFs, see [START_REF] Nguyen | An augmented system approach for LMI-based control design of constrained Takagi-Sugeno fuzzy systems[END_REF] for a counterexample. This is primarily due to the presence of a priori unknown time-derivatives of the MFs in the theoretical developments when using FLFs [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF]. Hence, the stability conservatism strongly depends on the way how such unknown time-derivatives are handled.

For simplicity, one could exploit the ideas related to properties of the MFs to deal with their time-derivatives. For instance, from the convex sum property (4), it follows that r i=1 ḣi (z)X = 0, for any matrix X. This implies that

r i=1 ḣi (z)P i = r i=1 ḣi (z)(X + P i ).
Assume there exist upper bounds of the time-derivatives of the MFs such that | ḣi (z)| ≤ φ i , for some positive scalars φ i , i ∈ Ω r . Then, it follows that

x r i=1 ḣi (z)(P i + X) x ≤ x P φ x (9)
where P φ = r i=1 φ i (P i + X). Using a null-term approach or Finsler lemma [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF], inequality (9) can be exploited for stability analysis as summarized in the following theorem.

Theorem 2.

[34] Given a T-S fuzzy system (5), and upper bounds on the time-derivatives of the MFs as

| ḣi (z)| ≤ φ i , i ∈ Ω r .
If there exist positive definite matrices P i ∈ R n×n , with i ∈ Ω r , and matrices X ∈ R n×n , M ∈ R n×n , N ∈ R n×n , satisfying the following linear matrix inequalities:

P i + X > 0, i ∈ Ω r (10)
P φ -M A i -A i M P i + M -N A i N + N < 0, i ∈ Ω r (11)
with P φ given by [START_REF] Kosko | Fuzzy systems as universal approximators[END_REF]. Then, the T-S fuzzy system (5) is asymptotically stable.

Remark 4. Differently from Theorem 1, Theorem 2 only allows for local stability analysis of system (5) since the system state has to satisfy | ḣi (z)| ≤ φ i , i ∈ Ω r . Hence, an implicit goal is also to maximize the domain of attraction included inside the state region defined by the above assumption [START_REF] Lee | Relaxed LMI conditions for local stability and local stabilization of continuous-time Takagi-Sugeno fuzzy systems[END_REF]. Note that exploiting different types of properties of fuzzy MFs and relaxing tools from robust control theory, several local stability conditions have been proposed in the FLFs literature [START_REF] Campos | A comparison of different upper-bound inequalities for the membership functions derivative[END_REF]. It is important to note also that unlike the continuous-time case, the stability analysis for discrete-time T-S fuzzy systems does not suffer from this kind of difficulty in dealing with the time-derivatives of MFs when using FLFs [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF], [START_REF] Guerra | LMI-based relaxed non-quadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form[END_REF].

Remark 5. In Theorem 2, matrices X, M , N , are considered as slack variables, used for relaxation purposes. By imposing X = 0, P i = P > 0, for ∀i ∈ Ω r , and M = -P , it is easy to prove that the result of Theorem 2 includes that of Theorem 1. This theoretically confirms that compared to quadratic Lyapunov functions, FLFs enable less conservative stability analysis, see also Remark 3.

For T-S fuzzy systems with z k = x k , k ∈ Ω n , i.e., the premise variables are explicitly the state variables, line integral Lyapunov functions can be exploited to avoid dealing with the time-derivatives of the MFs [START_REF] Rhee | A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design[END_REF]. Consider now a Lyapunov function of the following form:

V (x) = 2 Γ[0,x] r i=1 h i (ζ)P i ζ, dζ (12)
where Γ[0, x] is a path from the origin to the present state, dζ is an infinitesimal displacement, •, • denotes the inner product.

As shown in [START_REF] Rhee | A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design[END_REF], if the matrices P i are written as

P i = D 0 + D i (13)
where

D 0 =        0 d 12 • • • d 1n d 12 0 • • • d 2n . . . . . . . . . . . . d 1n d 2n • • • 0        , D i =        d αi1 11 0 • • • 0 0 d αi2 22 • • • 0 . . . . . . . . . . . . 0 0 • • • d αin nn       
, and α ik indicating which M k fuzzy set is used for the ith rule. Then, it follows from ( 12) and ( 13) that

∇V (x) = 2 r i=1 h i (x)P i x.
Exploiting this special structure of the Lyapunov function, the following theorem can be stated.

Theorem 3.

[34] Given a T-S fuzzy system (5) with z ≡ x. If there exist symmetric matrices P i ∈ R n×n , with i ∈ Ω r , and matrices M ∈ R n×n , N ∈ R n×n such that

P i > 0, i ∈ Ω r (14) -M A i -A i M P i + M -N A i N + N < 0, i ∈ Ω r (15)
with P i having the structure in [START_REF] Tanaka | Stability analysis and design of fuzzy control systems[END_REF]. Then, the T-S fuzzy system ( 5) is globally asymptotically stable.

Remark 6. A great advantage of Theorem 3 compared to Theorem 2 is that the assumption on the upper bounds of the timederivatives of the MFs is not required anymore for LMI-based stability analysis. Such an assumption is not always verified, especially in T-S fuzzy control context [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF]. However, the special structure of the matrices P i as shown in (13) may induce some conservatism. In fact, up to now it is still hard to get a definitive answer on which approach leads to less conservative stability conditions.

2) Exploiting the Knowledge of the Membership Functions for Stability Relaxations: The MFs used to "blend" the local linear submodels of the T-S fuzzy system (3) represent the nonlinearity of system (1). However, these MFs have been widely considered as system uncertainty, and only their convex sum property (4) has been exploited in most of the existing works based on quadratic, fuzzy and polynomial Lyapunov functions [START_REF] Sala | On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems[END_REF]. Whenever the shape of the MFs and its intrinsic timevarying characteristic are not explicitly taken into account in the stability analysis, the conservativeness issue still remains, see further details in [START_REF] Lam | A review on stability analysis of continuous-time fuzzy-model-based control systems: From membership-function-independent to membership-function-dependent analysis[END_REF], [START_REF] Sala | Relaxed stability and performance conditions for Takagi-Sugeno fuzzy systems with knowledge on membership function overlap[END_REF]- [START_REF] Narimani | Relaxed LMI-based stability conditions for Takagi-Sugeno fuzzy control systems using regional-membership-functionshape-dependent analysis approach[END_REF].

Several approaches have been proposed to consider explicitly the shape of the MFs in the stability analysis. These approaches can be classified into two following categories [START_REF] Yang | Membership-dependent stability conditions for type-1 and interval type-2 T-S fuzzy systems[END_REF]. First, the membership-function-approximation approaches exploit the MF information via alternative similar functions such as staircase MFs [START_REF] Lam | Quadratic-stability analysis of fuzzy-model-based control systems using staircase membership functions[END_REF], piecewise linear MFs [START_REF] Lam | Polynomial fuzzy-model-based control systems: stability analysis via piecewise-linear membership functions[END_REF]. Second, the membershipbound-dependent approaches exploit the bound information of MFs for stability analysis [START_REF]Relaxed stability and performance LMI conditions for Takagi-Sugeno fuzzy systems with polynomial constraints on membership function shapes[END_REF]. In addition, the MFs image space and the order relations among the MFs have been also exploited, see for instance [START_REF] Yang | Membership-dependent stability conditions for type-1 and interval type-2 T-S fuzzy systems[END_REF], [START_REF] Bernal | A membership-function-dependent approach for stability analysis and controller synthesis of Takagi-Sugeno models[END_REF], [START_REF] Campos | Using information on membership function shapes in asymptotically exact triangulation approaches[END_REF]. Note that membership-functionapproximation and image-space approaches generally lead to a higher number of convex stability constraints. Concerning membership-bound-dependent approaches, slack variables are usually introduced into the stability conditions for relaxation purposes. As an illustrative example, the following theorem presents the idea in [START_REF]Relaxed stability and performance LMI conditions for Takagi-Sugeno fuzzy systems with polynomial constraints on membership function shapes[END_REF] on using the MFs shape information to reduce further the stability conservatism of Theorem 3 by introducing new relaxation variables. More discussions on the membership-function-dependent stability analysis can be found in the recent survey [START_REF] Lam | A review on stability analysis of continuous-time fuzzy-model-based control systems: From membership-function-independent to membership-function-dependent analysis[END_REF].

Theorem 4. (adapted from [START_REF]Relaxed stability and performance LMI conditions for Takagi-Sugeno fuzzy systems with polynomial constraints on membership function shapes[END_REF] and [START_REF] Mozelli | A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems[END_REF]) Given a T-S fuzzy system (5) with z ≡ x. Consider the MFs vector h(x) =

h 1 (x) h 2 (x) . . . h r (x) such that h(x) Sh(x) + h(x) w + v ≤ 0 (16)
where S ∈ R r×r , w ∈ R r and v ∈ R are given. If there exist symmetric matrices P i ∈ R n×n , i ∈ Ω r , and matrices

Z ∈ R 2n×2n , M ∈ R n×n , N ∈ R n×n such that Z > 0, P i > 0, i ∈ Ω r (17)
Q i -β ii Z < 0, i ∈ Ω r (18)
Q i + Q j -(β ij + β ji )Z < 0, i, j ∈ Ω r , j > i (19)
with P i having the structure in [START_REF] Tanaka | Stability analysis and design of fuzzy control systems[END_REF],

β ij = S (ij) + w i + v and Q i = -M A i -A i M P i + M -N A i N + N .
Then, the T-S fuzzy system ( 5) is globally asymptotically stable.

Remark 7.

The key difference between Theorems 3 and 4 lies on the extra decision variable Z. Via this slack variable, the shape information of the MFs ( 16), also represented by the scalars β ij , can be exploited to reduce the stability conservatism in Theorem 4. This shape information can be used to deal with relatively complex MF shapes, i.e., with some minor modifications, multiple constraints on the MF shapes are easily handled. However, it usually requires a preliminary optimization step to find the tightest constraint verifying ( 16) that could be applied to the MFs of the studied system.

Besides the two above mainstreams, research efforts have been also devoted to reduce the stability conservatism caused by the sufficiency of fuzzy summations, i.e., the ways how MFs are dropped out to obtain a finite set of LMI conditions. Although numerous results have been proposed to deal with this source of conservativeness [START_REF] Wang | An approach to fuzzy control of nonlinear systems: Stability and design issues[END_REF]- [START_REF] Márquez | Asymptotically necessary and sufficient conditions for Takagi-Sugeno models using generalized non-quadratic parameter-dependent controller design[END_REF], the most prominent approach relies on Pólya's theorem. Based on checking the positiveness of multidimensional matrices, the authors in [START_REF] Sala | Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya's theorem[END_REF] applied Pólya's theorem to derive asymptotically necessary and sufficient LMI-based conditions for the stability and performance of T-S fuzzy systems. It should be stressed that as the homogeneous degree of a multiple summation fuzzy Lyapunov function increases, the conservatism of the stability conditions decreases thanks to the introduction of more degrees of freedom. However, Pólya's theorem based approaches are conceptual rather than implementable since the computational burden swiftly increases in a way that most numerical solvers crash. Different (state or output) feedback control schemes can be applied to T-S fuzzy models [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF], [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF], [START_REF] Guerra | LMI-based relaxed non-quadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form[END_REF]. The most commonly used control law is based on the so-called parallel distributed compensation (PDC) concept, for which the fuzzy controller shares the same fuzzy rules and sets as the T-S fuzzy model. As a result, a PDC controller is obtained from a convex blending of the linear local feedback gains and the MFs of the T-S fuzzy model. Within LMI-based control framework, this control approach was originally proposed in [START_REF] Tanaka | A robust stabilization problem of fuzzy control systems and its application to backing up control of a truck-trailer[END_REF] and was named "PDC" in [START_REF] Wang | An approach to fuzzy control of nonlinear systems: Stability and design issues[END_REF].

For illustration, consider a PDC control law of the form

u(t) = r j=1 h j (z)K j x(t) (20)
Then, the closed-loop T-S fuzzy system can be obtained from ( 3) and ( 20) as follows:

ẋ(t) = r i=1 r j=1 h i (z)h j (z) (A i + B i K j ) x(t) (21)
The basic result is based on the quadratic Lyapunov function [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF]. For the closed-loop stability analysis, its time-derivative along the trajectory of system ( 21) is required to be negative, i.e., V (x) = ẋ P x + x P ẋ < 0, which results in

r i=1 r j=1 h i (z)h j (z)He (A i + B i K j ) P < 0 (22)
Note that ( 22) is expressed in terms of BMI due to the coupling between the Lyapunov matrix P and the feedback gains K j , j ∈ Ω r . However, by applying a simple congruence transformation [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] to [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF], with X = P -1 > 0, it is easy to show that if the following LMI constraint holds:

r i=1 r j=1 h i (z)h j (z)He[A i X + B i Y j ] < 0 (23)
Then, the PDC control law (20) asymptotically stabilizes the T-S fuzzy system (3) with K j = Y j X -1 , for j ∈ Ω r .

Remark 8. A direct consequence of sharing the same MFs for the PDC control and the T-S fuzzy system is that double convex sums appear, see for example [START_REF] Precup | A survey on industrial applications of fuzzy control[END_REF]. Similar to the stability analysis discussed above, several approaches are available to reduce the design conservatism at the expense of a higher computational cost. The key difference between stability analysis and control design is that the latter usually requires additional matrix transformations and/or matrix inversions of decision variables to derive LMI-based formulations. Apart from some special cases, this makes the use of some classes of Lyapunov functions with special structures on their parameter matrices, for instance LILFs and PLFs, very challenging for control design purposes.

Remark 9. Compared to LILFs and PLFs, FLFs as given in ( 8) can be applied more easily to the synthesis problem [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF],

[42], [START_REF] Bernal | Generalized nonquadratic stability of continuous-time Takagi-Sugeno models[END_REF]. However, in this case, for T-S fuzzy systems (5) with z ≡ x, the time-derivatives of the MFs involve the system dynamics, i.e., ḣi (x) = ∂hi ∂x ẋ, which can include the control action u(t) to be designed. Hence, it is not obvious to check a posteriori the validity region of the designed controller [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF]. This is the biggest gap between stability analysis and control design when using fuzzy Lyapunov functions.

Remark 10. It should be stressed that various alternative control schemes with different degrees of design conservatism have been proposed in the T-S fuzzy literature, including non-PDC control [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF], [START_REF] Guerra | LMI-based relaxed non-quadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form[END_REF], [START_REF] Lee | Approaches to extended non-quadratic stability and stabilization conditions for discrete-time Takagi-Sugeno fuzzy systems[END_REF], [START_REF] Xie | Control synthesis of discrete-time T-S fuzzy systems based on a novel non-PDC control scheme[END_REF], fuzzy observer-based control with measurable premises [START_REF] Liu | New approaches to H∞ controller designs based on fuzzy observers for T-S fuzzy systems via LMI[END_REF], [START_REF] Tanaka | Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs[END_REF]- [START_REF] Yoneyama | Design of output feedback controllers for Takagi-Sugeno fuzzy systems[END_REF], with unmeasurable premises [START_REF] Nguang | H∞ fuzzy output feedback control design for nonlinear systems: an LMI approach[END_REF]- [START_REF] Dong | Observer-based output feedback control for discrete-time T-S fuzzy systems with partly immeasurable premise variables[END_REF], fuzzy static output feedback control [START_REF] Chen | Robust static output-feedback stabilization for nonlinear discrete-time systems with time delay via fuzzy control approach[END_REF]- [START_REF] Wei | Reliable output feedback control of discrete-time fuzzy affine systems with actuator faults[END_REF],

etc., and their numerous extensions. Since full state-information is generally not available in practice, observers are usually designed to reconstruct the system state for state-feedback control. T-S fuzzy observer designs with or without performance specifications have been largely developed using Lyapunov stability arguments and LMI techniques, see for instance [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF],

[78]- [START_REF] Guerra | H∞ LMI-based observer design for nonlinear systems via Takagi-Sugeno models with unmeasured premise variables[END_REF] and related references. Note that in the framework of T-S fuzzy systems whose premises are state variables, two following cases are distinguished. If the premise variables are measurable, then the approaches in [START_REF] Liu | New approaches to H∞ controller designs based on fuzzy observers for T-S fuzzy systems via LMI[END_REF], [START_REF] Tanaka | Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs[END_REF], [START_REF] Chen | Mixed H 2 /H∞ fuzzy output feedback control design for nonlinear dynamic systems: an LMI approach[END_REF], [START_REF] Bergsten | Observers for takagi-sugeno fuzzy systems[END_REF], [START_REF] Teixeira | On relaxed LMI-based designs for fuzzy regulators and fuzzy observers[END_REF] can be directly applied. The second case, in which the premise variables are not measurable, is much more challenging. This is due to a mismatch between the T-S fuzzy systems and the corresponding T-S fuzzy observers. In this situation, it is necessary to estimate the premise variables as shown in [START_REF] Nguang | H∞ fuzzy output feedback control design for nonlinear systems: an LMI approach[END_REF]- [START_REF] Dong | Observer-based output feedback control for discrete-time T-S fuzzy systems with partly immeasurable premise variables[END_REF], [START_REF] Li | Diagnostic observer design for T-S fuzzy systems: Application to real-time-weighted fault-detection approach[END_REF].

IV. PIECEWISE MULTIAFFINE SYSTEMS

This section discusses the use of PMA modeling for theoretical studies of the nonlinear system (1).

A. Description of PMA Systems

To introduce the PMA modeling, let us first consider the nonlinear system (1) with u ≡ 0 for simplicity, which results in the following autonomous system:

ẋ = f (x) (24)
Remark 11. It is important to stress that T-S fuzzy model-based approaches require rewriting system [START_REF] Zadeh | The evolution of systems analysis and control: A personal perspective[END_REF] in the form ẋ = A(z)x, where the premise vector z regroups all nonlinear terms involved in the matrix A(•). Up to now, no systematic procedure is available to select, among the infinite possibilities, the "best" parameterization f (x) = A(z)x guaranteeing to achieve stability and other performance specifications. Therefore, the issue on the non-uniqueness of T-S fuzzy representation still remains open [START_REF] Sala | On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems[END_REF]. As shown below, we do not have this source of conservatism with PMA model-based approaches since the form ( 24) is directly dealt with. However, PMA modeling leads to appropriation errors. At the present stage of progress, taking into account such errors for stability analysis of the nonlinear system [START_REF] Zadeh | The evolution of systems analysis and control: A personal perspective[END_REF] still remains open in PMA model-based framework.

Without loss of generality, we assume that x i ≤ x i ≤ x i , i ∈ Ω n , where x i and x i denote respectively the upper and lower bounds of the ith entry of x. As a consequence, the state x belongs to the set

R = [x 1 , x 1 ] × . . . × [x n , x n ].
Let us partition the state-space of system (24) as follows:

x j = χ [1] j < χ [2] j < . . . < χ [Nj +1] j = x j , j ∈ Ω n ( 25
)
Let K v = Ω N1+1 × . . . × Ω Nn+1 be the set of multi-indexes corresponding to all the vertices induced by the partition [START_REF] Bonissone | Industrial applications of fuzzy logic at General Electric[END_REF] and K r = Ω N1 × . . . × Ω Nn the set of multi-indexes corresponding to the regions. For i = (i 1 , . . . , i n ) ∈ K r , the region

χ [i1] 1 , χ [i1+1] 1 × . . . × χ [in] n , χ [in+1] n is denoted by R i and K i = {i 1 , i 1 + 1} × . . . × {i n , i n + 1} is the set of multi- indexes corresponding to all vertices of R i . For k ∈ K i , χ k is the vertex of R i whose jth component is defined as χ [kj ]
j , for j ∈ Ω n . For a hyper-rectangle region R i , for i ∈ K r , we consider the following set of fuzzy rules:

IF x 1 (t) is η [k1] 1 (x 1 ) and • • • and x n (t) is η [kn] n (x n ), THEN ẋ(t) is F k , k ∈ K i ,
where F k = f (χ k ) is the singleton vector, and the triangular membership function η

[kj ] j (x j ) with respect to x j (t), for j ∈ Ω n and k j ∈ Ω Nj +1 , is defined as

η [kj ] j (x j ) =                      x j -χ [kj -1] j χ [kj ] j -χ [kj -1] j , if x j ∈ χ [kj -1] j , χ [kj ] j and j ≥ 2 χ [kj +1] j -x j χ [kj +1] j -χ [kj ] j , if x j ∈ χ [kj ] j , χ [kj +1] j and j ≤ N j 0, otherwise. ( 26
)
Note that the MFs (26) satisfy the following properties [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF]:

η [kj ] j (x j ) ≥ 0, ij +1 kj =ij η [kj ] j (x j ) = 1, j ∈ Ω n .
Given x(t) with these membership functions, ẋ(t) can be inferred by taking the weighted average of F k as follows:

ẋ = k∈Ki η k (x)F k , η k (x) = n j=1 η [kj ] j (x j ) (27)
The following expression of x(t) can be derived from that of the triangular MFs [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF]:

x = k∈Ki η k (x)χ k ( 28
)
Based on ( 27) and ( 28), the parametric expression of PMA system on R = i∈Kr R i can be expressed as follows [START_REF] Nguyen | LMI-based stability analysis for piecewise multi-affine systems[END_REF]:

ẋ = k∈Kv η k (x)F k , x = k∈Kv η k (x)χ k (29)
For stability analysis, we assume that the equilibrium x ≡ 0 of system (29) corresponds to the vertex χ k0 of the state-space partition, for a given k 0 ∈ K v . Let K Z be the set of multi-indexes for regions containing the origin which is called zero-regions, and K N Z = K r \K Z (called non-zero regions). We denote also

K * i = K i \{k 0 }, for i ∈ K Z .
Remark 12. Observe in ( 29) that the weights with respect to x j , j ∈ Ω n , in the premises are computed by the multiplication of η

[kj ] j (x j ), k j ∈ Ω Nj +1 . Therefore, the fuzzy reasoning used here is characterized by normalized membership functions, multiplicative weights calculation, and weighted average aggregation [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF]. Note also that system [START_REF] Zadeh | The evolution of systems analysis and control: A personal perspective[END_REF] can be approximated by PMA model [START_REF] Lam | A review on stability analysis of continuous-time fuzzy-model-based control systems: From membership-function-independent to membership-function-dependent analysis[END_REF] with arbitrary accuracy on R by increasing the number of piecewise regions with [START_REF] Bonissone | Industrial applications of fuzzy logic at General Electric[END_REF].

Remark 13. PMA systems are characterized by the triangular MFs defined in [START_REF] Van Der Wal | Application of fuzzy logic control in industry[END_REF]. Other types of MFs for PMA systems, for instance Gaussian MFs or trapezoidal MFs, could be considered. However, the parametric expression (28) of x(t) is crucial for the stability analysis of PMA systems, which is directly derived from triangular MFs. Moreover, note also that triangular MFs are largely employed in fuzzy control systems and in practice [START_REF] Pedrycz | Why triangular membership functions?[END_REF].

Consider a particular case with n = 2. Then, the resulting second-order PMA model [START_REF] Lam | A review on stability analysis of continuous-time fuzzy-model-based control systems: From membership-function-independent to membership-function-dependent analysis[END_REF] with the MFs (26) corresponds to the parametric expressions of the singleton-type fuzzy systems studied in [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF], [START_REF] Sugeno | On improvement of stability conditions for continuous Mamdani-like fuzzy systems[END_REF]. In this case, by eliminating one of the triangular MFs from the parametric expression (29), the following state-space representation of PMA systems can be readily obtained:

     ẋ = A i (x)x + µ i , x ∈ R i , i ∈ K r µ i = k∈Ki η k (0)F k (30)
where µ i = 0, for i ∈ K Z , and η k (0) denotes the value of η k (x) for x = 0. The state-dependent matrix is of the form

A i (x) = η [k1] 1 (x 1 )S(k 1 , •) + η [k1+1] 1 (x 1 )S(k 1 + 1, •) (31)
for k ∈ K i and i ∈ K r . The explicit expressions of S(k 1 , •) and S(k 1 + 1, •), their derivations, and alternative equivalent state-space representations of ( 30)-( 31) can be found in [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF].

Remark 14. For second-order PMA systems, both representations ( 29) and ( 30) are strictly equivalent. Note also that system (30) is found to be a biaffine system. Moreover, with expression (31), the PMA system (30) can be viewed as a piecewise polytopic affine system with two vertex systems:

ẋ = S(k 1 , •)x + µ i and ẋ = S(k 1 + 1, •)x + µ i .

B. Literature Review on PMA Systems

Many advantages of using PMA systems for control purposes were highlighted in [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF]. First, a PMA model ( 29) is easily obtained from the mathematical expression (24) of a nonlinear system as shown above. Second, PMA models have a general approximation capacity of any smooth nonlinear systems. Third, PMA models are simply implemented by means of look-up tables (LUT) which are one of the most widely used practical tools in the industry for model approximation and control implementation, especially in automotive and aerospace engineering. Finally, PMA systems are fully parametric, i.e., both the state vector and its rate can be expressed by parametric expressions in [START_REF] Lam | A review on stability analysis of continuous-time fuzzy-model-based control systems: From membership-function-independent to membership-function-dependent analysis[END_REF]. As shown later, this enables a systematic framework to study PMA systems.

Despite these practical and theoretical advantages, up to 1999 there was no rigorous stability framework on PMA systems.

Almost all papers had been focused on the practical applications of PMA systems rather than their theoretical studies [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF]. By fully taking into account the information of the triangular MFs as functions of state variables, Sugeno first set the theoretical foundations on a quadratic Lyapunov stability framework for both continuous-time and discrete-time PMA systems [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF]. These results were then improved in [START_REF] Sugeno | On improvement of stability conditions for continuous Mamdani-like fuzzy systems[END_REF] to achieve necessary and sufficient stability conditions with respect to a common quadratic Lyapunov function [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF]. The basic idea of the stability analysis in [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF], [START_REF] Sugeno | On improvement of stability conditions for continuous Mamdani-like fuzzy systems[END_REF] consists in using both parametric expressions [START_REF] Lam | A review on stability analysis of continuous-time fuzzy-model-based control systems: From membership-function-independent to membership-function-dependent analysis[END_REF] and the state-space representation [START_REF] Johansson | Piecewise quadratic stability of fuzzy systems[END_REF] to analytically set V (x) > 0, V (x) < 0, for x = 0, as the stability conditions for each piecewise region. Following the same line, Taniguchi and Sugeno proposed in [START_REF] Taniguchi | Stabilization of nonlinear systems based on piecewise Lyapunov functions[END_REF] necessary and sufficient stability conditions with respect to a piecewise Lyapunov function of the form

V (x) = 1 2 x P i x + q i x + r i , i ∈ K r (32)
where the explicit expressions of P i , q i and r i , for i ∈ K r , can be found in [START_REF] Taniguchi | Stabilization of nonlinear systems based on piecewise Lyapunov functions[END_REF]. In addition, the conditions guaranteeing that function ( 32) is a proper Lyapunov function candidate are also discussed therein. Note that the Lyapunov function [START_REF] Feng | Stability analysis of discrete-time fuzzy dynamic systems based on piecewise Lyapunov functions[END_REF] can be considered as a piecewise approximation of an arbitrary Lyapunov function by second-order functions. These preliminary results of Sugeno's group clearly show the great potential of PMA model-based approaches to get rid of the conservativeness issue of T-S fuzzy theory, for which MFs are usually regarded as unknown parameters [START_REF] Sala | On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems[END_REF].

In parallel, without considering the knowledge on the MFs, numerical approaches were developed for stability analysis and control design of PMA control systems in [START_REF] Kim | A new computational approach to stability analysis and synthesis of linguistic fuzzy control system[END_REF], [START_REF] Kim | A new approach to numerical stability analysis of fuzzy control systems[END_REF]. Also based on Lyapunov stability arguments, there is however a fundamental difference compared to the results in [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF], [START_REF] Sugeno | On improvement of stability conditions for continuous Mamdani-like fuzzy systems[END_REF], [START_REF] Taniguchi | Stabilization of nonlinear systems based on piecewise Lyapunov functions[END_REF]. That is, these works essentially focused on the characteristics of PMA controllers performed on T-S fuzzy objective systems. As such, the stability analysis of PMA control systems is embedded into the conventional T-S fuzzy theory, which is not compatible with the original motivation of PMA control systems. Indeed, the final goal in PMA control theory is to provide a stability analysis framework of nonlinear systems by embedding them in PMA systems to avoid the inherent drawbacks of T-S fuzzy control systems [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF].

It should be stressed that all the above-mentioned results on PMA control systems suffer from two major drawbacks [START_REF] Nguyen | LMI-based stability analysis for piecewise multi-affine systems[END_REF]. First, the proposed stability conditions are expressed in terms of nonlinear matrix inequalities, which obviously induce numerical difficulties. Second and more importantly, these conditions are only applicable to second-order PMA systems. Note that statespace representation (30) of PMA systems was exclusively employed to study PMA systems in [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF], [START_REF] Kim | A new computational approach to stability analysis and synthesis of linguistic fuzzy control system[END_REF], [START_REF] Sugeno | On improvement of stability conditions for continuous Mamdani-like fuzzy systems[END_REF], [START_REF] Taniguchi | Stabilization of nonlinear systems based on piecewise Lyapunov functions[END_REF], [START_REF] Kim | A new approach to numerical stability analysis of fuzzy control systems[END_REF].

However, such a representation is very challenging for theoretical study due to its piecewise polytopic affine feature [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF], see also Remark 14. As a consequence, the extensions of these results to high-dimensional PMA systems are not obvious. Due to these restrictive drawbacks, PMA control systems have not yet received much attention from the (fuzzy) control community.

Only very recently, it is demonstrated in [START_REF] Nguyen | LMI-based stability analysis for piecewise multi-affine systems[END_REF] that the specific representation of PMA via parametric expressions (29) enables a simple and effective stability framework. The key idea relies on the fact that within a piecewise region, each of these expressions is uniquely determined by its vertex values and its restriction to the region is a convex combination of these values. Moreover, the piecewise feature of PMA systems is fully exploited via a piecewise Lyapunov function. This led to promising results on stability analysis of PMA systems of any order, and has opened new research perspectives as discussed in the subsequent sections.

C. LMI-Based Stability Analysis for General PMA Systems

Consider a piecewise quadratic Lyapunov function candidate parameterized as follows [START_REF] Johansson | Piecewise quadratic stability of fuzzy systems[END_REF]:

V (x) =    x P i x, for x ∈ R i , i ∈ K Z x Pj x, for x ∈ R j , j ∈ K N Z (33)
where x = x 1 and

   P i = M i T M i , for i ∈ K Z Pj = M j T Mj , for j ∈ K N Z (34)
The constraint matrices are constructed as follows [START_REF] Johansson | Piecewise quadratic stability of fuzzy systems[END_REF]:

Li = L i l i , Mi = M i m i (35) such that Li x ≥ 0, x ∈ R i , i ∈ K r , Mi x = Mj x, x ∈ R i ∩ R j , i, j ∈ K r ,
where l i = 0, m i = 0, i ∈ K Z .

Remark 15. Since the Lyapunov function [START_REF] Campos | New stability conditions based on piecewise fuzzy Lyapunov functions and tensor product transformations[END_REF] combines the power of quadratic functions near an equilibrium point with the flexibility of piecewise functions in the large, it can lead to less conservative results compared to those based on a common quadratic Lyapunov function [START_REF] Nguyen | LMI-based stability analysis for piecewise multi-affine systems[END_REF], [START_REF] Johansson | Piecewise quadratic stability of fuzzy systems[END_REF]. Indeed, the latter can be regarded as a special case of ( 33) by considering M i = I and

L i = 0.
Note that all free parameters of the Lyapunov function V (x) defined in [START_REF] Campos | New stability conditions based on piecewise fuzzy Lyapunov functions and tensor product transformations[END_REF] are collected in the symmetric matrix T and the expression of P i is linear in T . This feature allows for an LMI-based framework to study the stability of PMA systems as shown in the following theorem.

Theorem 5. [7, Theorem 1] Given an PMA system in [START_REF] Lam | A review on stability analysis of continuous-time fuzzy-model-based control systems: From membership-function-independent to membership-function-dependent analysis[END_REF]. If there exist a symmetric matrix T ∈ R N ×N , symmetric matrices with nonnegative entries U q ∈ R 2n×2n and W q ∈ R 2n×2n (for q ∈ K r ), matrices n+1) and Ŷ2j ∈ R 1×(n+1) (for j ∈ K N Z ), satisfying the LMI constraints ( 36), ( 37), ( 38) and [START_REF] Zhang | Relaxed stability conditions for continuous-time T-S fuzzy-control systems via augmented multi-indexed matrix approach[END_REF]. Then, the PMA system ( 29) is asymptotically stable. Moreover, the piecewise quadratic function V (x) defined in ( 33) is a Lyapunov function of this system.

Y 1i ∈ R n×n , Y 2i ∈ R 1×n (for i ∈ K Z ), Ŷ1j ∈ R (n+1)×(
P i -L i U i L i > 0, i ∈ K Z ( 36
) He Y 1i + L i W i L i /2 -Y 1i χ k + P i F k Y 2i -Y 2i χ k < 0, i ∈ K Z , k ∈ K * i (37) Pj -L j U j Lj > 0, j ∈ K N Z (38)
He Ŷ1j + L j W j Lj /2 -Ŷ1j χk + Pj Fk Ŷ2j -Ŷ2j χk < 0, j ∈ K N Z , k ∈ K j (39)
with P i and Pj defined in [START_REF] Mozelli | A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems[END_REF] and

Fk = F k 0 , χk = χ k 1 , k ∈ K j , j ∈ K N Z .
Remark 16. LMI conditions [START_REF] Rhee | A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design[END_REF] and [START_REF] Tognetti | Selective H 2 and H∞ stabilization of Takagi-Sugeno fuzzy systems[END_REF] are imposed for each region to guarantee that (33) is a proper Lyapunov function candidate. The set of LMIs [START_REF] González | A generalised integral polynomial Lyapunov function for nonlinear systems[END_REF] and [START_REF] Zhang | Relaxed stability conditions for continuous-time T-S fuzzy-control systems via augmented multi-indexed matrix approach[END_REF] ensures that the value of the Lyapunov function decreases along any trajectory of the PMA system [START_REF] Lam | A review on stability analysis of continuous-time fuzzy-model-based control systems: From membership-function-independent to membership-function-dependent analysis[END_REF].

Remark 17. For the stability results in Theorem 5, the parametric expressions of both x and ẋ in (28) can be fully exploited via

Finsler lemma [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF], see [START_REF] Nguyen | LMI-based stability analysis for piecewise multi-affine systems[END_REF] for details. Since expression ( 28) is equivalent to that of the triangular MFs ( 26), the information on the MFs can be thus easily taken into account in the theoretical studies of PMA systems to reduce the conservatism. Note that this is not the case of T-S fuzzy systems whose MFs are often considered as uncertainty [START_REF] Sala | On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems[END_REF].

D. Lyapunov-Based LUT Control Design

Following the same modeling procedure as in Section IV-A, the following PMA model of system (1) is straightforwardly obtained on R:

ẋ = k∈Kv η k (x)(F k + G k u), x = k∈Kv η k (x)χ k (40)
where G k = g(χ k ). Assume that the linearized system (A, B) of ( 1) around x = 0 is stabilizable. This means that there exists a linear feedback gain H such that (A + BH) is Hurwitz with

A = ∂f ∂x x=0 , B = ∂g ∂x x=0 (41)
This control gain H can be designed in advance by any linear control technique to guarantee some local closed-loop properties of (1). To stabilize the PMA system (40), let us consider the state-feedback control law of the form

u(x) = k∈Kv η k (x)υ k + Hx (42)
where the control vertex at the origin must be assigned as υ 0 = 0 and other input vertices υ k , with k ∈ K v \ {0}, have to be designed.

Remark 18. The control law ( 42) is composed of two parts. The parametric expression part k∈Kv η k (x)υ k is defined in the same way as x and ẋ, see [START_REF] Chen | Stability analysis and region-of-attraction estimation using piecewise polynomial Lyapunov functions: Polynomial fuzzy model approach[END_REF]. The incorporation of the linear feedback part Hx in ( 42) is obviously crucial to guarantee the closed-loop stability in the case where system ( 1) is open-loop unstable around x = 0. Note that (42) can be easily implemented with look-up-tables which is an outstanding feature for real-time control applications [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF].

Using similar arguments as for Theorem 5, the following theorem provides sufficient conditions to design LUT controller [START_REF] Lee | Local stability analysis of continuous-time Takagi-Sugeno fuzzy systems: a fuzzy Lyapunov function approach[END_REF] for nonlinear systems.

Theorem 6. Given an PMA system in (40) and a stabilizing feedback gain H of the linearized system (A, B) defined in [START_REF] Lee | Relaxed LMI conditions for local stability and local stabilization of continuous-time Takagi-Sugeno fuzzy systems[END_REF]. Assume there exist a symmetric matrix T ∈ R N ×N , symmetric matrices with nonnegative entries U q ∈ R 2n×2n

and

W q ∈ R 2n×2n (for q ∈ K r ), matrices Y 1i ∈ R n×n , Y 2i ∈ R m×n , Y 3i ∈ R 1×n (for i ∈ K Z ), Ŷ1j ∈ R (n+1)×(n+1) , Ŷ2j ∈ R m×(n+1) , Ŷ3j ∈ R 1×(n+1) (for j ∈ K N Z )
, and the control input vertices υ k ∈ R m×1 (for k ∈ K v \ {0}), satisfying the following matrix inequalities ( 43), ( 44), ( 45) and [START_REF] Guerra | LMI-based relaxed non-quadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form[END_REF]. Then, the control law (42) asymptotically stabilizes the PMA system [START_REF] Chen | Stability analysis and region-of-attraction estimation using piecewise polynomial Lyapunov functions: Polynomial fuzzy model approach[END_REF]. Furthermore, the piecewise quadratic function ( 33) is a Lyapunov function of the closed-loop system.

P i -L i U i L i > 0, i ∈ K Z (43) He 
   Γ i P i G q -Y 1i χ k + P i F k + Hυ k Y 2i + H -I/2 υ k -Y 2i χ k Y 3i 0 -υ k υ l /2 -Y 3i χ k     < 0, i ∈ K Z , q ∈ K i , k ∈ K * i , l ∈ K * i (44) Pj -L j U j Lj > 0, j ∈ K N Z (45)
He

    Γj Pj Ĝk -Ŷ1j χk + Pj Fk + Ĥυ k Ŷ2j + Ĥ -I/2 υ k -Ŷ2j χk Ŷ3j 0 -υ k υ l /2 -Ŷ3j χk     < 0, j ∈ K N Z , k ∈ K j , l ∈ K j (46)
where Ĥ = H 0 and

Ĝk

= G k 0 , k ∈ K j , j ∈ K N Z , Γ i = Y 1i + L i W i L i + H H /2, i ∈ K Z , Γj = Ŷ1j + L j W j Lj + Ĥ Ĥ /2, j ∈ K N Z .
Remark 19. The slack matrices U q and W q , for q ∈ K r , are introduced into the conditions of Theorems 5 and 6 through the S-procedure [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. This contributes to reduce the conservatism since the piecewise-region feature of PMA systems can be fully exploited via the constraint matrices defined in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF].

Remark 20. The design conditions in Theorem 6 are expressed in terms of BMIs due to the product υ k υ l involved in [START_REF] Campos | A comparison of different upper-bound inequalities for the membership functions derivative[END_REF] and [START_REF] Guerra | LMI-based relaxed non-quadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form[END_REF]. Note that for each vertex, there are 2 n associated BMI-based conditions to be verified. Hence, the value of υ k , for k ∈ K v \ {k 0 }, should be imposed identical for 2 n conditions concerning the same vertex to avoid sliding modes and chattering phenomena.

V. FUTURE PERSPECTIVES OF FUZZY CONTROL SYSTEMS

This paper provides a concise discussion on the evolution of fuzzy control systems. Through a selective list of references, we present the historical motivations and the current research progress of three types of fuzzy systems: Mamdani-type fuzzy systems, T-S fuzzy systems and PMA (or singleton-type fuzzy) systems. Great advances on both fundamental and application aspects of fuzzy control have been made with a huge number of available publications on the topic. However, many interesting and important issues still remain challenging, which provide fantastic opportunities for the research on fuzzy control systems in the future. Since we believe that within fuzzy control context, only fuzzy model-based approaches enable systematic frameworks for stability analysis and control design of nonlinear systems, below our personal perspectives are focused on T-S fuzzy systems and PMA systems.

A. Open Issues on Takagi-Sugeno Fuzzy Systems

A large research effort has been focused on reducing the conservativeness of stability analysis and control design. As discussed above, this has been done by searching for more general Lyapunov function candidates and clever manipulations to bring the problem to an LMI framework.

Although there are conditions considering the information on the shape of the MFs, see for instance [START_REF] Lam | A review on stability analysis of continuous-time fuzzy-model-based control systems: From membership-function-independent to membership-function-dependent analysis[END_REF], [START_REF] Sala | Relaxed stability and performance conditions for Takagi-Sugeno fuzzy systems with knowledge on membership function overlap[END_REF], we believe that more powerful methodologies could arise with the use of interval type-2 T-S fuzzy modeling techniques [START_REF] Mendel | Type-2 fuzzy sets and systems: An overview[END_REF]- [START_REF] Biglarbegian | On the stability of interval type-2 TSK fuzzy logic control systems[END_REF]. These fuzzy models allow to deal with uncertain grades of membership. Hence, type-2 T-S fuzzy systems are very useful in the cases where exact MFs are difficult to be chosen and/or there is a need to cope with large amounts of uncertainties [START_REF] Hagras | Type-2 FLCs: A new generation of fuzzy controllers[END_REF], [START_REF] Lam | Stability analysis of interval type-2 fuzzy-model-based control systems[END_REF].

However, for stability analysis and control design, the lower and upper MFs have to be simultaneously considered leading to more elaborated manipulations and computational burden [START_REF] Lam | Control design for interval type-2 fuzzy systems under imperfect premise matching[END_REF], [START_REF] Xiao | Stabilization of interval type-2 polynomial-fuzzy-model-based control systems[END_REF]. Possible contributions go toward building less conservative LMI-based conditions for interval type-2 T-S fuzzy models depending on MFs, reducing the gap between type-1 and type-2

T-S fuzzy modeling methodologies. For instance, membership-dependent stability conditions for both type-1 and type-2 T-S fuzzy systems were recently proposed in [START_REF] Yang | Membership-dependent stability conditions for type-1 and interval type-2 T-S fuzzy systems[END_REF]. To this end, it is considered that the MFs belong to a unified space. Then, an extrema-based method is proposed to construct a polyhedron convex hull to enclose the membership distribution in this space.

Despite the interests of the proposed results, much research effort should be still devoted to the stabilization problem, and even to the stability analysis to reduce further the gap between both types of fuzzy systems.

In our opinion, the design of T-S fuzzy controllers/observers that do not share the same MFs or premise variables with T-S fuzzy models is another important topic. This topic should deserve more thorough consideration since it can greatly improve the flexibility of the control/observer design in many cases [START_REF] Nguang | H∞ fuzzy output feedback control design for nonlinear systems: an LMI approach[END_REF]. Also, such a design scheme is involved in current research mainstreams on T-S fuzzy model-based approaches as shown in the two following illustrative examples.

• The first example is concerned with the design of nonlinear networked control systems (NCSs) in which network-induced imperfections, for instance network-induced delay and sampling issues, are intrinsic [START_REF] Li | Observer-based fuzzy control for nonlinear networked systems under unmeasurable premise variables[END_REF], [START_REF] Zhang | Network-induced constraints in networked control systems -A survey[END_REF]. One of the major difficulties in this case is the fact that, when considering the network-induced delay, the information transmitted over the network to the controller is usually delayed, causing a mismatch between the premise variables. This mismatch can be seen as extra constraints in the design of NCSs [START_REF] Oliveira | Improved Takagi-Sugeno fuzzy output tracking control for nonlinear networked control systems[END_REF]- [START_REF] Zhang | Networked fuzzy output feedback control for discrete-time Takagi-Sugeno fuzzy systems with sensor saturation and measurement noise[END_REF]. Moreover, it can be very restrictive to assume that the premise variables of the fuzzy systems and the fuzzy controllers/observers are synchronous if bilateral networks (controller-to-actuator and sensor-to-controller) have to be designed [START_REF] Zhang | Networked fuzzy output feedback control for discrete-time Takagi-Sugeno fuzzy systems with sensor saturation and measurement noise[END_REF]. Then, the asynchronous transmission issue should be considered.

• The second example is related to the fact that in any control system, fault detection, diagnosis and recovery is decisive to have a robust and resilient system operation. In this context, control reconfiguration plays a crucial role to obtain a fault tolerant control (FTC) that can effectively handle severe actuator/sensor faults while still guaranteeing a desired closed-loop performance [START_REF] Lan | Integrated design of fault-tolerant control for nonlinear systems based on fault estimation and T-S fuzzy modeling[END_REF]. To achieve this goal, it is crucial to have effective fault detection and isolation (FDI) schemes to support the control reconfiguration in a proper way [START_REF] Zhang | Bibliographical review on reconfigurable fault-tolerant control systems[END_REF]. One idea behind the control reconfiguration is to redesign the problem for the faulty system by choosing possibly a new structure and adapting the control law to this new scenario. A possible solution for this problem relies on the use of two-step design procedures for which FDI and FTC

C. Takagi -

 Takagi Sugeno Fuzzy Control DesignAn important application of stability theorems is to design stabilizing fuzzy controllers based on T-S fuzzy models of nonlinear plants. The control design of T-S fuzzy models are usually addressed with the following steps: (1) choose a specific form of the control law, (2) find the respective closed-loop T-S fuzzy representation, (3) apply a set of stability analysis conditions to the closed-loop representation, and (4) transform these conditions into LMI-based formulations. The challenge usually lies on the last step since without any transformation at this step, the design conditions are expressed in terms of bilinear matrix inequalities (BMIs) instead of LMIs, leading to numerical difficulties.

schemes are separately dealt with. To this end, virtual actuators/sensors can be used to mask the actuator/sensor faults [START_REF] Odgaard | A benchmark evaluation of fault tolerant wind turbine control concepts[END_REF]. More specifically, an FDI scheme via T-S fuzzy virtual actuator/sensor models have to be designed to achieve the reconfiguration goal. This requires the design of a T-S fuzzy observer-based controller, in which the T-S fuzzy controller and the T-S fuzzy observer possibly do not share the same MFs or it is necessary to estimate their unmeasurable premise variables.

B. Open Issues on Piecewise Multiaffine Systems

Lyapunov-based studies of PMA systems are much more limited compared to those of T-S fuzzy systems. As mentioned previously, this is due to theoretical challenges when dealing with this type of fuzzy systems, especially for high-order systems.

Although the outstanding contributions in [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF], [START_REF] Nguyen | LMI-based stability analysis for piecewise multi-affine systems[END_REF], [START_REF] Sugeno | On improvement of stability conditions for continuous Mamdani-like fuzzy systems[END_REF] throw new light on the research of PMA control systems, several important issues need to be solved in the future.

• How to fully take into account the knowledge of triangular MFs to derive tractable necessary and sufficient stability conditions for high-order PMA systems? Two of the authors have provided a discussion on this issue in [START_REF] Sugeno | On improvement of stability conditions for continuous Mamdani-like fuzzy systems[END_REF], [START_REF] Taniguchi | Stabilization of nonlinear systems based on piecewise Lyapunov functions[END_REF].

However, the stability conditions are only applicable to second-order PMA systems and expressed in terms of nonlinear matrix inequalities.

• How to develop an effective framework to design LUT controllers from the PMA stability results? BMI-based design conditions in Theorem 6 induce numerical difficulties and may be conservative, especially for high-order systems. PMA systems can be also used for LUT-based control implementation of feedback error learning (FEL) scheme [START_REF] Eciolaza | Piecewise bilinear models for feedback error learning: Online feedforward controller design[END_REF]. In FEL control, it is crucial to build an online pseudo-inverse model of the general nonlinear plant for feedforward control purposes.

Up to now, neural networks have been intensively used for this aim. However, due to computational and overfitting reasons, PMA systems are considered as a powerful alternative to neural networks. Despite the practical effectiveness of LUT-based FEL control, more rigorous stability analysis is required in the future.

• How to extend the theoretical results concerning the "classical" PMA system (40) to a wider class of systems such as time-delay PMA systems, descriptor PMA systems, switching PMA systems and so on? The research on this issue depends obviously on the progress of the two previous ones.

• Whether does the stability of PMA system [START_REF] Chen | Stability analysis and region-of-attraction estimation using piecewise polynomial Lyapunov functions: Polynomial fuzzy model approach[END_REF] imply that of the nonlinear system (1)? To solve this issue, the approximation errors should be characterized, and explicitly considered in the stability analysis as highlighted in [START_REF] Nguyen | LMI-based stability analysis for piecewise multi-affine systems[END_REF].

• For a given nonlinear system, how to obtain its "best" PMA model in terms of facilitating both accurate modeling and effective stability/performance analysis, and reducing the numerical complexity (namely the number of piecewise regions)?

The results in [START_REF] Zeng | Approximation accuracy analysis of fuzzy systems as function approximators[END_REF] may bring some interesting ideas to this open issue.

A part from the above open issues, many other challenges and perspectives concerning fuzzy control systems have been raised in numerous publications, for instance [START_REF] Sala | Perspectives of fuzzy systems and control[END_REF], [START_REF] Guerra | Fuzzy control turns 50: 10 years later[END_REF], [START_REF] Feng | A survey on analysis and design of model-based fuzzy control systems[END_REF], [START_REF] Precup | A survey on industrial applications of fuzzy control[END_REF] and references therein. To conclude the paper, we strongly believe that PMA model-based approaches could provide a promising future for fuzzy control, which would have great impacts on the control community. From the theoretical viewpoint, they allow overcoming the conservativeness issue of T-S fuzzy modelbased approaches [START_REF] Nguyen | LMI-based stability analysis for piecewise multi-affine systems[END_REF], [START_REF] Sugeno | On improvement of stability conditions for continuous Mamdani-like fuzzy systems[END_REF]. From the application viewpoint, a systematic LUT-based control approach is of special interest for real-time applications [START_REF] Sugeno | On stability of fuzzy systems expressed by fuzzy rules with singleton consequents[END_REF]. Therefore, PMA based approaches could provide a viable answer to the issue: "Only straightforward in theory plus straightforward in practice are great solutions to change a discipline", recently raised in [START_REF] Guerra | Fuzzy control turns 50: 10 years later[END_REF].