Dalel Jabri

Kevin Guelton
email: kevin.guelton@univ-reims.fr

Noureddine Manamanni

Abdelhafidh Jaadari
email: abdelhafidh.jaadari@univ-reims.fr

Cuong D Chinh

Robust Stabilization of Nonlinear Systems Based on a Switched Fuzzy Control Law

Keywords: Switched fuzzy models, Takagi-Sugeno, uncertainties, external disturbances, switched PDC control law, Linear Matrix Inequalities (LMI), switched Lyapunov function.. 

This paper deals with robust stabilization of nonlinear systems represented by uncertain and disturbed switched Takagi-Sugeno fuzzy systems. First, stabilization of uncertain switched fuzzy systems is considered without external disturbances. A stabilization criterion is proposed as sufficient Linear Matrix Inequality (LMI) conditions. These ones allow designing a switched parallel distributed compensation fuzzy control law based on a candidate switched Lyapunov function. Then, an extension to systems subject to external disturbances is provided based on a H-infinity criterion. To illustrate the effectiveness of the proposed stabilization criterion and controller design approaches, a designed numerical example is studied and some simulations are provided.

INTRODUCTION

With the growing complexity of some control engineering problems, control techniques drawn from linear theory have shown their limits. Among nonlinear theory, new control approaches have appeared in the last decades such as hybrid or fuzzy techniques. A hybrid dynamical system (HDS) consists of continuous (or discrete) time dynamics associated with discrete events following some logical or decision-making rules. For instance, power transmission and distribution, constrained robotic systems and intelligent vehicle highway systems may be considered as HDS. Among HDS, switched linear systems have attracted extensive research, see e.g. [START_REF] Chiou | Stability analysis for a class of switched large-scale time-delay systems via time-switched method[END_REF], [START_REF] Daafouz | Stability Analysis and Control Synthesis for Switched Systems: A Switched Lyapunov Function Approach[END_REF], [START_REF] Fang | Stabilization and Performance Analysis for a Class of Switched Systems[END_REF], [START_REF] Hetel | Stabilization of Arbitrary Switched Linear Systems With Unknown Time-Varying Delays[END_REF], [START_REF] Mansouri | Robust pole placement controller design in LMI region for uncertain and disturbed switched systems[END_REF], [START_REF] Liberzon | Basic problems in stability and design of switched systems[END_REF], [START_REF] Ni | Minimum Dwell Time for Stability and Stabilization of Switched Linear Systems, 7th World Congress on Intelligent Control and Automation[END_REF]). To obtain stability conditions, two techniques are usually employed. Some authors consider the dwell time concept [START_REF] Liberzon | Basic problems in stability and design of switched systems[END_REF], [START_REF] Chiou | Stability analysis for a class of switched large-scale time-delay systems via time-switched method[END_REF], [START_REF] Chiou | Stability analysis for a class of switched large-scale time-delay systems via time-switched method[END_REF]). In these works, authors proved that, when the linear subsystems are Hurwitz, the overall switched system is stable if the time between consecutive switching is sufficiently large. The second technique, generally based on Lyapunov theory, aims at designing a control law able to stabilize the overall linear switched system without considering a particular switching law [START_REF] Hetel | Stabilization of Arbitrary Switched Linear Systems With Unknown Time-Varying Delays[END_REF], [START_REF] Mansouri | Robust pole placement controller design in LMI region for uncertain and disturbed switched systems[END_REF]). Note that, all these studies consider HDS described as a collection of linear systems switching together. However, stability and stabilisation issue for nonlinear switched systems has been seldom treated in the literature [START_REF] Hespanha | Stability of switched systems with average of the dwell-time[END_REF], [START_REF] Palm | Fuzzy switched hybrid systems-Modeling and identification[END_REF]).

Independently to the works on HDS, other studies have focussed on fuzzy modelling and control approaches. Starting from basic fuzzy control techniques [START_REF] Mamdani | Application of fuzzy algorithms for control of a simple dynamic plant[END_REF]), the last three decades have shown Takagi-Sugeno (T-S) fuzzy modelling and control techniques arising [START_REF] Takagi | Fuzzy identification of systems and its application to modelling and control[END_REF], [START_REF] Tanaka | Fuzzy control systems design and analysis. A linear matrix inequality approach[END_REF]). Indeed, T-S fuzzy models have then attracted interest when dealing with nonlinear systems. These are constituted by a set of linear models interconnected by fuzzy membership functions. Thus, using a convenient convex polytopic transformation, a T-S model can match exactly an affine (bounded) nonlinear system in a compact set of the state space [START_REF] Tanaka | Fuzzy control systems design and analysis. A linear matrix inequality approach[END_REF]). Based on the polytopic structure of T-S models, the merit of T-S fuzzy control approaches is that they make possible the extension of some linear concept to the case of nonlinear systems [START_REF] Tanaka | Fuzzy control systems design and analysis. A linear matrix inequality approach[END_REF], [START_REF] Sala | Perspectives of fuzzy systems and control[END_REF], [START_REF] Bouarar | LMI Stability Conditions for Takagi-Sugeno Uncertain Descriptors[END_REF], (2010), [START_REF] Zerar | Linear fractional transformation based H-infinity output stabilization for Takagi-Sugeno fuzzy models[END_REF], [START_REF] Mansouri | Output feedback LMI tracking control conditions with H∞ criterion for uncertain and disturbed T-S models[END_REF], [START_REF] Lendek | Stability Bounds for Fuzzy Estimation and Control[END_REF]). Nevertheless, an inherent drawback remains since the number of fuzzy rules of a TS model increase exponentially with the number of nonlinearities constituting the matched nonlinear system [START_REF] Delmotte | Continous Takagi-Sugeno's models: reduction of the number of LMI conditions in various fuzzy control design techniques[END_REF]). This makes fuzzy controller design and implementation difficult as the complexity of the nonlinear system to be controlled increases.

To outline the problem of rules explosion in T-S modelling, some authors have proposed to combine the merit of switched systems with T-S ones to deal with nonlinear control problems (Othake et al. (2002), Othake et al. (2006[START_REF] Lam | Stability analysis of sampled-data fuzzy controller for nonlinear systems based on switching T-S fuzzy model[END_REF]). To do so, partitioning the state space of a nonlinear system allows defining a switched nonlinear system. Then, inside each partition, a T-S model can be obtained. So, as stated in [START_REF] Yang | Robust controller for a class of uncertain switched fuzzy systems[END_REF]), the resulting switched T-S system inherits some essential features of hybrid systems and maintains all the information and knowledge representation capacity of fuzzy systems. Few papers have studied stabilization issues of switched fuzzy systems based on quadratic approaches [START_REF] Palm | Fuzzy switched hybrid systems-Modeling and identification[END_REF][START_REF] Lam | Stability analysis of sampled-data fuzzy controller for nonlinear systems based on switching T-S fuzzy model[END_REF]), Yand et al. (2008), [START_REF] Ojleska | Switched Fuzzy Systems: Overview and Perspectives[END_REF], [START_REF] Yang | Robust controller for a class of uncertain switched fuzzy systems[END_REF]) or switching Lyapunov function (Othake et al. (2002), Othake et al. ( 2006)). Note that these studies only consider nominal systems and so, they are irrelevant when dealing with robustness of the designed controller. Therefore, a robust controller design has been proposed in [START_REF] Yang | Robust controller for a class of uncertain switched fuzzy systems[END_REF]) for uncertain switched T-S systems. Nevertheless, in the latter study, a classical quadratic Lyapunov approach has been employed leading to conservative results since it needs to check the existence of a common Lyapunov matrix for a set of linear matrix inequalities (LMI) constraints. Following the work on switched linear systems (Dafouz et al. (2002), [START_REF] Fang | Stabilization and Performance Analysis for a Class of Switched Systems[END_REF]), less conservative LMI conditions for T-S switched systems have been provided by employing switched Lyapunov function (Othake et al. (2006)). The aim of this paper is then to extend these works to the case of robust switched fuzzy Parallel Disturbed Compensation (PDC) controller design for the class of uncertain and disturbed switched T-S fuzzy systems.

The paper is organized as follows. In the first section, the class of uncertain and disturbed switched T-S fuzzy systems is depicted as well as the considered switched PDC control law and switched Lyapunov candidate function are presented. After some useful lemmas and notations, the second section presents the main result: a stabilization criterion is proposed as LMI conditions for uncertain switched T-S systems. Then, this result is extended to the class of uncertain switched fuzzy system subject to external disturbances using a H  criterion. Finally, a simulation example, followed by a conclusion, is provided to illustrate the efficiency of the proposed approaches.

PROBLEM STATEMENT

From a nonlinear system to its switched fuzzy representation:

Consider the following nonlinear system:

                              x t f x t f x t x t g x t g x t u t d x t t        (1)
where

  1 2 ( ) ( ) ( ) ( ) n n x t x t x t x t     ,         1 2 m m u t u t u t u t         and   p t    are
respectively the state, the input and the external disturbances vectors.

    n n f x t    ,     n m g x t   
are nonlinear matrices defining the nominal part of (1). Following the way proposed in (Othake et al. ( 2002)) for nominal systems, using the sector nonlinearity (SNL) approach [START_REF] Tanaka | Fuzzy control systems design and analysis. A linear matrix inequality approach[END_REF]) and a convenient state space partitioning, the uncertain and disturbed nonlinear system (1) can be rewritten as an uncertain and disturbed switched Takagi-Sugeno model described as follows:

                          1 1 q r Q q q i q i q i q i qi qi qi x t v x t h x t A A t x t B B t u t G t            (2)
where Q denotes the number of partitioned regions of the state space and q r is the number of rules in each region.

n n qi A    , n m qi B    and n p qi G   
are constant matrices with appropriate dimensions for all 1,..., i r  and 1,..., q Q  .

The matrices

  n n qi A t    ,   n m qi B t   
represent the uncertain norm-bounded (lebesgue measurable) matrices which can be rewritten such that:

        qi aqi a aqi qi bqi b bqi A t H F t N B t H F t N          (
    1 1 q r qi i h x t    and     q v x t are the switched laws defined by:             1 2 1 2 1 region , , , 0 region , , , q q q n q q q q q n q if x t R s s s v x t if x t R s s s           (4) Consider the state vector         1 ... n x x x t t t  , the th q region   1 2
, , ,

q q q n q R s s s  follows:     1 0 , 1 , . . . , 0 0 k kq k if x t s k n if x t          (5)
To illustrate this modelling approach, based on the switched laws (4), the state space partitions of a second order switched T-S system leads to four regions

  1 2 , q q q R s s depicted in Fig.1.
Note that, using the modelling methodology proposed by (Othake et al. ( 2002)), the switched T-S system (2) represents exactly the nonlinear system (1) on a compact set of the state space. That is to say that the robust controller design proposed in the sequel is valid on the whole state space if the nonlinearities of the uncertain system (1) are bounded (global SNL) or, in the contrary, on a restricted region (local SNL), see [START_REF] Ohtake | A construction method of switching Lyapunov function for nonlinear systems[END_REF]) for more details.

1 x 2 x   1 1,1 R   2 0,1 R   3 0, 0 R   Fig. 1.
Example of a second order state space partition.

To lead to the LMI conditions proposed in the next section, an extended state space system can be employed (Othake et al. ( 2006)). Following this way, let us consider a stable autonomous linear system such that:

    ˆx t Cx t   (6)
where

        1 2 ˆˆˆˆn n x t x t x t x t         is a state vector and n n C    is a Hurwitz matrix. Let       x t x t x t      
be an extended state vector, (2) can be extended with (6) leading to:

                          1 1 q r Q q q i q i q i q i qi qi qi x t v x t h x t A A t x t B B t u t G t                   (7) with 0 0 qi qi A A C         , 0 0 0 qi qi A A           , 0 qi qi B B         , 0 qi qi B B           and 0 qi qi G G         . Note that, qi A   and qi B   can be rewritten as follow:         qi aqi a aqi qi bqi b bqi A t H F t N B t H F t N               with 0 aqi aqi H H        , 0 bqi bqi H H        , 0 aqi aqi N N         and 0 bqi bqi N N         .
Note that, as shown in [START_REF] Ohtake | Switching fuzzy controller design based on switching Lyapunov function for a class of nonlinear systems[END_REF]), to lead to LMI conditions, it is convenient to choose

n n C I    
where  is an arbitrary positive scalar and

n n n n I     is a unit matrix.
Moreover, the trajectories of (2) with the initial state

  0 0 x x 
are equal to the first n trajectories of ( 7) with the

initial state   0 1 0 0 T T T n x x        .
Remark: The proposed control approach is dedicated to nonlinear systems (1) (instead of switched nonlinear ones) by rewriting them as switched TS systems (2). Other studies were focused on switched nonlinear systems regarded as sets of TS systems switching together, see e.g. [START_REF] Guelton | H-infinity decentralized static output feedback controller design for large scale Takagi-Sugeno systems[END_REF], Jabri et al. ( 2011)).

Switched PDC controller and switched Lyapunov candidate:

Now, in order to stabilize the switched T-S systems (7), consider the following switched Parallel Distributed Compensation (PDC) control law Ohtake et al. ( 2006)):

            1 1 q r Q q q i q i q q i u t v x t h x t K E x t      (8)
where

2 m n qi K R  
are the gain matrices and

2 2 n n q E R   are
non singular matrices defined such that:

        1 1 2 2 1 1 1 1 2 2 1 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 q q
q q r q r q rq rq q q q q q r q r q rq rq s s s s

s s s s E s s s s s s s s                                                                                    (9) with kq s , for 1,..., k n 
, defined above in (5).

Substituting ( 8) into (7), one obtains the following closedloop uncertain and disturbed system :

                          1 1 q r Q q q i q i q i q i qi qi qj q qi x t v x t h x t A A t B B t K E x t G t                   (10)
The goal is now to design the matrices qi K , for 1,..., i r  and 1,..., q Q  , ensuring the stability of the closed loop system (10). The following results will be obtained through the use of a candidate switched Lyapunov function given by:

                            1 11 1 2 1 1 2 2 12 22 2 1 2 , , , , , , , , , , , , T n T
n T Q Q Q Q n Q x t Px t x t R s s s x t P x t x t R s s s V x t x t P x t x t R s s s                          (11)
where

2 2 n n q P   
, for 1,..., q Q  , are positive definite real matrices.

Note that, to guarantee the continuity of the candidate Lyapunov function (11) on region boundaries [START_REF] Liberzon | Basic problems in stability and design of switched systems[END_REF]), q P can be rewritten such that:

T q q q P E PE  ( 12
)
where P is a definite positive matrix with q E defined above.

Let us assume that for each instant t , only one region can be activated, thus (11) can be rewritten as:

        1 Q T T q q q V x t x t E PE x t       (13)
Therefore, if there exist 0 P  such that ( 13) is strictly decreasing, the closed loop system (10) is stable. The main result will be provided in the next section in terms of LMIs.

Notations and Lemmas:

The following notations will be used in the sequel to clarify the mathematical expression:

        1 1 q r Q vh q qi qi q i A v x t h x t A     ,     1 , Q v q q q E v x t E            1 1 q q r r h h i i i i i i X Y h x t X h x t Y                      .
Note that, assuming only one region is allowable at each instant. Thus one has:

            1 1 1 Q Q Q v v q q q q q q q q q q X Y v x t X v x t Y v x t X Y                    
As usual, a star (*) indicates a transpose quantity in a matrix. The time t will be omitted when there is no ambiguity. I denote identity matrices with appropriate dimensions.

Moreover, the following lemmas will be used in the sequel to derive relaxed LMI stability conditions.

Lemma 1 [START_REF] Tuan | Parametrized linear matrix inequality techniques in fuzzy control design[END_REF]). The inequality :

        1 1 0 q q r r i j i j i j h z t h z t      ( 14
)
is verified if, for all 1,..., i r  and 1,..., j r  , j i  :

0 ii   (15)
and

  1 1 0 1 2 ii ij ji q r        (16)
Lemma 2 (Zhou and Khargonedkar (1988)). Let us consider X and Y two matrices of appropriate dimensions, there exists a positive scalar 0

  such that : 1 T T T T X Y Y X X X Y Y       (17)

ROBUST LMI BASED SWITCHED CONTROLLER DESIGN

In this section, one firstly proposes to study the stabilisation of the system (2) without considering external disturbances (i.e.   0 t   ). Using the above defined notations, the closed loop uncertain switched T-S system is given as follows:

          vh vh vh vh vh v x t A A B B K E x t            (18)
Sufficient LMI conditions for the design of a switched controller (8) guaranteeing the stability of ( 18) are proposed in the following theorem.

Theorem 1. The uncertain switched fuzzy system (2) without external disturbances (   0 t   ) is GAS (globally asymptotically stabilized) using the PDC switched fuzzy control law (8) if there exist the matrices 0

T X X   , qi M , the scalars 0 qi   , 0 qi   satisfying the following LMIs for all 1,..., q Q  , , 1,..., q i j r  and i j  : 0 ii   (19)
and

  1 1 0 1 2 ii ij ji q r       (20) with       1,1 1 * * 0 0 qij qij aqi q qi bqj qj qj N E X I N M I                   and   1,1 1 T T T T T T qij q qi q q qi q qi qj q q qj qi T T T T qi q qi qi q qj q qj qj q XE A E E A E X M B E E B M E H H E E H H E               
Then, the switched PDC controller gain matrices are obtained through the bijective change of variables

1 qi qi K M X   .
Proof. Consider the Lyapunov candidate (13), the closed loop system is asymptotically stable if

       0 T T T T v v v v x t E PE x t x t E PE x t         (21)
Considering (18), inequality ( 21) is verified for all  

x t  if:             0 T T vh vh vh vh vh v v v T v v vh vh vh vh vh v A A B B K E E PE E PE A A B B K E                     (22)
Multiplying left by T v E  and right by 1 v E  , ( 22) becomes:

        1 0 T T T T T T v vh vh v vh vh vh v v v h v h v v v h v h v h E A A E P K B B E P PE A A E PE B B K                     (23)
which can be rewritten in its extended form considering (3) as:

        1 1 0 T T T T T T v vh v v vh v vh vh v v vh vh T T T T T v avh a avh v v avh a avh v T T T T T vh bvh b bvh v v bvh b bvh vh E A E P PE A E K B E P PE B K E N F t H E P PE H F t N E K N F t H E P PE H F t N K                         (24)
Let us consider 1 X P   , left and right multiplying ( 24) by X , one obtains :

        1 1 0 T T T T T T v vh v v vh v vh vh v v vh vh T T T T T v avh a bvh v v avh b avh v T T T T T vh bvh b bvh v v bvh b bvh vh XE A E E A E X M B E E B M XE N F t H E E H F t N E X M N F t H E E H F t N M                         (25) Recall that     T a a F t F t I  and     T b b
F t F t I  , applying lemma 2 and the Schur complement, ( 25) is verified if:

      1,1 1 * * 0 0 0 vhh vhh avh q vh bvh vh vh N E X I N M I                      (26)
with

  1,1 1 T T T T T T vhh v vh v v vh v vh vh v v vh vh T T T T vh v vh vh v vh v vh vh v XE A E E A E X M B E E B M E H H E E H H E                   
Finally, applying Lemma 1 on (26), the proof is completed. ■ Now, the purpose is to extend theorem 1 to the design of robust controllers (8) stabilizing uncertain switched T-S systems (2) subject to external disturbances (   0 t   ). To do so, the following H  criterion is employed to minimize the effect of the external disturbances on the state dynamics:

        2 0 0 tf tf T T x t Wx t dt t t dt         (27)
where

2 2 n n W   
is a weighting positive definite real matrix and  is the disturbances attenuation level.

The result is summarized in the following theorem.

Theorem 2. The uncertain switched fuzzy system (2) subject to external disturbances is GAS using the PDC switched fuzzy control law (8) if there exist the matrices 0 following LMIs for all 1,..., q Q  , , 1,..., q i j r  and i j  :

Minimize  such that :

0 ii   (28) and   1 1 0 1 2 ii ij ji q r        (29) with     1 1 2 * * 0 0 0 0 0 0 0 qij q qij T T qi q q E X W G E PE I                    
and qij  defined in theorem 1.

Therefore, the robust switched PDC controller gain matrices are obtained using the bijective change of variables

1 qi qi K M X  
and the designed control law ensures a H  performance    .

Proof. Consider the Lyapunov candidate function ( 13) and the H  criterion given in ( 27). The close loop system ( 10) is stable and the H  performance  is guaranteed if:

               0 T T T T v v v v T T x t E PE x t x t E PE x t x t Rx t t t               ( 30
)
Considering the same steps as for the proof of theorem 1, ( 30) is verified for all  

x t and   t

 if:   2 * 0 0 0 vhh T T vh v v W G E PE I               (31)
with vhh  defined in (26) (proof of theorem 1) and 1 0 0

0 0 0 0 0 0 T v v XE RE X W             
.

Then, applying the Schur complement, ( 31) is verified if:

    1 1 2 * * 0 0 0 0 0 0 0 vhh vhh v T T vh v v E X W G E PE I                     (32)
Finally, applying lemma 2, one obtains the LMI stability conditions proposed in theorem 2. This ends the proof. ■

NUMERICAL EXAMPLE

Let us consider the following uncertain nonlinear system inspired from the nominal one given in Ohtake et al. ( 2006)):

                                            1 2 1 2 2 1 2 0.
                      (33) with   1 , x d d   and   2 , x d d   , 0.5 d  ,   3 3 2 2 1 2 1 2 1 2 1 2 1 2 5 5 3 f x x x x x xx xx x x         and       1 2 0.7 g x x t x t    .
The state space partition is chosen as in Fig. 1 with four regions

  1 1,1 R ,   2 0,1 R ,   3 1, 0 R and   4 0, 0 R
. Thus, the considered switched law is defined by:

    1 2 1 1 i f0 d ,0 d 0 o t h e r w i s e x x v x t             1 2 2 1 i f d 0 ,0 d 0 o t h e r w i s e x x v x t              1 2 3 1 i f 0 d , d 0 0 o t h e r w i s e x x v x t          and     1 2 4 1 i f d 0 , d 0 0 o t h e r w i s e x x v x t           .
Using the sector nonlinearity approach [START_REF] Tanaka | Fuzzy control systems design and analysis. A linear matrix inequality approach[END_REF]) and the previous state partition, the nonlinear system (33) can be constructed as an uncertain switched TS disturbed system given by:

                          1 1 Q r q q i q i q i q i qi qi qi x t v x t h x t A A t x t B B t u t G t            (34) with 11 13 0 1 -0.246 -0.246 A A         , 12 14 0 1 -1.25 -1.25 A A         , 21 23 0 1 -1.952 -0.246 A A         , 22 24 0 1 0.75 -1.25 A A         , 31 33 0 1 -0.246 -1.952 A A         , 32 34 0 1 -1.25 0.75 A A         41 43 0 1 -1.952 -1.952 A A         , 42 44 0 1 0.75 0.75 A A         ,   11 12 41 42 0 -0.45 B B B B     ,   13 14 43 44 0 -0.7 B B B B     ,   21 23 31 32 0 -0.7 B B B B     ,   22 24 33 34 0 -0.95 B B B B     , 11 12 21 22 31 32 41 42 1 1 G G G G G G G G              
and the external disturbances defined by

    qi aqi aqi A t H F t N   and     qi bqi bqi B t H F t N   with, for 1,..., 4 q  and 1,.., 4 i  ,   0.2 0.15 T aqi bqi H H   and   0.25 0.1 aqi bqi N N   .
Recall that , 1,..., 4 q E q  , are defined in (9) and lead, for the considered partition, to:

1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 E              , 2 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 E              , 3 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 E              , 4 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 E             
.

The membership functions are given by

    qi qi qi h x t w   with:       1 2 1 2 11 13 1 2 1.25 1.25 , , 0, 0 1.004 1.004 1 otherwise f x x x x x x x                    1 2 1 2 12 14 1 2 0.246 0.246 , , 0,0 1.004 1.004 0 otherwise x x f x x x x x                     1 2 1 2 21 23 1 2 0.75 1.25 , , 0,0 2.702 1.004 1 otherwise f x x x x x x x                     1 2 1 2 22 24 1 2
1.952 0.246 , , 0, 0 2.702 1.004 0 otherwise

x x f x x x x x                      1 2 1 2 31 33 1 2
1.25 0.75 , , 0, 0 1.004 2.702 In order to be able to apply the above proposed theorems, the extended form of ( 34) is built by adding a stable linear system (6) with C I   . Hence, the augmented switched fuzzy system yields:

1 otherwise f x x x x x x x                    1 2 1 2 32 34 1 2 0.246 1.952 , , 0, 0 1.004 2.702 0 otherwise x x f x x x x x                     1 2 1 2 41 43 1 2 0.75 0.75 , , 0, 0 2.702 2.702 1 otherwise f x x x x x x x                     1 2 1 2 42 44 1 2 1.952 1.952 , , 0,0 2.702 2.702 0 otherwise x x f x x x x x               
                          1 1 Q r q q i q i q i q i qi qi q x t v x t h x t A A t x t B B t u t G t                   (35)
with 11 13 0 1 0 0 -0.246 -0.246 0 0 0 0 1 0 0 0 0 1

A A                   , 12 14 0 1 0 0 -1.25 -1.25 0 0 0 0 1 0 0 0 0 1 A A                   , 21 23 0 1 0 0 -1.952 -0.246 0 0 0 0 1 0 0 0 0 1 A A                   , 22 24 0 1 0 0 0.75 -1.25 0 0 0 0 1 0 0 0 0 1 A A                   , 31 33 0 1 0 0 -0.246 -1.952 0 0 0 0 1 0 0 0 0 1 A A                   , 32 34 0 1 0 0 -1.25 0.75 0 0 0 0 1 0 0 0 0 1 A A                   , 41 43 0 1 0 0 -1.952 -1.952 0 0 0 0 1 0 0 0 0 1 A A                   , 42 44 0 1 0 0 0.75 0.75 0 0 0 0 1 0 0 0 0 1 A A                   , 11 12 41 42 0 0.45 0 0 B B B B                      , 13 14 43 44 0 0.7 0 0 B B B B                      , 21 23 31 32 0 0.7 0 0 B B B B                      , 22 24 33 34 0 0.95 0 0 B B B B                      and for 1, 2 q  and 1,.., 4 i  , 1 1 0 0 iq G               , 0.2 0.15 0 0 aqi bqi H H                 and   0.25 0.1 0 0 aqi bqi N N     .
In order to show the efficiency of theorem 1, let us consider the system (2) without external disturbances (i.e.   0 t   ).

The Matlab LMI toolbox is used to solve the LMI conditions leading to the synthesis of a switched fuzzy controller given by:

5

CONCLUSION

In this paper, the stabilization of a class of nonlinear systems represented by uncertain and disturbed switched Takagi-Sugeno fuzzy systems has been studied. The interest of this approach is to benefit from the well-known information of T-S systems with association to the characteristic of the switched systems. Moreover, to cope with uncertainties and external disturbances, LMI conditions for robust switched fuzzy PDC controller design have been obtained based on switched quadratic Lyapunov function and a H  criterion. Finally, a numerical example has been provided to illustrate the efficiency of the proposed switched fuzzy PDC controller design methodology.

AKNOWLEDGEMENT

This work has been supported by the GIS 3SGS within the COSMOS2 project, the FEDER and the « région Champagne-Ardenne » within the CPER MOSYP project, the French Ambassy in Tunisia and the CNOUS. The authors would like to thank Mr Marco Mutey for his valuable comments.

Fig. 4 .

 4 Fig. 2. Phase plan of the closed-loop uncertain switched fuzzy system.

Fig. 6 .

 6 Fig. 5. State's dynamics of the closed-loop uncertain and disturbed system, ( 1 x ) dotted line, ( 2 x ) solid line.

Fig. 7 .

 7 Fig. 7. Control law evolution of the closed-loop uncertain switched fuzzy disturbed system.

Fig. 8 .

 8 Fig. 8. Lyapunov function evolution of the closed-loop uncertain switched fuzzy disturbed system.

1, 0 R


Tahar Bouarar

Kevin Guelton
email: kevin.guelton@univ-reims.fr

Noureddine Manamanni

Robust non-quadratic static output feedback controller design for Takagi-Sugeno systems using descriptor redundancy

Keywords: Takagi-Sugeno models, Robust Static Output Feedback Controller, Descriptor Redundancy, Non-Quadratic Stabilization, LMI, H∞ criterion, Unmeasurable premise variables

This work concerns robust static output feedback controller (SOFC) design for uncertain and disturbed Takagi-Sugeno (TS) systems using an H-infinity criterion. The main result is based on a descriptor formulation of the closed-loop dynamics. The proposed approach allows avoiding appearance of crossing terms between the controller's and the TS system's input matrices leading to easier LMI formulation than existing studies in the literature. Moreover, the proposed SOFC design conditions don't require any restrictions on the output equation and allow dealing with unmeasurable premise variables. Indeed, taking advantage of the uncertain TS modeling, nonlinearities associated to unmeasurable premises variables can be reported from the nominal part to the uncertainties. To provide LMIs of less conservatism, the results are conducted in the non-quadratic framework. Finally, two numerical examples and a benchmark of a crane system are proposed to illustrate the efficiency of the SOFC design methodology.

Introduction

Takagi-Sugeno fuzzy systems [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF] have shown their interests since they allow extending some of the linear control concepts to the nonlinear cases (Takagi and Wang, 2001). Indeed, a TS fuzzy model is a collection of linear time invariant systems blended together with nonlinear membership functions. Therefore, convenient control approaches for such systems have been proposed through the concept of Parallel Distributed Compensation (PDC) (Takagi and Wang, 2001;[START_REF] Sala | Perspectives of fuzzy systems and control[END_REF]. PDC controllers design has been studied using a quadratic Lyapunov functions, see (Takagi and Wang, 2001;Takagi et al., 1996;[START_REF] Sala | Perspectives of fuzzy systems and control[END_REF] and references therein. These approaches remain conservative since they need to find common Lyapunov matrices for a set of linear matrix inequalities (LMI) constraints. Thus, many ways have been proposed to relax these conditions. For instance, relaxation schemes have been developed based on rewriting the closed-loop interconnection structure of the considered control plant [START_REF] Xiaodong | New approaches to H∞ controller design based on fuzzy observers for fuzzy T-S systems via LMI[END_REF][START_REF] Tuan | Parametrized linear matrix inequality techniques in fuzzy control design[END_REF]. Other works have considered piecewise Lyapunov functions [START_REF] Johansson | Piecewise quadratic stability of fuzzy systems[END_REF] and, more accurately with the fuzzy aggregation of TS models, through a non-quadratic approach (also called fuzzy Lyapunov approach) [START_REF] Tanaka | A multiple Lyapunov function approach to stabilization of fuzzy control systems[END_REF][START_REF] Guerra | LMI based relaxed nonquadratic stabilizations for nonlinear systems in the Takagi-Sugeno's form[END_REF][START_REF] Rhee | A new Lyapunov function approach for a Takagi-Sugeno fuzzy control system design[END_REF][START_REF] Feng | A Survey on Analysis and Design of Model-Based Fuzzy Control Systems[END_REF][START_REF] Lam | LMI-based stability and performance conditions for continuous-time nonlinear systems in Takagi-Sugeno's form[END_REF][START_REF] Chang | Relaxed stabilization conditions for continuous-time Takagi-Sugeno fuzzy control systems[END_REF][START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF]. For a review of relaxation issues, one can refer to [START_REF] Sala | On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems[END_REF].

Among control theory, regarding to output stabilization of TS fuzzy models, many works have been done for observer-based controller (OBC) design [START_REF] Tanaka | Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs[END_REF][START_REF] Ma | Analysis and design of fuzzy controller and fuzzy observer[END_REF][START_REF] Yoneyama | Output stabilization of Takagi-Sugeno fuzzy systems[END_REF][START_REF] Yoneyama | Design of output feedback controllers for Takagi-Sugeno fuzzy systems[END_REF][START_REF] Guerra | Conditions of output stabilization for nonlinear models in the Takagi-Sugeno's form[END_REF][START_REF] Mansouri | Output feedback LMI tracking control conditions with H∞ criterion for uncertain and disturbed T-S models[END_REF], dynamic output feedback controller (DOFC) design in both the quadratic and the non-quadratic case [START_REF] Li | Dynamic parallel distributed compensation for Takagi-Sugeno fuzzy systems: An LMI approach[END_REF][START_REF] Assawinchaichote | Output feedback control design for uncertain singularly perturbed systems: an LMI approach[END_REF][START_REF] Nguang | Robust H∞ output feedback control design for fuzzy dynamic systems with quadratic stability constraints: An LMI approach[END_REF][START_REF] Zerar | Linear fractional transformation based H-infinity output stabilization for Takagi-Sugeno fuzzy models[END_REF][START_REF] Guelton | Fuzzy Lyapunov LMI based output feedback stabilization of Takagi-Sugeno systems using descriptor redundancy[END_REF][START_REF] Guelton | Dynamic output feedback fuzzy Lyapunov stabilization of Takagi-Sugeno systems -a descriptor redundancy approach[END_REF]. Static Output Feedback Controllers (SOFC) are of some interests for practical applications since they only require available signals from the plant to be controlled and doesn't need any online differential equation solving, so reducing online computational cost [START_REF] Syrmos | Static output feedback -A survey[END_REF]. In [START_REF] Huang | Robust H∞ static output feedback control of fuzzy systems: An ILMI approach[END_REF] and [START_REF] Huang | Static output feedback controller design for fuzzy systems: An ILMI approach[END_REF], SOFC design methodologies for TS fuzzy systems are proposed in terms of Bilinear Matrix Inequalities instead of LMI. In [START_REF] Chadli | Static output feedback for Takagi-Sugeno systems: an LMI approach[END_REF], LMI conditions have been proposed. However, in these studies, some restrictive modeling assumptions have been made such that the output equation must be linear (full column rank matrix) and without direct transfers from the inputs to the outputs. Note that these restrictions are due to the occurrence of crossing terms in the closed-loop formulation [START_REF] Zerar | Linear fractional transformation based H-infinity output stabilization for Takagi-Sugeno fuzzy models[END_REF][START_REF] Syrmos | Static output feedback -A survey[END_REF][START_REF] Zhou | Robust optimal control[END_REF].

Moreover, the results proposed in these previous studies being quadratic, they may suffer from conservatism. Note also that, unmeasurable premises constitute an important problem in static output feedback since it is somewhat difficult to estimate unmeasured variables without introducing a dynamical equation (observer) in the feedback and so reduce the applicative interest of SOFC. This problem has been seldom treated in the literature dealing with SOFC for TS systems and has been the topic of a very recent work where premises selection in SOFC fuzzy structure has been proposed [START_REF] Tognetti | Reduced-order dynamic output feedback control of continuous-time T-S fuzzy systems[END_REF].

In [START_REF] Tanaka | A Descriptor System Approach to Fuzzy Control System Design via Fuzzy Lyapunov Functions[END_REF], a descriptor redundancy formulation has been employed to derive new nonquadratic LMI stability conditions for state feedback PDC controllers. One of the interests of such approach is that it allows decoupling crossing terms in the closed-loop dynamics formulation (Bouarar et al., 2010). Therefore, based on this property, LMI based non-quadratic robust DOFC design has been proposed in [START_REF] Guelton | Fuzzy Lyapunov LMI based output feedback stabilization of Takagi-Sugeno systems using descriptor redundancy[END_REF][START_REF] Guelton | Dynamic output feedback fuzzy Lyapunov stabilization of Takagi-Sugeno systems -a descriptor redundancy approach[END_REF]. Following this way, one of our preliminary studies has dealt with SOF controller design [START_REF] Bouarar | Static output feedback controller design for Takagi-Sugeno systems -A fuzzy Lyapunov LMI approach[END_REF]. In the latter, in aid of the redundancy property, non-quadratic LMI based SOFC design using a fuzzy Lyapunov approach for nominal TS systems has been proposed. However, this preliminary result isn't relevant in the case of unmeasurable premises. Note that the above described studies dealing with continuous time TS models, complementary works has been recently done in the discrete time framework (which is reputed more favorable than the continuous time case for LMI purposes). Indeed, the descriptor redundancy has been recently used for OBC or SOFC design for discrete-time fuzzy models [START_REF] Chang | A descriptor representation approach to observer-based H∞ control synthesis for discrete-time fuzzy systems[END_REF][START_REF] Chadli | LMI solutions for robust static output feedback control of Takagi-Sugeno fuzzy models[END_REF].

The goal of this paper is to propose a descriptor redundancy approach leading to strict LMI based robust non-quadratic SOFC design for the class of continuous time uncertain and disturbed TS systems without any restrictions on the output equations. Taking advantage of the uncertain plant, it will be emphasis through an example how this modeling approach is relevant to deals with unmeasured premise variables for SOFC design. Moreover, the non-quadratic framework will be considered with recent conservatism improvements (Mozelli et al., 2009b). This paper is organized as follows. First, convenient notations and lemma will be described. Then, in section 3, the problem statement of robust SOFC design for TS fuzzy models is presented with highlights on the descriptor redundancy approach and unmeasurable premises. Afterward, relaxed nonquadratic LMI based robust SOFC design conditions are proposed for the considered class of uncertain TS fuzzy models. Then, an extension to uncertain TS models subject to external disturbances is proposed by the use of a H ∞ criterion. Finally, two numerical examples are provided to show the effectiveness of the proposed approach and an engineering example of a crane system is proposed as a benchmark in simulation.

Notations and Lemma

In the sequel, when there is no ambiguity, the time t in a time varying variable will be omitted for space convenience. As usual, in a matrix, ( ) * indicates a symmetrical transpose quantity. Moreover I denotes an identity matrix with appropriate dimension. Let us consider ( ) i h z as scalar convex functions, the matrices i Y and ij T for,

{ } 1, , i r ∈ … and { } 1, , j l ∈ … , with appropriate dimensions, we will denote ( ) 1 r h i i i Y h z Y = = ∑ , ( ) ( ) 1 1 r r hh k i ik k i T h z h z T = = = ∑∑
. Finally, one denotes ( )

1 r i i i h d h z X X dt = ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ = ∑ and ( ) ( ) 1 1 1 r i i i h d h z X X dt - - = ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ = ∑ i .
The following lemmas will be useful in the following sections to lead to and to relax LMI conditions.

Lemma 1 [START_REF] Zhou | Robust stabilization of linear systems with norm-bounded timevarying uncertainty[END_REF]:

For real matrices X and Y with appropriate dimensions and a positive scalar ε , we have:

1 T T T T X Y Y X X X Y Y ε ε - + ≤ + (1)
Lemma 2 [START_REF] Tuan | Parametrized linear matrix inequality techniques in fuzzy control design[END_REF]: Consider the Parameterized Matrix Inequality (PMI) given by: ( ) ( )

1 1 0 r r k i i k k i h z h z = = Ω < ∑∑ (2)
with ( )

0 i h z ≥ and ( ) 1 1 r i i h z = = ∑ .
The PMI (2) is verified if the matrix inequalities (3) and (4) hold:

0 ii Ω < , for all , 1, 2,..., i j r = (3)
( )

1 1 0 1 2 ii ij ji r Ω + Ω + Ω < - , for 1 i j r ≤ ≠ ≤ , (4)
Note that there exist numerous relaxation schemes in the literature which can be used without affecting the mathematical development of the results proposed below, for a review on these techniques, see [START_REF] Sala | On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems[END_REF]. Nevertheless, the one presented in lemma 2 [START_REF] Tuan | Parametrized linear matrix inequality techniques in fuzzy control design[END_REF] constitutes a good compromise between conservatism and computational cost since it doesn't require additional slack variables.

Problem statement

Let us consider the class of uncertain and disturbed TS fuzzy systems described by:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
1 1 r i i i i i i i r i i i i i i i x t h z t A A t x t B B t u t F t y t h z t C C t x t D D t u t G t ϕ ϕ = = ⎧ = + Δ + + Δ + ⎪ ⎪ ⎨ ⎪ = + Δ + + Δ + ⎪ ⎩ ∑ ∑ (5)
where r represents the number of fuzzy rules. ( )

1 1 r i i h z t = = ∑ . n n i A × ∈ , n m i B × ∈ , q n i C × ∈ , q m i D × ∈ , d n i F × ∈ , d q i G × ∈ are real matrices. ( ) n n i A t × Δ ∈ , ( ) n m i B t × Δ ∈ , ( ) q n i C t × Δ ∈ and
( ) q m i D t × Δ ∈
) ( ) ( ) T s s i i f t f t I ≤ .
Let us consider the following non-PDC SOFC:

( ) ( ) ( ) ( ) ( ) ( ) 1 5 1 1 r r i i i i i i u t h z t L h z t W y t - = = ⎛ ⎞ ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ∑ ∑ (6)
where 1 m q L × ∈ and 5

q q i W × ∈
are real gain matrices to be synthesized.

Remark 1: To be coherent with the aim of SOFC which is to control a dynamical system from the only knowledge of measurable signals, the premises vector ( ) z t is required to depend only on the inputs ( ) u t , the outputs ( ) y t and, eventually on measurable state variables. Therefore, in the case where unmeasurable variables ( ) z t exists in the nonlinear model to be controlled, it is still possible to report these nonlinearities from the nominal part to the uncertainties of the relevant uncertain TS model (5) to design a robust non-PDC SOFC (6) composed of membership functions ( ) ( ) i h z t which are strictly independent of ( ) z t . Therefore, the robust static output feedback controller design proposed in the sequel cope with unmeasurable premises variables. The benefit of uncertain TS modeling in the case of unmeasured variables will be illustrated through the example of a crane system provided in section 6.3.

The classical way to write a closed-loop dynamics consists on substituting (6) into (5) leading, with the above defined notations, to:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 5 5 1 1 1 5 5 h h h h h h h h h h h h h h h h x t A B L W I D L W C x t B L W I D L W G t F t ϕ ϕ - - - - - - ⎛ ⎞ = + - ⎜ ⎟ ⎝ ⎠ + - + (7)
where ( )

h h h A A A t = +Δ , ( ) h h h B B B t = +Δ , ( ) h h h C C C t = +Δ and ( ) ( ) h h h D D D t y t = +Δ .
Hence, the closed-loop dynamics (7) involves numerous crossing terms between the gains ( )

1 5 h h L W - and the system's matrices h C , h C Δ , h B , h B Δ , h D and h D Δ .
This leads to a strong difficulty to obtain convenient LMI conditions for the design of SOFC (6). Indeed, previous studies on SOFC design have reduced the problem by making restrictive assumptions such as C common and column full rank matrices, 0 h D = and without uncertainties in the output equation (Huang and Nguang, 2006; Huang and Nguang, 2007; Chadli et al., 2002). In the sequel, we will see how this problem of crossing terms in the closed-loop dynamics may be overcame.

In (Guelton et al., 2008; Guelton et al., 2009), LMI based design for DOFC controllers has been proposed in aid of a descriptor approach. These ones are based on an interesting property called the descriptor redundancy (Tanaka et al., 2007; Bouarar et al., 2010; Chen, 2004). In the present study, taking advantage of a redundancy formulation for decoupling crossing terms occurring in (7), equations

(5) and ( 6) can be easily rewritten as:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 h h h h h h h h h h x t A A t x t B B t u t F t y t y t C C t x t D D t u t G t ϕ ϕ ⎧ = +Δ + +Δ + ⎪ ⎨ = - + + Δ + + Δ + ⎪ ⎩ (8) and ( ) ( ) ( ) ( ) 1 5 0 h h u t u t L W y t - = - + (9)
Note that the redundancy consists on introducing virtual dynamics in the output equations of (8) and in the control law (9). Then, considering the extended state vector ( )

( ) ( ) ( ) T T T T x t x t y t u t ⎡ ⎤ = ⎣
⎦ , the closed-loop dynamics can be expressed by the following descriptor:

( ) ( ) ( ) ( ) ( ) hh h h Ex t A A t x t F t ϕ = +Δ + (10) with 0 0 0 0 0 0 0 0 I E ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , ( )
1 5 0 0 h h hh h h h h A B A C I D L W I - ⎡ ⎤ ⎢ ⎥ = - ⎢ ⎥ ⎢ ⎥ - ⎣ ⎦ , ( ) ( ) () ( ) ( ) 0 0 0 0 0 h h h h h A t B t A t C t D t Δ Δ ⎡ ⎤ ⎢ ⎥ Δ = Δ Δ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ and 0 h h h F F G ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ .
Therefore, ( 5) is stabilized via the control law ( 6) if ( 10) is stable. Thus, the goal is now to provide sufficient LMI stability conditions allowing to find the matrices h L and 5 h W ensuring the stability of ( 10)

. Unlike previous studies on static output feedbackfor continuous time TS models (Huang and Nguang, 2006; Huang and Nguang, 2007; Chadli et al., 2002) where the quadratic stability conditions are not strictly LMI, writing the closed-loop system (10) by the use of descriptor redundancy allows to avoid appearance of crossing terms between the state space matrices and the controller's ones. Therefore, the benefit of this descriptor formulation will be emphasized in the following section since it makes easier to obtain strict LMI based non-quadratic stability conditions without restrictions on the output equation of (5).

LMI based conditions for SOFC design

The main result is presented in this section. Let us first focus on the non-quadratic stabilization of uncertain TS systems (5) but without external disturbances ( ( ) (with ( ) 0 t ϕ = ) is asymptotically stable via the non-PDC SOFC (6) if there exist, for all combinations of 1, 2,..., i r =

0 t ϕ = ).
, 1 i j r ≤ ≠ ≤ and 1, 2,..., k r = , the matrices ( )

1 1 0 T j j W W = > , 5 j W , 7 j W , 8 j W , 9 j W , i L , ij R and the scalars 1a ij ε , 1c ij ε , 7b ij ε , 7d ij ε , 8b ij ε , 8d ij ε , 9b ij ε and 9d ij
ε such that the following LMI conditions are satisfied:

0 ii Γ < (11)
( )

1 1 0 1 2 ii ij ji r Γ + Γ + Γ < - ( 12
) 1 0 k i j W R + ≥ (13)
where

T ij ij ij ij ij ij H ⎡ ⎤ ϒ + Ζ Γ = ⎢ ⎥ Ζ -Ρ ⎢ ⎥ ⎣ ⎦ , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (
)

1 1 7 7 1 1 8 1 7 5 5 8 8 9 7 9 8 9 9 r T T T k i j j i i j j i k ij k T T T T T ij j i i j i j j j i j j i T T T T T j i j j i i j j j AW W A BW W B W R W B CW DW W W DW W D W B W W D L W W W φ = ⎡ ⎤ + + + - + * * ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ϒ = + + -- + + * ⎢ ⎥ ⎢ ⎥ - +- -- ⎢ ⎥ ⎣ ⎦ ∑ , ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 7 8 9 1 7 8 9 0 0 0 0 T T T T a a a b b b b b c c c d d d ij i i ij ij ij i i ij i i ij i i T d d d d ij ij ij i i H H H H H H H H H H H ε ε ε ε ε ε ε ε ⎡ ⎤ + + + + + ⎢ ⎥ ⎢ ⎥ = + ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 1 7 1 8 7 8 9 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a i j b i j c i j b i j ij d i j d i j b i j d i j N W N W N W N W N W N W N W N W ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Ζ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ and 1 7 1 8 7 8 9 9 a b c b d d b d ij ij ij ij ij ij ij ij ij diag I I I I I I I I ε ε ε ε ε ε ε ε ⎡ ⎤ Ρ = ⎣ ⎦ .
Proof: Let us consider the non-quadratic candidate fuzzy Lyapunov function given by:

( ) ( ) ( ) ( ) ( ) 1 T h v x t x t E W x t - = (14)
For ( 16) being a Lyapunov function one needs

( ) ( ) 0 v x t > , ( ) ( ) 0 v x t < .
Thus, classically for descriptor systems, see e.g. [START_REF] Taniguchi | Fuzzy descriptor systems and nonlinear model following control[END_REF][START_REF] Bouarar | LMI Stability Conditions for Takagi-Sugeno Uncertain Descriptors[END_REF], one needs:

( ) ( ) 1 T h h E W W E - - = (15)
Therefore, let us consider

1 2 3 4 5 6 7 8 9 h h h h h h h h h h W W W W W W W W W W ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦
. Multiplying (15), left by T h W and right by h W , one has

T h h W E EW = which leads to ( ) 1 1 0 T h h W W = > (ensuring ( ) ( ) 0 v x t > ), 2 0 h W = and 3 0 h W = .
Then, the closed-loop system (10) is stable if:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 0 T T T h h h v x t x t E W x t x t E W x t x t E W x t - - - = + + < i (16)
Considering ( 10), ( 16) is obviously satisfied if:

( ) ( )( ) ( ) ( ) ( ) ( ) 1 1 0 T T T hh h h h hh h h A A t W W A A t E W - - - + Δ + + Δ + < i (17)
Multiplying left by T h W and right by h W and since 0

T h h W E EW = > , (17) yields: ( ) ( ) ( ) ( ) ( ) 1 0 T T T h h h h h h h h h h h W A A t A A t W EW W W - + Δ + + Δ + < i (18)
It is well-known that ( )

1 h h h h W W W W - = - i
, see e.g. [START_REF] Bouarar | Stabilization of uncertain Takagi-Sugeno descriptors: a fyzzy Lyapunov approach[END_REF]. Thus ( 18) can be rewritten as:

( )

0 hhh hh h t EW Ψ +ΔΨ - < (19)
with

T T hhh h hh hh h W A A W Ψ = + and ( ) ( ) ( ) T T hh h h h h t W A t A t W ΔΨ = Δ + Δ .
Extending hhh Ψ with matrices defined in (10), it yields:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 7 7 8 1 4 7 5 5 8 8 1 3,3 9 5 4 7 6 9 8 T T T h h h h h h h h T T T T T hhh h h h h h h h h h h h h h T T T T T h h h h h h h h h h h h h h A W W A B W W B W B C W W DW W W DW W D W B L W W W W W D L W - ⎡ ⎤ + + + * * ⎢ ⎥ ⎢ ⎥ Ψ = + - + -- + + * ⎢ ⎥ ⎢ ⎥ + - - + + - Ψ ⎢ ⎥ ⎣ ⎦ (20) where ( ) ( ) ( ) ( ) ( ) ( ) 1 1 3,3 5 6 6 5 9 9 T T T T hhh h h h h h h h h L W W W W L W W - - Ψ = + - -
Let us recall that, due to the nature of the candidate Lyapunov function ( 14), 4 5 9 , ,...,

)

1 1 7 7 8 1 7 5 5 8 8 9 7 9 8 9 9 T T T h h h h h h h h T T T T T hh h h h h h h h h h h h h T T T T T h h h h h h h h h A W W A B W W B W B C W D W W W D W W D W B W W D L W W W ⎡ ⎤ + + + * * ⎢ ⎥ ⎢ ⎥ Ψ = + + -- + + * ⎢ ⎥ ⎢ ⎥ - + - -- ⎢ ⎥ ⎣ ⎦ (21)
Now, extending ( )

hh t ΔΨ , it yields: ( ) ( ) ( ) () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (1,1) (2,1) 8 8 9 9 0 hh T T T T d d d d d d hh hh h h h h h h h h T T T T T T T T b b b d d d h h h h h h h h t t t W N f t H H f t N W W N f t H W N f t H ⎡ ⎤ ΔΨ * * ⎢ ⎥ ⎢ ⎥ ΔΨ = ΔΨ + * ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (22) with ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (1,1) 1 1 7 7 T T T T T T T a a a a a a b b b b b b hh h h h h h h h h h h h h h h h h t W N f t H H f t N W W N f t H H f t N W ΔΨ = + + + and ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (2,1) t W N f t H H f t N W H f t N W ΔΨ = + +
Expression ( 22) can be bounded using lemma 1 such that:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (1,1) (2,2) 1 1 9 9 9 9 9 9 0 0 0 hh hh hh hh T T T T b b b d d d h h h h h h h h h h h h t W N N W W N N W ε ε - - ⎡ ⎤ ΔΨ * * ⎢ ⎥ ΔΨ ≤ ΔΨ = ΔΨ * ⎢ ⎥ ⎢ ⎥ + ⎢ ⎥ ⎣ ⎦ (23)
with:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (1,1) 1 7 8 9 1 7 1 1 1 1 1 7 7 7 T T T T a a a b b b b b c c c d d d hh hh h h hh hh hh h h hh h h hh h h T T T a a a b b b hh h h h h hh h h h h H H H H H H H H W N N W W N N W ε ε ε ε ε ε ε ε - - ΔΨ = + + + + + + + and ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 (2,2) 8 9 1 1 1 8 8 8 1 1 7 7 7 8 8 8 T T T T T d d d d c c c b b b hh hh hh h h hh h h h h hh h h h h T T T T d d d d d d hh h h h h hh h h h h H H W N NW W N NW W N N W W N N W ε ε ε ε ε ε - - - - ΔΨ = + + + + +
Note that (23) can be rewritten as:

( )

1 T hh hh hh hh hh H - ΔΨ = + Ζ Ρ Ζ (24) with ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 7 8 9 1 7 8 9 0 0 0 0 T T T T a a a b b b b b c c c d d d hh h h hh hh hh h h hh h h hh h h T d d d d hh hh hh h h H H H H H H H H H H H ε ε ε ε ε ε ε ε ⎡ ⎤ + + + + + ⎢ ⎥ ⎢ ⎥ = + ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 1 7 1 8 7 8 9 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a h h b h h c h h b h h hh d h h d h h b h h d h h N W N W N W N W N W N W N W N W ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Ζ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦
ε ε ε ε ε ε ε ε ⎡ ⎤ Ρ = ⎣ ⎦ .
Therefore, from ( 21) and ( 24), ( 19) is verified if the following condition holds:

( )

1 0 T hh hh hh hh h hh h h H E W - Ψ + + Ζ Ρ Ζ - < (25)
Applying the Schur complement on (25), one obtains:

0 T hh hh h h hh h hh hh H EW ⎡ ⎤ Ψ + - Ζ < ⎢ ⎥ Ζ -Ρ ⎢ ⎥ ⎣ ⎦ (26)
Let us now focus on ( )

1 1 1 r h k k k W h z W = = ∑ included in the term h EW -
. From the convex property of the membership functions ( )

k h z one has ( ) 1 1 r k k h z = = ∑ , thus ( ) 1 0 r k k h z = = ∑
and so, for any fuzzy matrices ( ) ( )

1 1 r r i j i j i j h z h z R = = ∑∑ , one has ( ) ( ) ( ) 1 1 1 0 r r r i j k i j i j k h z h z h z R = = = = ∑∑∑
. Therefore, one can write:

( )( ) ( ) 1 1 1 1 1 r r h k k i j k k i j k k W h z W R W R φ = = = + ≥ + ∑ ∑ (27) with, for 1,..., k r = , 1 0 k i j W R + ≥ and
where k φ are the lower bounds of ( )

k h z .
Finally, from ( 26) and ( 27), after applying lemma 2, ( 26) is satisfied if the conditions proposed in theorem 1 hold. That ends the proof. ■

Remark 2: Equation ( 27) improve the proposed relaxation for non-quadratic TS based stability conditions proposed in (Mozelli et al., 2009b). Indeed, in the latter study a common slack decision variable R has been introduced instead of fuzzy distributed ones ij R , for 1,..., i r = and 1,..., j r = .

Moreover, it also improve the SOFC design conditions proposed in [START_REF] Bouarar | Static output feedback controller design for Takagi-Sugeno systems -A fuzzy Lyapunov LMI approach[END_REF] for TS systems without uncertainties where a particular case of the present relaxation is considered with 1 r ij R W = -. Note also that it can be argue that introducing fuzzy distributed slack variables may increase the number of decision variables and so the computational cost. Nevertheless, with the growing devices computational capabilities, this concern may be considered as less of a drawback and, if a solution cannot be found from theorem 1 due to computational crashes, it is still possible to check if a solution exists by setting ij R R = common.

H∞ controller design

This section aims at extending the previous results to the case of TS fuzzy systems with external disturbances. Hence, considering ( ) 0 t ϕ ≠ and using a H ∞ criterion, the objective is now to stabilize (5) such that the influence of the external disturbance ( ) t ϕ on the output behavior is minimized. Let us consider the following H ∞ criterion (Takagi and Wang, 2001):

( ) ( ) ( ) ( ) ( ) 2 0 0 T T y t y t t t dt λ ϕ ϕ ∞ - ≤ ∫ (28) Recall that ( ) ( ) ( ) ( ) T T T T x t x t y t u t ⎡ ⎤ = ⎣
⎦ , thus (28) can be rewritten as:

( ) ( ) ( ) ( ) ( ) 2 0 0 T T x t Qx t t t dt λ ϕ ϕ ∞ - ≤ ∫ ( 29
)
with 0 0 0 0 0 0 0 0

Q I ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ .
In that case, the stability of the closed-loop system (10) is guaranteed under the constraint (29) if the LMI conditions summarized in the following theorem hold.

Theorem 2: Consider, for all 1, 2,..., k r = , k φ the lower bounds of ( ) k h z . The TS fuzzy model ( 5) is asymptotically stabilized via the non-PDC SOFC (6) and guarantees the attenuation level λ η = if there exist, for all combinations of 1, 2,..., i r =

, 1 i j r ≤ ≠ ≤ and 1, 2,..., k r = , the matrices ( )

1 1 0 T j j W W = > , 5 j W , 7 j W , 8 j W , 9 j W , i L , ij R and the positive scalars 1a ij ε , 1c ij ε , 7b ij ε , 7d ij ε , 8b ij ε , 8d ij ε , 9b ij ε
and 9d ij ε such that the following LMI conditions are satisfied.

Minimize 0 η > such that:

0 ii Θ < (30) ( ) 1 1 0 1 2 ii ij ji r Θ + Θ + Θ < - (31) 1 0 k i j W R + ≥ (32)
where

( ) ( ) ( ) ( ) 5 0 0 0 0 0 0 0 0 0 0 0 0 ij ij ij ij ij j T T i i H W I F G I η * ⎡ ⎤ ⎢ ⎥ ϒ + * * * ⎢ ⎥ ⎢ ⎥ Θ = ⎢ ⎥ Ζ -Ρ ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ - ⎢ ⎥ ⎣ ⎦
and with the matrices ij ϒ , ij H , ij Ζ and ij Ρ defined in theorem 1.

Proof:

The stability of the closed-loop system (10) is guaranteed, under the constraint (29), if:

( ) ( ) ( ) ( ) ( ) ( ) 2 0 T T v x t x t Qx t t t λ ϕ ϕ + - < (33)
That is to say if:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 1 1 1 1 1 2 0 T T T T hh h h h hh h h T T T T T h h h h x t A A t W W A A t EW Q x t t F W x t x t W F t t t ϕ ϕ λ ϕ ϕ - - - - - ⎛ ⎞ ⎜ ⎟ + Δ + + Δ + + ⎝ ⎠ + + - < i ( 34
)
which is obviously satisfied if:

( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 2 * 0 T T T hh h h h hh h h T h h A A t W W A A t EW Q F W I λ - - - - ⎡ ⎤ + Δ + + Δ + + ⎢ ⎥ < ⎢ ⎥ - ⎢ ⎥ ⎣ ⎦ i (35)
Multiplying left by 0 0

T h W I ⎡ ⎤ ⎢ ⎥ ⎣ ⎦
and right by 0

0 h W I ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ , one obtains: ( ) ( ) ( ) ( ) ( ) 1 2 * 0 T T T T h h h h h h h h h h h h h T h W A A t A A t W EW W W W QW F I λ - ⎡ ⎤ + Δ + + Δ + + ⎢ ⎥ < ⎢ ⎥ - ⎢ ⎥ ⎣ ⎦ i ( 36
)
Following the same way as for the proof of theorem 1, ( 36) is satisfied if (32) holds as well as:

( ) ( )

1 2 * 0 T T hh hh hh hh hh h h T h H WQ W F I λ - ⎡ ⎤ ϒ + + Ζ Ρ Ζ + ⎢ ⎥ < ⎢ ⎥ - ⎣ ⎦ (37)
Note that ( )

5 5 0 0 0 0 0 0 0 0 T T h h h h W QW W W ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦
, using the Schur complement and lemma 2, (30) and (31) yield.

That ends the proof. ■

Discussion on non-quadratic approaches limits:

The LMI conditions proposed in theorems 1 and 2 depend on the lower bounds of ( )

k h z for 1,..., k r = .
Even if it is often pointed out as a criticism to fuzzy Lyapunov approaches since these parameters may be difficult to choose, a way to obtain these bounds has been proposed in [START_REF] Tanaka | A multiple Lyapunov function approach to stabilization of fuzzy control systems[END_REF] in some special cases. Moreover, let us recall that this approach remains one of the least conservative in terms of LMI based design (Mozelli et al., 2009b). In [START_REF] Tanaka | A Descriptor System Approach to Fuzzy Control System Design via Fuzzy Lyapunov Functions[END_REF][START_REF] Guerra | A way to improve results for the stabilization of continuous-time fuzzy descriptor models[END_REF][START_REF] Guelton | Dynamic output feedback fuzzy Lyapunov stabilization of Takagi-Sugeno systems -a descriptor redundancy approach[END_REF], a fuzzy Lyapunov candidate function has been reduced leading to relaxed quadratic stability.

Indeed, some elements in the Lyapunov matrix can be set common in order to make the LMIs free of membership function's lower bounds. In the present study, this remains on setting 1 W common matrices in the previous theorems. However, following the latter way, the 'price' to pay for more practical applicability is obviously an increase of the conservatism. An elegant way has also been recently

proposed to overcome the knowledge of the membership function derivative bounds in [START_REF] Bernal | Generalized non-quadratic stability of continuous-time Takagi-Sugeno models[END_REF][START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF]. However, in these studies, the design goal has been reduced to a local view point and lead to complex LMI formulation which are, at this time only available for standard stability analysis and stabilization. Another approach based on line integral Lyapunov functions, which is not investigated in this paper, has been proposed to avoid appearance of membership function derivatives in non-quadratic stability conditions [START_REF] Rhee | A new Lyapunov function approach for a Takagi-Sugeno fuzzy control system design[END_REF][START_REF] Guelton | Comments on fuzzy control systems design via fuzzy Lyapunov functions IEEE Transactions on Systems Man[END_REF], Mozelli 2009a). Therefore, some further research efforts will have to be done to extend these approaches to SOFC design. However, Let us point out that the goal of the present study is not to reach the difficulties of non-quadratic approaches but to overcome and derive LMI based SOFC design for a general class of TS systems (5) without constraining assumptions on output equation and including the most commonly used and effective non-quadratic approach for conservatism reduction.

Numerical examples

In this section, in order to show the efficiency and the applicability of the proposed fuzzy approaches, three examples are considered. The aim of the first one is to compare the conservatism of the approach proposed in theorem 2 without uncertainties, the quadratic stability conditions discussed in the above section ( 1 W set as a common matrix in theorem 1 and 2) and the non-quadratic conditions proposed in our preliminary study [START_REF] Bouarar | Static output feedback controller design for Takagi-Sugeno systems -A fuzzy Lyapunov LMI approach[END_REF] through a numerical example. Then, a second example is devoted to show the effectiveness of the SOFC based design on a 4 th order and 4 rules numerical uncertain and disturbed TS system containing nonlinearities in the output equation. Finally, the benchmark of a crane system is considered in simulation as a third example to illustrate the validity of the proposed approach on an engineering application with unmeasurable premises variables reported as uncertainties.

Example 1:

Let us consider the following uncertain and disturbed TS fuzzy model:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 2 1 i i i i i i i i i i x t h z t Ax t Bu t F t y t h z t C x t D u t G t ϕ ϕ = = ⎧ = + + ⎪ ⎪ ⎨ ⎪ = + + ⎪ ⎩ ∑ ∑ (38) with 1 5 4 1 2 A -- ⎡ ⎤ = ⎢ ⎥ -- ⎣ ⎦ , 2 2 4 10 2 A -- ⎡ ⎤ = ⎢ ⎥ - ⎣ ⎦ , 1 0 10 B ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ , 2 0 3 B ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ , 1 2 10 5 1 C - ⎡ ⎤ = ⎢ ⎥ - ⎣ ⎦ , 2 3 20 7 2 C - ⎡ ⎤ = ⎢ ⎥ -- ⎣ ⎦ , 1 3 1 D ⎡ ⎤ = ⎢ ⎥ - ⎣ ⎦ , 2 2 0.5 D - ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ , 1 2 0 0.25 F F ⎡ ⎤ = = ⎢ ⎥ ⎣ ⎦ , 1 0.5 0.5 G - ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ , ( ) ( ) ( ) 1 2 T x t x t x t = ⎡ ⎤ ⎣ ⎦ , ( ) ( ) ( ) 1 2 T y t y t y t = ⎡ ⎤ ⎣ ⎦ , ( ) ( ) 1 z t y t = and ( ) ( ) ( ) ( ) ( ) ( ) 2 1 1 2 cos 1 h z t y t h z t = = - .
Using the Matlab LMI Toolbox, the attenuation level value corresponding to the quadratic approach ( 1 W set as common matrix and

1 ij R W = -in theorem 2) is 1.2758 λ =
. This solution can be improved since the non-quadratic LMI conditions are reputed of less conservatism. Nevertheless, in that case, the respectively lower bounds 1 φ and 2 φ of ( )

1
h z and ( ) 2 h z , which are difficult to choose in practice, are required. For the sake of generality, one proposes to study the influence of these bounds on the conservatism of the proposed LMI conditions regarding to the above discussed quadratic result and the non-quadratic conditions proposed in our preliminary study [START_REF] Bouarar | Static output feedback controller design for Takagi-Sugeno systems -A fuzzy Lyapunov LMI approach[END_REF]. Thus, the attenuation level has been computed from theorem 2 of the present study and theorem 2 in [START_REF] Bouarar | Static output feedback controller design for Takagi-Sugeno systems -A fuzzy Lyapunov LMI approach[END_REF] for . These results are summarized in Fig. 1. From the latter, one can conclude that, to stabilize (38) without uncertainties, the proposed non-quadratic approach (theorem 2) leads to the lower conservatism results since the H ∞ performances are improved.

Example 2:

Let us consider the following 4 th order-4 rules uncertain and disturbed TS system with nonlinear output equation:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 4 1 4 1 i i i i i i i i i i i i i i x t h z t A A t x t B B t u t F t y t h z t C C t x t D D t u t G t ϕ ϕ = = ⎧ = + Δ + + Δ + ⎪ ⎪ ⎨ ⎪ = + Δ + + Δ + ⎪ ⎩ ∑ ∑ (39) where ( ) ( ) ( ) ( ) ( ) 1 2 3 4 T x t x t x t x t x t = ⎡ ⎤ ⎣ ⎦ , ( ) ( ) ( ) ( ) ( ) 1 2 3 4 T y t y t y t y t y t = ⎡ ⎤ ⎣ ⎦ , ( ) ( ) 1 z t y t ≡ , ( ) ( ) ( ) ( ) 1 1 1 sin 2 y t h z t + = , ( ) ( ) ( ) ( ) 1 2 1 cos 2 y t h z t + = , ( ) ( ) ( ) ( ) 2 1 3 cos 1 2 y t h z t - = , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (
)

4 1 2 3 1 h z t h z t h z t h z t = - + +
and the nominal system's matrices:

1 -10 0.5 -1 0 0.5 3 0 1

1 1 -4 -1 0.5 -1 0 -6 A ⎡ ⎤ ⎢ ⎥ - - ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 2 -3 -1 -1 0 1 3 1.5 0.5 1.5 0 -3.5 0 -0.5 0.5 0 -8 A ⎡ ⎤ ⎢ ⎥ - - ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 3 -5 0 1 0 0.2 4 0 0 1.5 0.5 -1 -1 0.55 -0.5 0 -4 A ⎡ ⎤ ⎢ ⎥ - ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , -15 0.5 1 0 0 5 2.5 0 0.5 0 -1.5 0 -1 0.4 0 -7 A ⎡ ⎤ ⎢ ⎥ - - ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 1 2.5 1 2.8 0 B ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 2 1.5 0.5 5.5 0 B ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ - ⎢ ⎥ ⎣ ⎦ , 3 0.5 0 1 0 B ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 4 0.5 0.15 5.5 0.5 B - ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ - ⎢ ⎥ ⎣ ⎦ , 1 2 1 0 0 0 3 0 0 1 0.5 0 0 0 1 0 0 C - ⎡ ⎤ ⎢ ⎥ - ⎢ ⎥ = ⎢ ⎥ -- ⎢ ⎥ - ⎣ ⎦ , 2 3 2 0 0 1 2 0 0 5.5 0.5 0 0 0.5 1 0 0 C ⎡ ⎤ ⎢ ⎥ - - ⎢ ⎥ = ⎢ ⎥ - ⎢ ⎥ - ⎣ ⎦ , 3 2.5 1.5 0 0 0 1.5 0 0 2 1 0 0 0.5 2 0 0 C - ⎡ ⎤ ⎢ ⎥ - ⎢ ⎥ = ⎢ ⎥ - ⎢ ⎥ - ⎣ ⎦ , 4 1 0.8 0 0 0 0.5 0 0 5.5 0.6 0 0 2.5 2 0 0 C ⎡ ⎤ ⎢ ⎥ - ⎢ ⎥ = ⎢ ⎥ - ⎢ ⎥ ⎣ ⎦ , 1 0.3 0.1 0.1 0.1 D ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 2 0.1 0.2 0 0 D ⎡ ⎤ ⎢ ⎥ - ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 3 0.15 0.2 0.25 0 D ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 4 0.35 0.4 0.15 0 D ⎡ ⎤ ⎢ ⎥ - ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 1 0 0.25 0 0 F ⎡ ⎤ ⎢ ⎥ - ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 2 0.25 0.25 0 0 F - ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 3 0.2 0.5 0.2 0 F ⎡ ⎤ ⎢ ⎥ - ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 4 0.5 0.2 0.6 0 F - ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 1 0.3 0.3 0 0 G - ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 2 0.35 0.2 0.1 0 G ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 3 0.2 0.35 0 0.2 G - ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 4 0.3 0 0.45 0.1 G ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦
, as well as the uncertain matrices ( ) ( )

a a i i a i A t H f t N Δ = , ( ) ( ) b b i i b i B t H f t N Δ = , ( ) ( )
c c i i c i C t H f t N Δ = and
( ) ( ) respectively an external force vector on the cart and an external torque on the pendulum and the parameters given table 1.

d d i i d i D t H f t N Δ = with: 1 0 0.1 0.25 0.1 a H ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 2 0 0.1 0.15 0.1 a H ⎡ ⎤ ⎢ ⎥ - ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 3 0 0.15 0.05 0.1 a H ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 4 0 0.2 0.25 0.3 a H ⎡ ⎤ ⎢ ⎥ - ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 1 0 0.2 0 0 b H ⎡ ⎤ ⎢ ⎥ - ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 2 0 0.1 0 0 b H ⎡ ⎤ ⎢ ⎥ - ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 3 0 0.25 0 0 b H ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 4 0 0.2 0 0.1 b H ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 1 0.1 0.1 0 0.1 c H ⎡ ⎤ ⎢ ⎥ - ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 2 0.1 0.1 0 0.1 c H - ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 3 0 0.2 0 0 c H ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 4 0.2 0.15 0 0 c H - ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 1 0.5 0.5 0 0.2 d H ⎡ ⎤ ⎢ ⎥ - ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 2 0.1 0.1 0.1 0.1 d H - ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 3 0.5 0 0.25 0.1 d H - ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , 4 0.25 0.2 0.3 0 d H - ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , [ ] 1 0.1 0.1 -0.1 0.2 a N = , [ ] 2 -0.1 0.1 0.1 -0.2 a N = , [ ] 3 -0.1 -0.1 0.1 0 a N = , [ ] 1 0 -0.2 0.3 -0.4 a N = , 1 0.1 b N = , 2 0
By considering the inertia matrix ( )

cos cos a c M c b θ θ θ ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ and its inverse ( ) 1 2 2 cos 1 cos cos b c M c a ab c θ θ θ θ - - ⎡ ⎤ = ⎢ ⎥ - - ⎣ ⎦ , (
0 0 1 0 0 0 0 1 0 0 s d s d A k b c g k b k c acg k c θ η θ η θ η θ η θ η θ η θ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ - - ⎢ ⎥ - ⎣ ⎦ , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 3 2 2 2 1 2 0 0 0 0 0 0 0 0 , 0 00 0 00 a a a A H t N bc c θ δ θ η θ θ α δ θ η θ α δ θ η θ η θ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ Δ = = Δ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ - ⎣ ⎦ , ( ) ( ) a t δ θ Δ = , 0 1 0 0 T a N ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , ( ) ( ) ( ) ( ) ( ) 3 2 2 2 2 1 2 0 0 a H bc c θ η θ α η θ α η θ η θ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ - ⎣ ⎦ , ( ) ( ) ( ) 3 1 0 0 B b c θ η θ η θ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ - ⎣ ⎦ , ( ) ( ) ( ) ( ) 3 1 1 0 0 0 0 F b c c a θ η θ η θ η θ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ - ⎢ ⎥ - ⎣ ⎦ , 1 0 0 0 0 1 0 0 C ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦
⎧ = + Δ + + ⎪ ⎨ ⎪ = ⎩ ∑ (46) with 2 1 2 2 2 2 2 2 2 2 2 2 2 2 0 0 1 0 0 0 0 1 0 0 s d s d k b k b c g A ab c ab c ab c k c k c acg ab c ab c ab c β β β β β β ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = - - ⎢ ⎥ - - - ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ - - - ⎣ ⎦ , 2 2 2 2 2 2 2 2 2 0 0 1 0 0 0 0 1 0 0 s d s d k b k b c g A ab c ab c ab c k c k c acg ab c ab c ab c ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = - - ⎢ ⎥ - - - ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ - - - ⎣ ⎦ , 2 2 2 2 2 2 2 2 2 2 2 2 2 0 0 1 0 0 0 0 1 0 0 s d s d k b k b c g A ab c ab c ab c k c k c acg ab c ab c ab c ρ β β β ρ β β β ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = - - ⎢ ⎥ - - - ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ - - - ⎣ ⎦ , 2 4 2 2 2 2 2 2 0 0 1 0 0 0 0 1 0 0 s d s d k b k b c g A ab c ab c ab c k c k c acg ab c ab c ab c ρ ρ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = - - ⎢ ⎥ - - - ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ - - - ⎣ ⎦ , 2 2 2 2 2 2 2 2 2 2 2 2 2 0 0 1 0 0 0 0 1 0 0 s d s d k b k b c g A ab c ab c ab c k c k c acg ab c ab c ab c β β β β β β β β β ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = - - ⎢ ⎥ - - - ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ - - - ⎣ ⎦ , 2 6 2 2 2 2 2 2 0 0 1 0 0 0 0 1 0 0 s d s d k b k b c g A ab c ab c ab c k c k c acg ab c ab c ab c β β β ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = - - ⎢ ⎥ - - - ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ - - - ⎣ ⎦ , 2 2 2 2 2 2 2 2 2 2 2 2 2 0 0 1 0 0 0 0 1 0 0 s d s d k b k b c g A ab c ab c ab c k c k c acg ab c ab c ab c βρ β β β β β ρ β β β ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = - - ⎢ ⎥ - - - ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ - - - ⎣ ⎦ , 2 8 2 2 2 2 2 2 0 0 1 0 0 0 0 1 0 0 s d s d k b k b c g A ab c ab c ab c k c k c acg ab c ab c ab c βρ β β ρ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = - - ⎢ ⎥ - - - ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ - - - ⎣ ⎦ , 3 2 2 2 2 0 0 b B B ab c c ab c β β ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = = ⎢ ⎥ - ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ - ⎣ ⎦ , 2 4 2 2 0 0 b B B ab c c ab c ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = = ⎢ ⎥ - ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ - ⎣ ⎦ , 5 7 2 2 2 2 0 0 b B B ab c c ab c β β β ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = = ⎢ ⎥ - ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ - ⎣ ⎦ , 6 8 2 2 0 0 b B B ab c c ab c β ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = = ⎢ ⎥ - ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ - ⎣ ⎦ , 1 3 2 2 2 2 2 2 2 2 0 0 0 0 b c F F ab c ab c c a ab c ab c β β β β ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = = - ⎢ ⎥ - - ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ - - ⎣ ⎦ , 2 4 2 2 2 2 0 0 0 0 b c F F ab c ab c c a ab c ab c ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = = - ⎢ ⎥ - - ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ - - ⎣ ⎦ , 5 7 2 2 2 2 2 2 2 2 0 0 0 0 b c F F ab c ab c c a ab c ab c β β β β β β ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = = - ⎢ ⎥ - - ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ - - ⎣ ⎦ , 6 8 2 2 2 2 0 0 0 0 b c F F ab c ab c c a ab c ab c β β ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = = - ⎢ ⎥ - - ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ - - ⎣ ⎦ , 2 1 2 2 2 2 2 2 0 0 a bc H ab c c ab c α β α β ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ - ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ - ⎣ ⎦ , 2 2 2 2 2 2 0 0 a bc H ab c c ab c α α ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ - ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ - ⎣ ⎦ , 2 3 2 2 2 2 2 2 0 0 a bc H ab c c ab c α ρ β α ρ β ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ - ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ - ⎣ ⎦ , 2 4 2 2 2 2 0 0 a bc H ab c c ab c α ρ α ρ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ - ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ - ⎣ ⎦ , 2 5 2 2 2 2 2 2 0 0 a bc H ab c c ab c α β α β β ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ - ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ - ⎣ ⎦ , 2 6 2 2 2 2 0 0 a bc H ab c c ab c α α β ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ - ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ - ⎣ ⎦ , 2 7 2 2 2 2 2 2 0 0 a bc H ab c c ab c α ρ β α βρ β ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ - ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ - ⎣ ⎦ , 2 7 2 2 2 2 0 0 a bc H ab c c ab c α ρ α βρ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ - ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ - ⎣ ⎦ .
For simulation purpose, the maximal angular velocity has been set as show respectively the system's state, the control signal and the disturbance evolutions for the initial conditions ( )

0 1 0 0 4 T x π ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ .
It can be conclude that the synthesized robust non-PDC SOFC is correctly controlling the crane system in spite of the presence of external disturbances.

Conclusion

In this paper, the problem of robust static output feedback stabilization for continuous time uncertain and disturbed Takagi-Sugeno models has been considered. A non-PDC static output feedback control law has been proposed and its design has been involved through a fuzzy Lyapunov approach. Thanks to the descriptor redundancy, crossing terms have been avoided in the closed-loop dynamic formulation and so LMI conditions have been obtained without any assumptions on the output equation of the considered TS model. Then, a H ∞ criterion has been employed to derive conditions which ensure a minimal attenuation level of external disturbances. It has been shown that the proposed SOFC based design lead to low conservatism results regarding to previous works. Moreover, it has been emphasis that, thank to the uncertain TS modeling, such approach is suitable for the design of a robust non-PDC static output feedback controller without the need of unmeasurable variables estimations. Finally, two academic examples has been considered to illustrate the efficiency the proposed fuzzy Lyapunov based SOFC design. Table 1. Model parameters of the crane system (example 3).

Figures and tables :

Parameters Designation Value

(

  decision matrices that are free of choice. A way to run to LMI conditions is to choose 4 non-singular and 1 0 h W > . Consequently, (20) becomes:

  to design a robust H ∞ controller that guarantee the attenuation of the external disturbances ( ) t ϕ affecting the system output and state vector. The following SOFC gain matrices, guaranteeing a minimized H ∞ attenuation level 0

  Fig 2, 3 and 4 show respectively the system's state and output responses, the control signal and the disturbance evolutions for the initial conditions (

  A convenient non-PDC SOFC (6) has been designed through theorem 2 and the Matlab LMI toolbox with the bounds of the membership functions sets as 10 i φ = -. The result is given by the following gain matrices for a minimal attenuation level 0). Simulations where performed with and without external disturbance to highlight the efficiency of the H ∞ attenuation. Fig6 and 7

Fig. 1 :

 1 Fig. 1: Attenuation level λ for several values of 1 φ (example 1).

Fig. 2 :Fig. 3 :Fig. 4 :

 234 Fig. 2: Time responses of the T-S system's states with external disturbances (example 2)

Fig. 5 .

 5 Fig.5. Crane system (example3).

Fig. 6 .Fig. 7 .

 67 Fig.6. Evolution of the states of the crane system (example 3).

  40) can be rewritten as:

	2 1 cos	2	(	2	sin	2	cos sin	1	2	cos	)
	2 1 cos	2	(	cos	sin		2 2	cos sin	cos	cos	1	cos	2	)

7 T T T T b b b c c c d d d hh h h h h h h h h h h h h

1 2 1 2 1

Acknowledgements

This work was supported by the French Ministry of Research and the "Région Champagne-Ardennes" within the CPER MOSYP. The authors would also like to thanks Miss Dolly Prane for her support within this study.

SOFC is correctly stabilizing the system and attenuates the external disturbances effect with a minimized H ∞ attenuation level 0.8533 λ = .

Example 3:

Let us consider the crane system depicted in figure 5 with the parameters and variables given in table 1.

From the well-known Lagrange formalism, the motion equations of the crane system are given by:

Note that the robot's velocities ( ) r t and ( ) t θ aren't considered available for measurement. Therefore, from the dynamical model ( 41) and to cope with the goal of designing a robust non-PDC SOFC which doesn't require unmeasurable premise variables (see remark 1), the nonlinear terms

leading to define the 8 following membership functions as ( ) ( ) ( ) ( )