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ISA Transactions 49 (2010) 447461

Contents lists available at ScienceDirect

ISA Transactions

journal homepage: www.elsevier.com/locate/isatrans

Robust fuzzy Lyapunov stabilization for uncertain and disturbedTakagiSugeno descriptors

T. Bouarar, K. Guelton , N. ManamanniCReSTIC, EA3804, University of Reims, Moulin de la House BP1039, 51687 Reims Cedex 2, France

a r t i c l e i n f o

Article history:

Received 17 November 2009Received in revised form15 June 2010

Accepted 23 June 2010

Available online 20 July 2010

Keywords:

TakagiSugeno

Redundancy

Descriptors

Robust fuzzy control

Non-quadratic

Fuzzy Lyapunov function

LMI

H criterion

a b s t r a c t

In this paper, new robust H controller design methodologies for TakagiSugeno (TS) descriptors is

considered. Based on Linear Matrix Inequalities, two different approaches are proposed. The first oneinvolves a classical closed-loop dynamics formulation and the second one a redundancy closed-loopdynamics approach. The provided conditions are obtained through a fuzzy Lyapunov function candidateand a non-PDC control law. Both the classical and redundancy approaches are compared. It is shownthat the latter leads to less conservative stability conditions. The efficiency of the proposed robustcontrol approaches for TS descriptors as well as the benefit of the redundancy approach are shownthrough an academic example. Then, to showthe applicability of the proposedapproaches, thebenchmarkstabilization of an inverted pendulum on a cart is considered.

2010 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction

In the past three decades, TakagiSugeno (TS) fuzzy mod-els [1] have attracted a great of deal interest in the controlcommunity since they are able to approximate some nonlinearsystems based on fuzzy logic paradigm. Indeed, a TS fuzzy modelis a collection of a set of linear ones blended together by nonlinearmembership functions. Moreover, it is well known that an affinenonlinear system can be exactly matched on a compact set of thestate space using for instance the sector nonlinearity approach [2].The main interest of such an approach is that it allows extendingsome of the linear control concepts to the case of nonlinear analy-sis. Taking benefit of that property, TS models have been applied

in various processes[36] and,regardingto theoretical results, sev-eral works have dealt with stability analysis and controller de-sign; see e.g. [711]. These are often based on the well-knownParallel Distributed Compensation (PDC) paradigm [10] and stabil-ity conditions are obtained from the Lyapunov theory in terms ofLinearor Bilinear Matrix Inequalities (LMI, BMI). Sufficient LMIsta-bility conditions have been proposed using a quadratic Lyapunovfunction (see e.g. [2,12] and the references therein). Nevertheless,these studies lead to conservatismsincethe existenceof a common

Corresponding author. Tel.: +33 3 26 91 32 61; fax: +33 3 26 91 31 06.E-mail address: [email protected] (K. Guelton).

Lyapunov matrix has to be checked to ensure the stability of theconsidered systems; see [13] for a review of the conservatismsources. Many ways have been investigated to reduce the con-servatism. Some of them propose to relax quadratic conditionsusing transformations within the summation structure of theclosed-loop TS systems [14,15]. Some others propose to introduceadditional decision variables in the LMI problem [16,17]. More re-cently, another type of Lyapunov function candidate has been pro-posed. In this way, LMI stability conditions and controller synthesisbased on thepiecewise Lyapunov function have been proposed [18,19] but remain irrelevant when analyzing a TS model obtainedfrom the sector nonlinearity approach [2]. With the same inten-tion of reducing the conservatism of LMI conditions, non-quadratic

Lyapunov functions (NQLF) have been consideredfor non-PDC con-troller design in [2025]. Due to its adequacy with the sector non-linearity approach, NQLF has proved a tremendous success withthe TS control community since they share the same fuzzy struc-ture as the model to be analyzed.

The above referenced papers are mostly focusing on standardTS systems (explicit systems) which remain to Ordinary Dif-ferential Equations. A wider class of dynamical systems calleddescriptors, are constituted by a set of Algebraic Differential Equa-tions [2630]. These are useful to represent implicit or singularsystems as well as many physical systems like, for instance, me-chanical [5,6,28] and electrical processes [29]. Despite numerousworks dealing with standard TS systems analysis, fewer stud-ies have been done concerning TS descriptors. Therefore, some

0019-0578/$ see front matter 2010 ISA. Published by Elsevier Ltd. All rights reserved.doi:10.1016/j.isatra.2010.06.003
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448 T. Bouarar et al. / ISA Transactions 49 (2010) 447461

closed problems for standard systems are still open for descrip-tors. Quadratic stability of such systems has been firstly studiedin [31,32]. Robust quadratic stability conditions for uncertain TSdescriptor systems have been proposed in terms of BMI [33] orLMI [3436]. Some relaxed quadratic conditions introducing fuzzyinferred slack variables have been proposed in [37]. And more re-cently, with the intention to reduce the conservatism of LMI baseddescriptor stability analysis, NQLF based analysis have been firstlyproposed in [38].

Thepurpose of this paper reachestwo objectives. The first oneisto extend the previous works to robust fuzzy Lyapunov based non-quadratic controller design for the class of uncertain and disturbedTS descriptors. On the other hand, based on a redundancy prop-erty, new approaches have been proposed to relax and reduce thecomputational cost of LMI conditions for standard TS fuzzy mod-els [39,40]. The main interest of such approaches is that they facil-itate achieving LMI conditions since they avoid the appearance ofcrossing terms in the closed-loop dynamics. Thus, the second con-tribution of this paper aimsat extending redundancyapproaches tothe stability analysis and robust controller design of TS descrip-tors. Moreover the superiority of redundancy approaches will bedemonstrated regarding to classical ones.

The paper is organized as follows. At first, the problem state-ment of classical and redundancy closed-loop descriptor dynam-ics, using a non-PDC control law, is proposed in the next section.In Section 3, both the classical and redundancy non-quadratic ap-proaches for non-PDC controller design are derived in terms ofLMI for the class of uncertain descriptors without external distur-bances. Afterward, an H criterion is employed in Section 4 toextend the proposed results to robust controllers design ensuringthe attenuation of external disturbances. Finally, in the last sectiontwo examples are provided. The first one considers an academicnonlinear system devoted to compare the proposed classical andredundancy approaches in terms of conservatism as well as to il-lustrate the performances of the proposed robust controllerdesign.Thesecondexampleis devoted to show theapplicability of thepro-posed approaches on a realistic nonlinear system. Therefore, thestudy of the well-known benchmark of an inverted pendulum ona cart is proposed.

2. Uncertain TS descriptor systems and problem statement

Let us consider the following bounded nonlinear uncertain anddisturbed descriptor systems represented by:

(E(x(t)) + E(x(t)))x(t) = (A(x(t)) + A(x(t)))x(t) + (B(x(t))

+ B(x(t)))u(t) + W(x(t))(t) (1)

with x(t) Rn, u(t) Rm and (t) Rd respectively the state,the input and the unknown bounded external disturbances vec-tors; E(x(t)) Rnn,A(x(t)) Rnn, B(x(t)) Rnm and W(x(t)) Rdn are norm bounded known nonlinear matrices describing

the nominal part of the considered system; E(x(t)) Rnn,A(x(t)) Rnn and B(x(t)) Rnm are unknown Lebesguemeasurable matrices describing the model uncertainties.

A convenient way to tackle the stabilization of(1) is to rewriteit as a TS fuzzy model. There is many ways to obtain a TS modelfrom a nonlinear one. Note that a well-known systematic way towrite a TS model representing (1) is called the sector nonlinearityapproach [2]. It allows a TS model matching exactly a nonlinearone on a compact set of the state space. In order to cope withthe nonlinear model structure (1), a TS fuzzy model of descriptorsystems has been firstly proposed in [31,32]. This one includesspecific membership structures respectively for the left and theright hand side of the nonlinear descriptor model (1). In this case,one can define respectively l and r the numbers of fuzzy rules in

the left andthe right hand side of the resultingTS fuzzy descriptorgiven by:

lk=1

vk(z(t))(Ek + Ek(t))x(t) =

ri=1

hi(z(t))((Ai + Ai(t))x(t)

+ (Bi + Bi(t))u(t) + Wi(t)) (2)

where vk(z(t)) 0, hi(z(t)) 0 are the membership functions

verifying the following convex sum properties

l

k=1 vk(z(t)) = 1

and ri=1 hi(z(t)) = 1,Ai Rnn, Bi Rnm and Wi Rdnare time invariant matrices, Ek(t) R

nn

, Ai(t) Rnn

andBi(t) R

nm are unknown Lebesgue measurable uncertaintymatrices bounded such that Ek(t) = H

kef

ke (t)N

ke , Ai(t) =

Hiafi

a (t)Nia and Bi(t) = H

ibf

ib(t)N

ib with H

ke , H

ia, H

ib, N

ke , N

ia and N

ib

are known real matrices and fke (t),fi

a (t) and fi

b (t) are unknown

time varying normalized functions such thatf{i,k}T

{e,a,b}(t)f{i,k}

{e,a,b}(t) I.

Remark 1. In this study, one assumes that (2) is regular and im-pulse free [28].

Remark 2. For more details on the well-known TS fuzzy modelrepresentation of nonlinear systems and how to obtain it, thereader can refer to [2]. Moreover, an example is proposed in thelast section of this paper to illustrate how to obtain an uncertainand disturbed TS fuzzy descriptor (2) from a nonlinear system ofthe form (1) using the well-known sector nonlinearity approach.

A modified PDC (Parallel Distributed Compensation) controllaw has been proposed for the quadratic stabilization of TS de-scriptors [32]. In that case, the designed fuzzy controller sharesthe same membership functions regarding to the considered fuzzymodel. Note that, in order to derive non-quadratic stability con-ditions for standard TS fuzzy models, a Lyapunov dependentnonlinear matrix must be introduced in the PDC scheme for LMIpurpose [20,24,33,39]. In the present study, to deal with TS fuzzydescriptors non-quadratic stabilization, one proposes the follow-ing modified non-PDC control law:

u(t) =

l

k=1r

i=1vk(z(t))hi(z(t))Fik

l

k=1

ri=1

vk(z(t))hi(z(t))X1ik

1x(t) (3)

where Fik and X1ik > 0 are real gain matrices with appropriate di-

mensions to be synthesized.

Notations. Along this paper, in order to improve the readabilityof the involved mathematical expressions, the following notationswill be used. Let us consider, for i = 1, . . . , r and k = 1, . . . , l,the scalar membership functions hi(z(t)) and vk(z(t)), thematricesYk, Gi, Tik and Lijk with appropriate dimensions, we will denote:

Yv =

l

k=1

vk(z(t))Yk,

Gh =

ri=1

hi(z(t))Gi,

Thv =

lk=1

ri=1

vk(z(t))hi(z(t))Tik

and

Lhhv =

lk=1

ri=1

rj=1

vk(z(t))hi(z(t))hj(z(t))Lijk.

Asusual a star () indicatesa transpose quantity in a symmetricmatrix.

Following previous studies on descriptor systems [3133], thestability is investigating by considering an extended state vector

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T. Bouarar et al. / ISA Transactions 49 (2010) 447461 449

x(t) =xT(t) xT(t)

T. Thus (2) can be rewritten with the above-

defined notations as:

Ex(t) = Ahvx(t) + Bhu(t) + Wh(t) (4)

with

E = I 00 0 , Ahv =

0 IAh + Ah(t) Ev Ev (t) ,

Bh =

0

Bh + Bh(t)

and Wh =

0

Wh

.

Following the same way, the control law (3) can be rewritten as:

u(t) = Khvx(t) (5)

with Khv =

Fhv (X1hv )

1 0

.Note that two ways are possible to express the closed-loop

dynamics. The first one, usually employed in previous studies [3135,38], is called classical closed-loop dynamics in the presentstudy. That one is obtained by substituting (5) in (4) and is givenby:

Ex(t) = (Ahv BhKhv )x(t) + Wh(t). (6)

In this paper, one proposes another way to express the closed-loop dynamics of TS fuzzy descriptors. That one is called theredundancy closed-loop dynamics. It is obtained by introducinga virtual dynamics in the modified non-PDC control law (5). That isto say, (5) can be rewritten as:

0u(t) = u(t) + Khvx(t) (7)

where 0 Rmm is a zero matrix.Thus, considering a new extended state vector x(t) =

x

T(t)

uT(t)T

, combining (4) and (7), the redundancy closed-loop dyna-mics can be expressed as:

Ex(t) = Ahv x(t) + Wh(t) (8)

with

E =

E 00 0

, Ahv =

Ahv BhKhv I

and Wh =

Wh

0

.

Remark 3. Let us point out that the classical closed-loop dyna-mics (6) involves crossing terms between the gain and the inputmatrices Khv and Bh which constitute a source of conservatismwhen designing a fuzzy controller. For more details and acomplete review of conservatism sources, see [13]. Unlike theclassical closed-loop dynamics, the redundancy closed-loop dy-namics (8) allows decoupling these matrices and so, it leads toless conservatism. This point will be demonstrated and shown inwhat follows. Moreover, note finally that, apart from our prelimi-nary study [38], to the best of the authors knowledge, there are noexisting results in the literature for TS descriptor stabilization inthe non-quadratic framework.

The goal now is to provide Linear Matrix Inequalities (LMI) sta-bility conditions allowing to design a controller (3) stabilizing (2).In the following sections, both sufficient stability conditions us-ing the classical closed-loop dynamics (6) and the redundancyclosed-loop dynamics (8) will be investigated, compared and dis-cussed.

3. LMI based stabilization for uncertain and disturbed TS

descriptors

In this section, non-quadratic stability conditions will beproposed using first the classical closed-loop dynamics (6), then

the redundancy closed-loop dynamics (8). The following lemmawill be useful to prove the LMI conditions proposed in the sequel.

Lemma 1 ([41]). For any real matrices X and Y with appropriatedimensions, there exist a positive scalar such that the followinginequality holds:

XTY + YTX XTX + 1YTY. (9)

3.1. Stabilization based on the classical closed-loop dynamics

LMI non-quadratic stability conditions have been firstly derivedfrom a fuzzy Lyapunov function (FLF) for uncertain TS descriptorsin our preliminary study [38] using a classical closed-loopdynamics described by (6) without external disturbances ((t) =0). In [24], LMI stability conditions of less conservatism have beenproposed for standard TS fuzzy systems in the non-quadraticframework. Based on this approach, the following theoremimproved the LMI conditions proposed in [38] for uncertain TSdescriptor systems.

Theorem 1. Assume that z(t) = 1, . . . , r h(z(t)) and

= 1, . . . , l, v (z(t)) . The uncertain TS descriptor (2) isglobally asymptotically stable via the non-PDC control law (3), if thereexist the matrices X1jk = (X

1jk)

T > 0,X3ij ,X4ij > 0 (or < 0), R1 =

RT1, R2 = RT2 and Fjk, the positive scalars

1ijk,

2ijk,

3ijk and

4ijk such

that the following LMIs are satisfied for all i,j = 1, . . . , r and k =1, . . . , l:

ijk < 0 (10)

X1jk + R1 0 (11)

X1jk + R2 0 (12)

where ijk is as given in Box I.

Proof. Let us consider the following candidate fuzzy Lyapunov

function (FLF):

V(x(t)) = xT(t)E(Xhhv )1x(t). (13)

In what follows, for space convenience, the time t in a time varyingvariable will be omitted when there is no ambiguity.

From (13), one needs:

E(Xhhv )1 = (Xhhv )

TE > 0. (14)

This condition leads, as classical for descriptor systems (see e.g.

[38]), to Xhhv =

X1hv 0

X3hh X4hh

with X1hv = (X

1hv)

T > 0. Moreover,

(Xhhv)1 exists if the matrix X4hh is invertible, i.e. if X

4hh > 0 or

X4hh < 0. Note that the fuzzy interconnection structure of X1hv,X

3hh

and X4hh is chosen for LMI purpose (see below Eq. (22)).

Then, the closed-loop system (6) is stable if:

V(x) = xT

E(Xhhv)1x +xTE(Xhhv )

1 x +xTE

(Xhhv )1x

< 0. (15)

According to (14) and (6), (15) yields:

(Ahv BhKhv)T(Xhhv)

1

+ ((Xhhv )1)T(Ahv BhKhv ) + E

(Xhhv)1x < 0. (16)

Multiplying left and right respectively by XT

hhv and Xhhv , andconsidering (14), (16) becomes:

XT

hhv(AT

hv KT

hvBT

h) + (Ahv BhKhv )Xhhv

+ E(Xhhv )(Xhhv )1Xhhv < 0. (17)

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ijk =

(1,1)ijk () () () () 0

NiaX1

jk 1

ijkI 0 0 0 0

NibFjk 0 2

ijkI 0 0 0

NkeX3ij 0 0

3ijkI 0 0

(X4ij )T +AiX

1jk EkX

3ij BiFjk 0 0 0

(5,5)ijk ()

0 0 0 0 NkeX4ij

4ijkI

(1,1)ijk = X

3ij + (X

3ij )

T

r

=1

(X1k + R1) +

l=1

(X1i + R2)

and

(5,5)ijk = (X

4ij )

TETk EkX4ij +

1ijkH

ia(H

ia)

T + 2ijkHib(H

ib)

T + 3ijkHke (H

ke )

T + 4ijkHke (H

ke )

T.

Box I.

Now, since

(Xhhv )1 =d

dt

(Xhhv)

1Xhhv

(Xhhv )1 (Xhhv)

1

=

(Xhhv )1Xhhv(Xhhv )1

+ (Xhhv)1 Xhhv (Xhhv )

1

(Xhhv )1

= (Xhhv )1 Xhhv(Xhhv )

1 (18)

inequality (17) becomes:

XT

hhv (AT

hv KT

hvBT

h) + (Ahv BhKhv)Xhhv E

Xhhv < 0 (19)

which can be extended, with the matrices defined in (4) and (5),under the condition in Box II.Applying Lemma 1, (20) is satisfied as in Box III.Then, applying the Schur complement, one obtains the inequalityin Box IV.Note that the minimal interconnection structure for (22) is a triple

sum (hhv). This justify the choice made on the interconnectionof the Lyapunov matrices X1hv ,X

3hh and X

4hh. Therefore, since the

membership functions verify the convex sum properties, one has:

X1hv =

rj=1

lk=1

hjvkX1

jk +

rj=1

lk=1

hjvkX1

jk

=

lk=1

rj=1

hjvk

r

=1

hX1k +

l=1

vX1

j

. (23)

Moreover, following the relaxation scheme proposed in [24], oneconsiders R1 and R2 real constant matrices. Therefore, one has

r=1 h(z(t))R1 = 0 and

l =1 v (z(t))R2 = 0 and so, without

loss of generality, (23) can be rewritten such that:

X1hv =

lk=1

rj=1

hjvk

r

=1

h(X1k + R1) +

l=1

v (X1

j + R2)

. (24)

Then, let us consider, for i = 1, . . . , r, i the lower bounds ofhi(z(t)) and, for k = 1, . . . , l, k the lower bounds ofvk(z(t)), (24)can be bounded such that:

X1hv

lk=1

rj=1

hjvk

r

=1

(X1k + R1)

+

l

=1 (X

1j + R2)

(25)

for which, the condition (11) and (12) are necessary.

Now, from (22) and (25), one has

V(x)

ri=1

rj=1

lk=1

hihjvkijk < 0 (26)

with ijk defined in (10).Finally, (26) is sufficiently satisfied if(10) holds. That ends the

proof.

Remark 4. In previous works [38], one has considered:

X1hv

lk=1

rj=1

hjvk

r1=1

(X1k X

1rk)

+

l1 =1

(X1

j X1

jl )

(27)

instead of(25) to derive LMI stability conditions. As shown in [24],this kind of boundary remains conservative and may be easilyimproved. Therefore, extending this works to descriptors systems,Theorem 1 provides less conservative results since (25) obviouslyinclude (27). Indeed, R1 and R2 being free slack matrices, (27) is aparticular case of(25) where R1 = X

1rk and R2 = X

1jl . Note also

that the quadratic cases [34,35,37] are included in Theorem 1 byconsidering X1jk = X

1 common matrix for all i,j and R1 = R2 =

X1.

Remark 5. For i = 1, . . . , r and k = 1, . . . , l, hi(z(t)) andvk(z(t)) are required to be at least C

1. This is obviously satisfiedfor fuzzy models constructed viaa sectornonlinearityapproach [2]if the system (1) is at least C1 or, for instance when membershipfunctions are chosen with a smoothed Gaussian shape.

3.2. Stability conditions basedon redundancyclosed-loop dynamics

Now, LMI conditions for non-quadratic controller (3) designfor uncertain TS descriptor (2) (without external disturbances)being established by Theorem 1 based on the classical closed-loop dynamics (6) approach, one proposes to extend them byconsidering the redundancy closed-loop dynamics (8). The resultis proposed in the following theorem.

Theorem 2. Assume that, z(t) {1, . . . , r} h(z(t)) and {1, . . . , l}, v (z(t)) . The uncertain TS descriptorsystem (2) (with (t) = 0) is globally asymptotically stable viathe non-PDC control law (3) if there exist, for i = 1, . . . , r and fork = 1, . . . , l, the matrices X1

jk

= (X1

jk

)T > 0,X4

ij

,X5

ij

> 0 (or 0 (or < 0), R1 = RT1, R2 = R

T2 and Fik,

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(X3hh)T +X3hh

X1hv ()(X

4hh)

T +AhX1hv EvX

3hh

BhFhv + Hhaf

ha (t)N

haX

1hv

Hve fv

e (t)Nve X

3hh H

hbf

hb (t)N

hb Fhv

(X4hh)

TETv EvX4hh

(X4hh)T(Nve )

T(fve )T(t)(Hve )

T Hve fv

e (t)Nve X

4hh

< 0 (20)

Box II.

(X

3hh)

T +X3hh + (1

hvv )1(X1hv)

T(Nha )TNhaX

1hv

+ (2hhv )1FThv (N

hb )

TNhb Fhv

+ (3hhv )1(X3hh)

T(Nve )TNve X

3hh

X1hv

()

(X4hh)T +AhX

1hv EvX

3hh BhFhv

(X

4hh)

TETv EvX4hh +

1hhvH

ha (H

ha )

T

+ 2hhv Hhb (H

hb )

T + (4hhv)1(X4hh)

T(Nve )TNve X

4hh

+ 3hhv Hve (H

ve )

T + 4hhvHve (H

ve )

T

< 0 (21)

Box III.

X3hh + (X3hh)

T X1hv () () () () 0

NhaX1hv 1hhv I 0 0 0 0

Nhb Fhv 0 2

hhvI 0 0 0

Nve X3hh 0 0

3hhv I 0 0

(X4hh)T +AhX

1hv

EvX3hh BhFhv

0 0 0

(X4hh)TETv EvX4hh+ 1hhvHha (Hha )T + 2hhv Hhb (Hhb )T

+ 3hhvHve (H

ve )

T + 4hhv Hve (H

ve )

T

()

0 0 0 0 Nve X4hh

4hhvI

< 0 (22)

Box IV.

the positive scalars 1ijk, 2ijk,

3ijk,

4ijk,

5ijk,

6ijk and

7ijk such that the

following LMI conditions are satisfied for all i,j = 1, . . . , r andk = 1, . . . , l:

ijk < 0 (28)

X1jk + R1 0 (29)

X1jk + R2 0 (30)

where ijk is as in Box V.

Proof. Let us consider the following candidate fuzzy Lyapunovfunction:

V (x(t)) = xT(t)E

Xhhv

1x(t) (31)

with

E

Xhhv1

=

XhhvT

E > 0. (32)

Considering that x(t) =xT(t) xT(t) uT(t)

T, (32) imposes that

Xhhv =

X1hv 0 0X4hh X5hh X6hh

X7hv X8hv X

9hv

with X1hv = X1hvT > 0.

Note that,

Xhhv

1exists if (X5hh > 0 or X

5hh < 0) and (X

9hv > 0 or

X9hv < 0).The TS descriptor (2) with (t) = 0 is stabilized by (3) if:

V(x) = xT

E

Xhhv

1

x + xTE

Xhhv

1x

+ xTE ( Xhhv)1 x < 0. (33)

Now, from (33), following the same path as for the proof of Theo-rem 1 (see Eqs. (15)(20)), after applying Lemma 1, one obtains theinequality in Box VI. (1,1), (2,2) and (3,3) in Box VI are defined as

follows:

(1,1) = X4hh +

X4hhT

X1hv +

1hhv1

X1hvT

NhaT

NhaX1hv

+

2hhv1

X4hh

T Nve

TNve X

4hh

+

3hhv1

X7hvT

Nhb

TNhbX

7hv ,

(2,2) = EvX5hh

X5hh

TETv + BhX

8hv +

X8hvT

BTh

+ 1hhv Hha

Hha

T+

2hhv + 4hhv

Hve

Hve

T+

3hhv + 5hhv

Hhb

HhbT

+

4hhv1

X5hhT

NveT

Nve X5hh

+

5hhv1

X8hvT

NhbT

NhbX8hv

+ 6hhv

Hve

THve +

7hhv (H

hb )

THhb

and

(3,3) = X9hv + (X9hv )

T + (6hhv )1X6hh

T (Nve )TNve X

6hh

+

7hhv1

(X9hv )T

NhbT

NhbX9hv .

Then, applying the Schur complement, (34) becomes the inequalityin Box VII.Now, similarly to the proof ofTheorem 1 (see inequality (24)), X1hvcan be bounded such that:

X1hv

lk=1

rj=1

hjvk

r

=1

X1k + R1

+

l

=1

X1j + R2

(36)

with,for = 1, . . . , r, = 1, . . . , l X1k +R1 0andX1

j +R2 0.

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ijk =

(1,1)ijk () () () () 0 0 0 0 0

NiaX1

jk 1ijkI 0 0 0 0 0 0 0 0

NkeX4ij 0

2ijkI 0 0 0 0 0 0 0

NibX7

jk 0 0 3ijkI 0 0 0 0 0 0

(5,1)ijk 0 0 0

(5,5)ijk () () () 0 0

0 0 0 0 NkeX5ij 4ijkI 0 0 0 0

0 0 0 0 NibX8

jk 0 5ijkI 0 0 0

(8,1)ijk 0 0 0

(8,5)ijk 0 0 X

9jk + (X

9jk)

T () ()

0 0 0 0 0 0 0 NkeX6ij

6ijkI 0

0 0 0 0 0 0 0 NibX9

jk 0 7ijkI

(1,1)ijk = X

4ij + (X

4ij )

T

r

=1

(X1k + R1) +

l =1

(X1

j + R2)

,

(8,1)ijk = (X

6ij )

T + Fik +X7

jk,

(5,1)ijk = AiX

1jk EkX

4ij + (X

5ij )

T + BiX7

jk, (8,5)ijk = (X

6ij )

TETk + (X9

jk)TBTi +X

8jk and

(5,5)ijk = EkX

5ij (X

5ij )

TETk + BiX8

jk + (X8

jk)TBTi +

1ijkH

ia(H

ia)

T + (2ijk + 4ijk)H

ke (H

ke )

T

+ (3ijk + 5ijk)H

ib(H

ib)

T + 6ijk(Hke )

THke + 7ijk(H

ib)

THib .

Box V.

(1,1) () ()AhX1hv EvX4hh + (X5hh)T + BhX7hv (2,2) ()

(X6hh)T + Fhv +X

7hv

X6hhT

ETv + (X9hv)

TBTh +X8hv (3,3)

< 0 (34)

Box VI.

(1,1)hhv () () () () 0 0 0 0 0

NhaX1hv

1hhvI 0 0 0 0 0 0 0 0

Nve X

4hh 0

2hhv I 0 0 0 0 0 0 0

NhbX7hv 0 0

3hhv I 0 0 0 0 0 0

(5,1)hhv 0 0 0

(5,5)hhv () () () 0 0

0 0 0 0 Nve X5hh

4hhvI 0 0 0 0

0 0 0 0 NhbX8hv 0

5hhvI 0 0 0

(8,1)hhv 0 0 0

(8,5)hhv 0 0 X

9hv + (X

9hv )

T () ()

0 0 0 0 0 0 0 Nve X6hh

6hhv I 0

0 0 0 0 0 0 0 NhbX9hv 0

7hhv I

< 0 (35)

with (1,1)hhv = X

4hh +

X4hh

T X1hv,

(5,1)hhv = AhX

1hv EvX

4hh +

X5hh

T+ BhX

7hv ,

(8,1)hhv = (X

6hh)

T + Fhv + X7hv,

(8,5)hhv = (X

6hh)

TETv +

X9hv

TBTh + X

8hv and

(5,5)hhv = EvX

5hh

X5hhT

ETv + BhX8hv +

X8hvT

BTh + 1hhv H

ha

Hha T

+ 7hhv Hhb

THhb +

2hhv + 4hhv

Hve Hve

T+

3hhv

+ 5hhvHh

bHh

bT + 6

hhvHv

eT Hv

eBox VII.

Thus, considering (35) and (36), the TS descriptor (2) with(t) = 0 is stabilized by (3) if the LMI conditions of Theorem 2hold. That ends the proof.

Remark 6. To ensure the stability of the considered closed-loopdynamics, one has to check the existence of

Xhhv =

X1hv 0

X3hh X4hh

in Theorem 1 or

Xhhv =X

1hv 0 0

X4

hh X5

hh X6

hh

X7hv X8hv X

9hv

in Theorem 2. Note that the redundancy closed-loop dynamicsapproach is introducing some additional slack decision variablesleading to reduce the conservatism of LMI conditions rather thanclassical closed-loop dynamics approach. Moreover, it can beeasily shown that the classical closed-loop dynamics approachis a particular case of the redundancy closed-loop dynamicsone. Indeed, according to the fuzzy Lyapunov function (31) andits symmetric condition (32), the matrices X4ij ,X

5ij ,X

6ij ,X

7jk,X

8jk and

X9jk, for i,j = 1, 2, . . . , r and k = 1, 2, . . . , l, are slack (free ofchoice) decision variables. Indeed, the only necessary condition for(31) to be a Lyapunov candidate function is X1jk = (X

1jk)

T > 0 for

j = 1, 2, . . . , r and k = 1, 2, . . . , l. Thus, replacing the matricesX4ij ,X5ij and X

7jk respectively by X

3ij ,X

4ij and Fjk, then considering

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X6ij = 0,X8

jk = 0 and X9

jk = 0, one obtains the conditions ofTheorem 1 from the ones ofTheorem 2.

Remark 7. Descriptor redundancy has been firstly used in [39] forstandard state space TS fuzzy models without uncertainties. Inthat case, the authors show that it allows reducing the computa-tional cost of LMI based design since it reduces the number of LMIsregarding to classical approaches. Note that, when dealing about

descriptor systems with different membership structure for theleft and the right hand side of the TS fuzzy model, i.e. vi(z) =hi(z), the numberof LMI tobe solvedremains the samein boththecases (Theorems 1 and 2). Therefore, in thepresent study, thebene-fit of the descriptor redundancy is not to reduce the computationalcost but to reduce the conservatism.

4. H based fuzzy controller design

The above LMI stability conditions, proposed in both the Theo-rems1 and 2, standfor (t) = 0. This section aims at extendingtheprevious results to robust non-PDC controller design for uncertainand disturbed TS descriptors using an H criterion. The goal isto stabilize (1) such that the influence of the external disturbance(t) regarding to the state dynamics is minimized. To do so, let us

consider the following H criterion:tft0

xT(t)Qx(t) 2tf

t0

T(t)(t) 0 (37)

where t0 is the initial time, tf is the final time, is the attenuationlevel and Q > 0 is a weighting symmetric matrix.

Recall that two ways have been investigated for TS descriptorsstabilization. The first one involved the classical closed-loopdynamics (6) considering the extended state vector x(t) =xT(t) xT(t)

Tand the second one involved the redundancy

closed-loop dynamics (8) with the extended state vector x(t) =x

T(t) uT(t)

T. Thus, once again, two paths can be prospected for

the H analysis of the closed-loop dynamics.

4.1. H non-PDC controllerdesign based on the classical closed-loopdynamics

Letus consider theextended state vectorx(t) = [xT(t) xT(t)]T,the H criterion (37) can be rewritten as:tf

t0

xT(t)Q x(t) 2

tft0

T(t)(t) 0 (38)

with Q =

Q 00 0

.

A robust H non-PDC controller (3) design methodology con-sidering the classical closed-loop dynamics (6) is summarized inthe following theorem.

Theorem 3. Assume that, z(t), = 1, . . . , r, h(z(t))

and = 1, . . . , l, v (z(t)) . The uncertain TS descriptorsystems (2) (with (t) = 0) is globally asymptotically stable via thenon-PDC control law (3)guaranteeing the H performance , ifthere

exist the matrices X1jk = X1

jk

T> 0,X3ij ,X

4ij > 0 (or < 0), R1 =

RT1, R2 = RT2, and Fjk, the positive scalars =

2, 1ijk, 2

ijk, 3

ijk, 4

ijk

such that the following LMIs are satisfied for all i,j = 1, . . . , r andk = 1, . . . , l: ijk ()X1jk 0 0 0 0 0

0 0 0 0 WTi 0

Q1 00 I

< 0 (39)

X1jk + R1 0 (40)

X1jk + R2 0 (41)

with ijk defined in Theorem 1.

Proof. The disturbed classical closed-loop dynamics (6) is stableunder the H performance (38) ifV(x) +x

TQ x 2T < 0 (see

Box VIII).

Multiplying (42) respectively left by diagX

T

hhv I

and right by

diagXhhv I

, one obtains the inequality in Box IX.

Now, following the same path as for the proof of Theorem 1, afterapplying the Schur complement and using the change of variable

= 2

, one obtains the conditions of Theorem 3. That ends theproof.

4.2. H non-PDC controller design based on the redundancy closed-

loop dynamics

Now, let us consider the extended state vector x(t) =xT(t)

xT(t) uT(t)T

, the H criterion (37) can be rewritten as:tft0

xT(t)Qx(t) 2tf

t0

T(t)(t) 0 (44)

with

Q = Q 0 00 0 0

0 0 0 .

A robust H non-PDC controller design methodology based on theredundancy closed-loop dynamics is summarized in the follow-ing theorem.

Theorem 4. z(t), = 1, . . . , r, h(z(t)) and = 1, . . . ,l, v (z(t)) . The uncertain TS descriptor systems (2) (with(t) = 0) is globally asymptotically stable via the non-PDC controllaw (3)guaranteeing the H performance , if there exist the matrices

X1jk = (X1

jk)T > 0,X4ij ,X

5ij > 0 (or < 0),X

6ij ,X

7jk,X

8jk,X

9jk >

0 (or < 0), R1 = RT1, R2 = R

T2 and Fik the positive scalars =

2, 1ijk, 2ijk,

3ijk,

4ijk,

5ijk,

6ijk and

7ijk such that the LMI conditions

in Box X are satisfied for all i,j = 1, . . . , r and for k = 1, . . . , l.

Proof. The redundancy closed-loop dynamics (8) subject to ex-

ternal disturbances is stable under the Hcriterion (38) ifV(x) +xTQx 2T < 0, that is to say if:

AThvX1hhv + (

X1hhv)T Ahv + E

(Xhhv)1 + Q ()

WTh (Xhhv)

1 2I

< 0. (48)

Multiplying (48) left and right respectively by

XThhv 00 I

and

Xhhv 00 I

, one obtains the inequality in Box XI.

Now, following the same path as for the proof of Theorem 2,after applying the Schur complement and using the change of vari-able = 2, one obtains the conditions of Theorem 4. That endsthe proof.

Remark 8. Following the same argument as given in Remark 6,Theorem 3 is a particular case ofTheorem 4. Therefore, LMI condi-tions ofTheorem 4 provide the less conservatism results. This willbe emphasis in the next section through an academic example.

Remark 9. The LMI conditions proposed in Theorems 14 are de-pending on the lower bounds of hi(z(t)) for i = 1, . . . , r andvk(z(t)) for k = 1, . . . , l. It is often pointed out as a criticism tofuzzy Lyapunov approach since these parameters may be difficultto choose in practice. Note that a way has been proposed to com-pute these bounds in the case of nominal standard state space TSfuzzy models (without uncertainties nor disturbances) [20]. Nev-ertheless, in the presence of uncertainties or external disturbances,this methodology failed. Indeed, it is not possible to predict, prioryto the controller designed, the dynamical behavior of the uncer-

tain bounded variables and so their influences on the membershipfunction dynamics cannot be strictly investigated. In the case of

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(Ahv BhKhv )

T(Xhhv )1 + ((Xhhv)

1)T(Ahv BhKhv) + E

(Xhhv )1 + Q ()

WT

h(Xhhv)1 2I

< 0 (42)

Box VIII.

X

T

hhv (Ahv BhKhv )T

+ (Ahv BhKhv)Xhhv +XT

hhv E

(Xhhv)1Xhhv +X

T

hhvQ Xhhv ()W

T

h 2I

< 0 (43)

Box IX.

ijk ()X1jk 0 0 0 0 0 0 0 0 0

0 0 0 0 WTi 0 0 0 0 0

Q1 00 I

< 0 (45)

X1jk + R1 0 (46)

X1jk + R2 0 (47)

Box X.

XThhv

AThv +Ahv Xhhv + X

Thhv E

( Xhhv )1 Xhhv + X

Thhv Q

Xhhv ()

WTh 2I

< 0 (49)

Box XI.

non-quadratic stabilization subject to uncertainties and/or distur-bances, what can be done is to assume wider values of the mem-bership function derivative lower bounds regarding to the oneobtained in the nominal case (expected to include the uncertaindynamics influences).

Another way to cope with this problem is to provide a slightlymodified version of Theorems 13 or 4, but leading to quadratic

results. Indeed, to avoid appearance of the unknown membershipderivative bounds, one can set X1 = R1 = R2 common in-stead ofX1hv in the Lyapunov functions (13), respectively (31). Thus,

the termsr

=1 (X1k + R1) +

l=1 (X

1i + R2) can be re-

moved from LMIs. This way has been firstly investigated in thecase of nominal TS descriptor stabilization in [37] but, obviously,the obtained results are more conservative than non-quadratic ap-proaches. Note finally that, a new way to deal with the problem ofmembership function derivatives in a local view point have beenpropose in [42]. Nevertheless, this promising result is, at this time,only suitable for the stability analysis of standard TS systemsand need more investigation and research efforts before being ex-tendedto the case of TS descriptor based robustcontroller design.

5. Simulation results and discussion

In this section, two examples, a numerical one and a realisticone, are proposed to show the efficiency of the above-proposedresults. The first one is devoted to show the benefit of the re-dundancy approach regarding to classical ones in terms of con-servatism. Thus, the feasibility fields and H performances will beinvestigated through an academic uncertain TS fuzzy descriptor.Then, a second example is provided toshow the validityof the pro-posed approaches on a realisticnonlinear system: an inverted pen-dulum on a cart.

5.1. Example 1: conservatism comparison of the proposed approaches

In order to compare the conservatism of the proposed ap-

proaches, let us consider the following academic nonlinear de-scriptor system [34]:

E(x(t))x(t) = A(x(t))x(t) + B(x(t))u(t) + W(t) (50)

with

E(x(t)) =

1 1

1 cos2x2(t)

,

B(x(t)) = 1 +1

1 +x21

(t)

a cos2x2(t) 2 , W = 0.25 0.25T

and

A(x(t)) =

0 cos2x2(t) 1

1 +x21(t)

3

23 + b

1 +

1

1 +x21(t)

sinx2(t)

x2(t)

.

Note that (50) contains one nonlinear term e(x2(t)) = cos2x2(t)

in its left hand side and three ones, a1(x2(t)) = cos2x2(t), a2

(x1(t)) =1

1+x21(t)and a3(x2(t)) =

sin(x2(t))

x2(t)in its right hand side.

Using the sector nonlinearity approach [2], x1(t) R and x2(t) R, one can write:

e(x2(t)) = a1(x2(t)) = cos2x2(t)= (1 cos2x2(t)) 0 + cos

2x2(t) 1 (51)

a2(x1(t)) =1

1 +x21(t)

=

1

1

1 +x21(t)

0 +

1

1 +x21(t) 1 (52)

a3(x2(t)) =sinx2(t)

x2=

x2(t) sinx2(t)

x2(t)(1 )

+sinx2(t) x2(t)

x2(t)(1 ) 1 (53)

with = minsinx2(t)

x2(t).

This leads to l = 2 and r = 8 for the left and the right part ofthe TS fuzzy model. Then, to ensure the stability of the descriptor

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system, lr(5r + 1) = 656 and 2lr(4r + 1) = 1056 LMI conditionshave to be verified respectively through the above-proposed Theo-rems 1 and 2. Consequently, this lead to a high computational costmaking unfruitful a controller design from LMI conditions with ac-tual computers. In order to reduce this computational cost, somenonlinear terms can be put into uncertainties. Indeed, it is possi-ble to consider some nonlinear terms that are weakly influencingthe global dynamics as uncertainties. In this case, the stabilization

problem remains toa robustcontroller designleading to reducethenumber of fuzzy rules [35]. For example, we consider the nonlin-ear terms depending on the state variable x2(t) as bounded uncer-tainties. Thus, the nonlinear term to be split is a2(x1(t)) =

1

1+x21

(t).

Thus, the descriptor (50) maybe rewritten as an uncertain descrip-tor such that:

(E + E(t))x(t) =

2i=1

hi(x1)((Ai + Ai(t))x(t)

+ (Bi + Bi(t))u(t)) + W(t) (54)

with

h1(x1(t)) = 1 1

1 +x2

1(t)

, h2(x1(t)) =1

1 +x2

1(t)

,

E =

1 1

11

2

, A1 =

0

1

2

3

23 +

b

2(1 )

,

A2 =

0

1

2

3

23 + b(1 )

, B1 =

2

a

2 2

,

B2 =

1

a

2 2

, E(t) =

0 0

01

2f1(t)

,

A1(t) =0

1

2f1(t)

0 b1 +

2f2(t)

,

A2(t) =

0 12f1(t)

0 b(1 + )f2(t)

and

B1(t) = B2(t) =

0

a

2f1(t)

.

According to (51) and (53), one can argue that, although thedescriptor (54) contains uncertainties, it is paradoxically repre-senting exactly the nonlinear descriptor (50) with the nonlinear

functions f1(t) and f2(t) given by:

f1(t) = 2 cos2x2(t) 1 (55)

and

f2(t) =1

1 +

1 + 2

sinx2(t)

x 2(t)

. (56)

Finally, in order to apply the LMI given in the above theorems, onehas to rewrite the uncertain matrix

E(t) = Hefe(t)Ne, Ai(t) = Hiaf

ia (t)N

ia et

Bi(t) = Hibfb(t)N

ib

(57)

with

He = Hia =1 0

0 1

, Hib =0

1

, Ne =0 0

0 12

,

1

0.5

0

-0.5

-1

-1.5

b

-2

-2.5

-3

-3.5-4 -3 -2 -1 0

a

1 2 3 4

Theorem 2

Theorem 1

Fig. 1. Feasibility fields obtained from Theorems 1 and 2 (Example 1).

N1a =

01

2

0 b

1 +

2

, N2a = 0

1

2

0 b(1 + ) and

N1b = N2b =

1

2a.

Note that now, considering the uncertain descriptor (54) andsolving Theorem 1 or 2, a controller design may be obtained withrespectively lr(5r+ 1) = 22 and 2lr(4r+ 1) = 36 LMI conditions.Therefore, the computational cost is now reasonably reduced.

First of all, in order to illustrate the benefit in terms of conser-vatism of the redundancy closed-loop dynamics based approachregarding to the classical closed-loop dynamics ones, one pro-poses to study the respective feasibility fields ofTheorems 1 and 2for a

4 4

, b

3.5 1

with 1 = 2 = 1. These are

presented in Fig. 1 and have been obtained using the Matlab LMI

Toolbox. As expected, the respective feasible area of Theorems 1and 2 confirm Remark 6.

Note that, as shown in Fig. 1, a controller cannot be synthesizedfor a = 3 and b = 0 from Theorem 1 when a solution existswith Theorem 2. As for non-PDC controller design example, thefollowing matrices and scalars give this solution:

F11 =

8.879 2.061

, F21 =

9.0071 1.9393

,

R1 =

46.5741 20.0709

20.0709 16.4484

,

X111 = X121 =

47.2233 20.4178

20.4178 16.6888

,

X4

11 = X4

12 =5.0855 2.9506

0.3231 1.0085

,

X421 = X422 =

10.4604 6.91912.9734 1.1802

,

X511 = X512 =

11.6118 5.90416.7604 19.8665

,

X521 = X522 =

8.0278 1.1601

3.9982 15.7893

,

X611 = X612 =

3.4569

0.8358

, X621 = X

622 =

3.5657

0.7555

,

X711 = X721 =

5.41111.1152

T

, X811 = X821 =

2.24283.1637

T

,

X911 = X921 = 0.2394,

1111 =

1121 = 0.3608,

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0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5

Time

6 7 8 9 10

0

0.5u(t)

1

x2

(t)

-2

-1

0

1

-4

-2x1

(t)

0

Fig. 2. Evolution of the statevector (without external disturbance) and the control

signal (Example 1).

1

211 =

1

221 = 0.9369,

2

111 =

2

121 = 0.976,2211 =

2221 = 1.7785,

3111 =

3121 = 10.5924,

3211 = 3221 = 4.6813,

4111 =

4121 = 4.1892,

4211 = 4221 = 4.7994,

5111 =

5121 = 6.5847,

5211 = 5221 = 3.7544,

6111 =

6121 = 0.935,

6211 = 6221 = 1.4624,

7111 =

7121 = 0.749 and

7211 = 7221 = 1.1434.

Fig.2 shows theevolutions of thestatevector andthe control signal

for the initial condition x(0) =

2 1.5T

. Note that, as shownin Fig. 3, the hypothesis made on lower bounds of membership

function derivatives are a posteriori verified in simulation since

min(h1(x1(t))) = 0.2933 > 1 and min(h2(x1(t))) =0.8969 > 1.

Another way to confirm that the redundancy closed-loopdynamics approaches are less conservative than classical closed-

loop dynamics ones is to compare there H performancesregarding to external disturbances. Thus, the attenuation level

values have been computed from Theorems 3 and 4 for several

values of a

2 0

, with b = 0, 1 = 2 = 1 andQ = I22. These are depicted in Fig. 4. As expected, the obtainedH performances from Theorem 4 are always better than the one

obtained from Theorem 3.

Then, as example of robust controller design, the following

solution ofTheorem 4 has been found via the Matlab LMI Toolboxfor 1 = 2 = 1, a = 3, b = 0 and Q = I22. The followingmatrices and scalars give that solution:

R1 =

0.1585 0.0542

0.0542 0.0524

,

F11 = F21 =

0.0649 0.0216

,

X111 = X121 =

0.1585 0.0542

0.0542 0.0524

,

X411 =

14.2628 0.0397

0.0126 0.0125

,

X421 =

159.1288 0.0397

0.0126 0.0125

,

X412 =

178.848 0.08170.0111 0.0177

,

1

0.5

0dh

1(t)/d(t)

-0.5

0.5

0

-0.5dh

2(t)/d(t)

-1

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5

Time

6 7 8 9 10

Fig. 3. Evolution of h1(x1(t)) and h2(x1(t)) (Example 1, controller designed from

Theorem 2).

-2 -1.8 -1.6 -1.4 -1.2 -1

a

-0.8 -0.6 -0.4 -0.2 00

0.5

1

1.5

2

2.5

3

3.5

4

4.5Theorem 4

Theorem 3

Fig.4. Comparisonof attenuationlevelsfor several valuesof parametera (Example

1, Theorems 3 vs. 4).

X422 =

178.746 0.08170.0111 0.0177

,

X511 =

14.2343 14.27550.0207 0.1055

,

X521 =

159.1002 159.14140.0207 0.1055

,

X512 = 178.7551 178.76720.0054 0.0609

,X522 =

178.6532 178.66520.0054 0.0609

, X611 = 10

5

4.093

1.087

,

X621 = 105

4.062

1.087

, X612 = X

622 = 10

5

4.214

0.938

,

X711 = X721 =

0.0649

0.0216

T, X811 = X

821 = 10

4

0.2684

0.3861

T,

X911 = X921 = 2.5069 10

6, 1111 = 1121 = 2.7341 10

6,

1211 = 1221 = 1.6802 10

4, 2111 = 2121 = 0.0045,

2211 = 2221 = 0.0095,

3111 =

3121 = 0.0983,

3211 = 3221 = 0.0492,

4111 =

4121 = 0.0204,

4211 = 4221 = 0.0198,

5111 =

5121 = 6.9223 10

5,

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0

-1x1

(t)

-2

x

2(t)

0

-0.5

-1

u(t)

0.4

0.6

0

0.2

-0.20 1 2 3 4 5

Time

6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

Fig. 5. Evolution of the state vector (with external disturbance) and the control

signal (Example 1).

1

0.5

0dh

1(t)/d(t)

-0.5

0.5

0

-0.5dh

2(t)/d(t)

-1

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5

Time

6 7 8 9 10

Fig. 6. Evolution of h1(x1(t)) and h2(x1(t)) (Example 1, controller designed from

Theorem 4).

5211 = 5221 = 3.3657 10

4, 6111 = 6121 = 1.169 10

5,

6211 = 6221 = 2.5353 10

4, 7111 = 7121 = 8.0248 10

6,

7211 = 7221 = 1.6947 10

4 and = 3.7166.

For simulation purpose, the external disturbance signal (t) hasbeen considered as a Gaussian random signal with a unit variance.

Considering the initial condition x(0) =

1.5 1

T

, Fig. 5shows respectively the convergence of the state vector system, the

control signal. Once more, as shown in Fig. 6, the condition madeon h1(x1(t)) and h2(x1(t)) are a posteriori verified in simulationsince min(h1(x1(t))) = 0.2737 > 1 and min(h2(x1(t))) =0.7509 > 1.

5.2. Example 2: stabilization of an inverted pendulum on a cart

Let us now consider the benchmark of an inverted pendulumon a cart given by Fig. 7. The motion equations obtained from theNewtons second law are given by [11]:

x1(t) = x2(t)lm

1

3+ sin2x1(t)

+

4

3lM

x2(t)

= (m + M)gsinx1(t) mlx22(t) sinx1(t) cosx1(t)

cosx1(t)u(t)

(58)

u(t)M

Fig. 7. Inverted pendulum on a cart (Example 2).

where M = 1 kg and m = 0.1 kg are respectively the masses ofthe cart and the pendulum, l = 0.5 m the length of the rod, x1(t)is the angle that the pendulum makes with the vertical, x2(t) isthe pendulum angular velocity and g = 9.8 m s2 is the gravityconstant.

Note that, the nonlinear term sin2x1(t) in the left hand sideof (58) is often neglected to derive a standard TS model with areduced number of rules to obtain a feasible solution of classi-cal quadratic stability conditions [9,11,43]. In this paper, takingbenefit of a descriptor representation, one proposes to avoid thisapproximation. Indeed, it is well known that descriptors are con-

venient for modelling mechanical systems [5,6]. Moreover, let usconsider that the velocity signal x2(t) is not available from mea-surements. Then, as described in [11], the nonlinear function x22(t)can be removed from the nominal part and put into the uncertainpart of the system. Thus, let us now considerx22(t)

2 with themaximal angular velocity of the inverted pendulum, one can write

x22(t) = f(x2(t)) with f(x2(t)) =x22(t)

2and f2(x2(t)) 1. Thus,

the following uncertain descriptor matching the dynamical system(58) can be considered:

E(x1(t))x(t) = (A(x(t)) + A(x(t)))x(t) + B(x1(t))u(t) (59)

where xT(t) =xT1(t) x

T2(t)T

is the state vector,

E(x1(t)) =1 0

0 lm

13

+ e(x1(t))

+ 43

lM

,

A(x(t)) =

0 1

(m + M)g1a (x1(t)) 0

,

A(x(t)) =

0 0

ml2f(x2(t))1

a (x1(t))2

a (x1(t)) 0

,

B(x1(t)) =

0

2a (x1(t))

with the nonlinear terms e(x1(t)) = sin2x1(t),

1a (x1(t)) =

sinx1(t)

x1(t)

and 2a (x1(t)) = cosx1(t).

Using the sector nonlinearity approach [2], a TS model can beobtained as shown in [11] by splitting the above-defined nonlinearterms. Note that for x1(t) =

2

, the system (58) is locallyuncontrollable thus the angular displacements will be reduced to

x1(t)

0 0

with 0

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