Observer-Based Controller Design for a Class of Takagi ... · Observer-based controller design for a class of T-S descriptor systems 137 The outline of the paper is as follows. The

Nonlinear Analysis and Differential Equations, Vol. 5, 2017, no. 3, 135 - 155HIKARI Ltd, www.m-hikari.com

https://doi.org/10.12988/nade.2017.61192

Observer-Based Controller Design for a Class

of Takagi-Sugeno Descriptor Systems

Boutayna Bentahra1,2, Jalal Soulami1,2∗, El Mahfoud El Bouatmani1,2,Abdellatif El Assoudi1,2 and El Hassane El Yaagoubi1,2

1Laboratory of High Energy Physics and Condensed Matter, Faculty of ScienceHassan II University of Casablanca, B.P 5366, Maarif, Casablanca, Morocco

2ECPI, Department of Electrical Engineering, ENSEMHassan II University of Casablanca, B.P 8118, Oasis, Casablanca Morocco

∗Corresponding author

Copyright c© 2016 Boutayna Bentahra, Jalal Soulami, El Mahfoud El Bouatmani, Ab-

dellatif El Assoudi and El Hassane El Yaagoubi. This article is distributed under the Creative

Commons Attribution License, which permits unrestricted use, distribution, and reproduc-

tion in any medium, provided the original work is properly cited.

Abstract

This paper deals with the design problem of observer-based controller fora class of nonlinear descriptor systems described by Takagi-Sugeno (T-S) fuzzy descriptor models with unmeasurable premise variables. Themain idea of the developed approach is based on the separation betweendynamic and static relations in the T-S descriptor model. Stability con-ditions are established with Lyapunov theory in order to guarantee theconvergence of the closed-loop state systems. The gains of the observerand controller are obtained by solving a set of linear matrix inequal-ities (LMIs). In practice, the computation of solutions of descriptorsystems requires the combination of an ordinary differential equation(ODE) routine together with an optimization algorithm. The main re-sult of this paper consists in showing that the observer-based controllerproblem for a class of T-S descriptor systems can be achieved by usinga fuzzy controller based on an ODE structure only. Finally, numeri-cal simulations are given to show the good performances of the fuzzydesigned observer-based controller.

Keywords: Takagi-Sugeno descriptor model, unmeasurable premise vari-ables, observer-based controller, linear matrix inequality

136 B. Bentahra et al.

1 Introduction

During the last decades, fuzzy T-S models [1] have been widely used for anal-ysis and controller synthesis of nonlinear systems. The great deal of interest ofsuch approach relies on the fact that once the T-S fuzzy models are obtained,some analysis and design tools developed in the linear case can be used, whichfacilitates observer or/and controller synthesis for complex nonlinear systemssee for example [2] and the references therein.In this paper, we are concerned with the design of stabilizing output feed-back control using T-S fuzzy observer. This theory which combines the couple(observer, controller system) is variously called observer-based controller ordynamic output feedback controller or separation principle. Several works onT-S fuzzy control and state observation for nonlinear systems described byordinary differential equations (ODEs) exist in the literature. Indeed, the sta-bility and the stabilization of such systems are mainly investigated throughthe direct Lyapunov method [3], [4]. Likewise, many researches have beenproposed to design observers [5], [6], [7], [8]. The observer-based controllersynthesis for such nonlinear systems has received considerable attention and itis still an active area of research. Indeed, it has been studied extensively withgreat success [9], [10], [11], [12] and [13]. Notice that, generally an interestingway to solve the various problems raised previously (control and observation)is to write the convergence conditions on the LMI form [14].In this paper, we deal with descriptor nonlinear systems variously called im-plicit systems or singular systems or differential-algebraic equations (DAEs).In fact, these systems have been widely used in the modeling of dynamic pro-cesses to describe the behavior of many chemical and physical processes seefor example [15], [16], [17] and [18] and references therein. This formulationincludes both dynamic and static relations. The numerical simulation of suchdescriptor models usually combines an ODE numerical method together withan optimization algorithm. However, the problems of control and observationdesign for nonlinear descriptor systems described by T-S models are mainlyinvestigated in the literature. We may cite [19], [20], [21], [22], [23].Based on a new design methodology through judicious use of the Young’s in-equality, the main contribution of this paper consists to propose an observer-based controller design for a class of T-S descriptor models with unmeasurablepremise variables. The developed result is based on the separation between thedynamic and static relations in the T-S descriptor model. The asymptotic sta-bility of the closed-loop state systems is studied by using the Lyapunov theoryand the stability conditions are given in terms of LMIs. Besides, the proposeddynamic controller for a class of T-S descriptor systems can be synthesized byonly an ODE structure.

Observer-based controller design for a class of T-S descriptor systems 137

The outline of the paper is as follows. The class of the fuzzy T-S structureof nonlinear descriptor systems is introduced in Section 2. The main resultabout fuzzy observer-based controller design for a class of T-S descriptor mod-els with unmeasurable premise variables is stated in sections 3. The controland observer gains are found directly from LMI formulation. In section 4, weillustrate the performance of the proposed observer-based controller in simu-lation through a rolling disc descriptor model.

Throughout the paper, some notations used are fair standard. For example,X > 0 means the matrix X is symmetric and positive definite. XT denotesthe transpose of X. The symbol I (or 0) represents the identity matrix (orzero matrix) with appropriate dimension.q∑

i,j=1

µiµj =q∑

i=1

q∑j=1

µiµj;

(X ∗Y Z

)=

(X Y T

Y Z

)and δZij = Zi −Zj.

2 T-S descriptor systems and problem state-

ment

In this paper, the aim consist to consider the problem of observer-based con-troller design for a class of nonlinear descriptor systems described by Takagi-Sugeno (TS) structure with unmeasurable premise variables. For this objec-tive, the following class of nonlinear descriptor systems is considered:

Ex = A(X1)x+B(X1)uy = Cx

(1)

where x = [XT1 XT

2 ]T ∈ Rn is the state vector with X1 ∈ Rn1 is the vectorof differential variables, X2 ∈ Rn2 is the vector of algebraic variables withn1 + n2 = n, u ∈ Rm is the control input, y ∈ Rp is the measured output.A(.) ∈ Rn×n, B(.) ∈ Rn×m are continuous functions which depend only onthe vector of differential variables X1. C ∈ Rp×n and E ∈ Rn×n such thatrank(E) = n1 are real known constant matrices with:

E =

(I 00 0

); C =

(C1 0

)(2)

To design a T-S fuzzy observer-based controller, we need a T-S fuzzy model forthe nonlinear descriptor systems (1). In general, there are two approaches forconstructing fuzzy models: identification (fuzzy modeling) using input-outputdata and derivation from given nonlinear system equations. In this paper, weuse the second approach which derives a fuzzy model from given nonlineardynamical models (1).


By the sector nonlinearity approach [2], the nonlinear descriptor system (1)can be exactly represented by the T-S fuzzy descriptor systems: Ex =

q∑i=1

µi(ξ)(Aix+Biu)

y = Cx

(3)

where Ai ∈ Rn×n, Bi ∈ Rn×m are real known constant matrices with:

Ai =

(A11i A12i

A21i A22i

); Bi =

(B1i

B2i

)(4)

with constant matrices A22i are supposed invertible. q is the number of sub-models. The premise variable ξ which depends here only on X1 is supposedunmeasurable. The µi(ξ) (i = 1, . . . , q) are the weighting functions thatensure the transition between the contribution of each sub model:

Ex = Aix+Biuy = Cx

(5)

They have the following properties:q∑

i=1

µi(ξ) = 1

0 ≤ µi(ξ) ≤ 1

(6)

The main object of this paper is to design an observer-based controller for theT-S descriptor system (3). For this we make the following assumption [15]:

Assumption 2.1 : Suppose that:

• (E, Ai) is regular, i.e. det(sE − Ai) 6= 0 ∀s ∈ C

• All sub-models (5) are impulse controlable and stabilisable.

• All sub-models (5) are impulse observable and detectable.

The main idea of the developed approach for the proposed controller designis based on the separation between dynamic and static relations in the T-Sdescriptor model (3). Indeed, from (2) and (4), systems (5) can be rewrittenas the well known second equivalent form [15]:

X1 = A11iX1 + A12iX2 +B1iu0 = A21iX1 + A22iX2 +B2iuy = C1X1

(7)


From (7) and using the fact that A−122i exist, the algebraic equations can be

solved directly for algebraic variables, to obtain:

X2 = −A−122iA21iX1 − A−1

22iB2iu (8)

Substitution of the resulting expression for X2 in equation (7) yields the fol-lowing model:

X1 = MiX1 +NiuX2 = QiX1 +Riuy = C1X1

(9)

where Mi = A11i − A12iA

−122iA21i

Ni = B1i − A12iA−122iB2i

Qi = −A−122iA21i

Ri = −A−122iB2i

(10)

In descriptor form, sub system (9) takes the following form:Ex = Mix+ Niuy = Cx

(11)

where

Mi =

(Mi 0Qi −I

); Ni =

(Ni

Ri

)(12)

So, using (6) and (12), system (3) can be rewritten in the following equivalentmodel:

X1 =q∑

i=1

µi(ξ)(MiX1 +Niu)

X2 =q∑

i=1

µi(ξ)(QiX1 +Riu)

y = C1X1

(13)

In order to investigate the observer-based controller for the T-S descriptorsystem (3), firstly the following control law established in [23] is adopted:

u(x) = u(X1, X2) = −q∑

i=1

µi(ξ)(Ki1X1 +Ki2X2) (14)

where the gains Ki1, Ki2 can be determined in order that the closed-loopsystem (3)-(14) is asymptotically stable.


In order to be able to state the result as LMIs, the authors in [23] demonstratedthat the fuzzy state feedback controller (14) can be rewritten in the followingequivalent expression:

u(x) = −q∑

i=1

µi(ξ)KiX1 = u(X1) (15)

where

Ki = (I +Ki2Ri)−1(Ki1 +Ki2Qi) (16)

Thus, to prove the global asymptotic stability of the closed-loop system (3)-(14), it suffices to prove that its equivalent closed-loop system (13)-(15) isglobally asymptotically stable. The following result has been studied in [23].

Theorem 2.2 [23]: The closed-loop system (13)-(15) is globally asymp-totically stable if there exist matrices Y1 > 0, Ui, i = 1, . . . , q verifying thefollowing LMIs:

∆ii < 0 i = 1, . . . , q∆ij + ∆ji < 0 i < j s.t. µi ∩ µj 6= ∅

(17)

where

∆ij = MiY1 + Y T1 M

Ti −NiUj − UT

j NTi (18)

The fuzzy local feedback gains Ki, i = 1, . . . , q are given by:

Ki = UiY−11 (19)

Thereby, from (16) the control gains Ki1 and Ki2 are given by (see [23]):

[Ki1 Ki2] = Ki(θi)+ (20)

where (θi)+ design the pseudo inverse matrix of θi which is expressed as:

θi =

[I

Qi −RiKi

](21)

On the other hand, the following observer for system (3) is employed:

˙X1 =

q∑i=1

µi(ξ)(MiX1 +Niu− Li(y − y))

X2 =q∑

i=1

µi(ξ)(QiX1 +Riu)

y = C1X1

(22)


where (X1, X2), y and ξ denote the estimated state vector, output vector andpremise variables vector respectively. Li are the gains of the observer whichcan be determined so that x converges asymptotically towards x.The convergence condition of the observer (22) can be formulated by the fol-lowing theorem.

Theorem 2.3 [22]: The state error between the T-S descriptor model (3)and its observer (22) converges asymptotically towards zero, if there existsmatrices Y2 > 0 and Y2 > 0, matrices Wi, i = 1, . . . , q and a positive scalar β,such that the following LMIs hold: Θj ∗ ∗

δMTjiY

T2 MT

i Y2 + Y2Mi ∗δNT

jiYT2 NT

i YT2 −βI

< 0 ∀(i, j) ∈ 1, . . . , q2 (23)

where

Θj = MTj Y2 + Y2Mj −WjC1 − CT

1 WTj + I (24)

The local observer gains Li, i = 1, . . . , q are derived from:

Li = Y −12 Wi (25)

3 Main result

In this paper, the aim is to propose a novel approach for the separation prin-ciple theory of a class of T-S descriptor systems (3). A fuzzy dynamic outputfeedback controller for T-S descriptor system (3) is defined by:

u(x) = u(X1, X2) = −q∑

i=1

µi(ξ)(Ki1X1 +Ki2X2) (26)

in order that the closed-loop system be asymptotically stable where Ki1 andKi2 are the control gains to be determined.In order to be able to state the main result in the form of LMIs, the key idealies in the use of the equivalent control law of (26) given by:

u(X1, X2) = −q∑

i=1

µi(ξ)KiX1 = u(X1) (27)

where Ki is defined in (16) which can be determined by Theorem 3 below andthe control gains Ki1 and Ki2 used in (26) can be determined by equation (20).


Accordingly, consider the following augmented closed-loop system:

X1 =q∑

i=1

µi(ξ)(MiX1 +Niu(X1))

X2 =q∑

i=1

µi(ξ)(QiX1 +Riu(X1))

˙X1 =

q∑i=1

µi(ξ)(MiX1 +Niu(X1)− Li(y − y))

X2 =q∑

i=1

µi(ξ)(QiX1 +Riu(X1))

u(X1) = −q∑

i=1

µi(ξ)KiX1

(28)

Thus, to prove the global asymptotic stability of the system (3)-(26), it sufficesto prove that its equivalent system (28) is globally asymptotically stable.The following theorem provides the main result of this paper.

Theorem 3.1 : The closed-loop T-S model described by (28) is globallyasymptotically stable if for fixed scalars ε1 > 0 and ε2 > 0, there exist matricesY1 > 0, Y2 > 0, Ui, Wi, i = 1, . . . , q verifying the following LMIs:

Λ11 ∗ ∗ ∗ ∗ ∗0 Λ22 ∗ ∗ ∗ ∗I 0 −ε1Y2 ∗ ∗ ∗0 Λ42 0 −ε−1

1 Y2 ∗ ∗Λ51 0 0 0 −ε2I ∗0 Y2 0 0 0 −ε−1

2 I

< 0 i, j = 1, . . . , q

Π11 ∗ ∗ ∗ ∗ ∗0 Π22 ∗ ∗ ∗ ∗I 0 −ε1Y2 ∗ ∗ ∗0 Π42 0 −ε−1

1 Y2 ∗ ∗Π51 0 0 0 −ε2I ∗0 Y2 0 0 0 −ε−1

2 I

< 0 i, j < k; µiµjµk 6= 0

(29)

where

Λ11 = ∆jj

Λ51 = δMjiY1 − δNjiUj

Λ22 = MTi Y2 − CT

1 WTj + Y2Mi −WjC1

Λ42 = WjC1

Π11 = ∆jk + ∆kj

Π51 = δMjiY1 − δNjiUk + δMkiY1 − δNkiUj

Π22 = 2MTi Y2 − CT

1 (Wj +Wk)T + 2Y2Mi − (Wj +Wk)C1

Π42 = (Wj +Wk)C1

(30)

with ∆ij is defined in (18).


The dynamic controller gains Ki and Li are given by:Ki = UiY

−11

Li = Y −12 Wi

(31)

Proof of Theorem 3 : Denoting the state estimation error by:

e =

(e1e2

)=

(X1 −X1

X2 −X2

)(32)

It follows from (13), (22) and (27) that the observer error dynamic is given bythe differential-algebraic equation:

e1 =q∑

i,j=1

µi(ξ)µj(ξ)(ΓijX1 − LiC1e1)

−q∑

i,j=1

µi(ξ)µj(ξ)(MiX1 −NiKjX1)

e2 =q∑

i,j=1

µi(ξ)µj(ξ)ΩijX1 −q∑

i,j=1

µi(ξ)µj(ξ)(QiX1 −RiKjX1)

(33)

which is equivalent to the following system:

e1 =q∑

i,j=1

µi(ξ)µj(ξ)(ΓijX1 − LiC1e1)

−q∑

i,j=1

µi(ξ)µj(ξ)(MiX1 −NiKjX1)

−q∑

i,j=1

µj(ξ)(µi(ξ)− µi(ξ))(MiX1 −NiKjX1)

e2 =q∑

i,j=1

µi(ξ)µj(ξ)ΩijX1 −q∑

i,j=1

µi(ξ)µj(ξ)(QiX1 −RiKjX1)

−q∑

i,j=1

µj(ξ)(µi(ξ)− µi(ξ))(QiX1 −RiKjX1)

(34)

where Γij = Mi −NiKj

Ωij = Qi −RiKj(35)

Let Zi = Mi, Ni, Qi, Ri, and using the fact that:

q∑i=1

(µi(ξ)− µi(ξ))Zi =q∑

i,k=1

µi(ξ)µk(ξ)δZik (36)

where δZik = Zi −Zk.


Hence, the system (34) becomes:e1 =

q∑i,j,k=1

µi(ξ)µj(ξ)µk(ξ)(ΦijkX1 + (Mi − LkC1)e1)

e2 =q∑

i,j,k=1

µi(ξ)µj(ξ)µk(ξ)(ΨijkX1 +Qie1)

(37)

where Φijk = δMki − δNkiKj

Ψijk = δQki − δRkiKj(38)

Thus, the global asymptotic stability of (28) can be proved in an equivalentfashion for:

˙X1 =

q∑i,j,k=1

µi(ξ)µj(ξ)µk(ξ)(ΓkjX1 − LkC1e1)

X2 =q∑

i,j,k=1

µi(ξ)µj(ξ)µk(ξ)ΩkjX1

e1 =q∑

i,j,k=1

µi(ξ)µj(ξ)µk(ξ)(ΦijkX1 + (Mi − LkC1)e1)

e2 =q∑

i,j,k=1

µi(ξ)µj(ξ)µk(ξ)(ΨijkX1 +Qie1)

(39)

which can be rewritten as follows:xa =

q∑i,j,k=1

µi(ξ)µj(ξ)µk(ξ)Σijkxa

xb =q∑

i,j,k=1

µi(ξ)µj(ξ)µk(ξ)Υijkxa

(40)

where

xa =

(X1

e1

), xb =

(X2

e2

)(41)

Σijk =

(Γkj −LkC1

Φijk Mi − LkC1

)Υijk =

(Ωkj 0Ψijk Qi

)(42)

Hence, to prove the convergence to zero of the state variable system (40), itsuffices to prove that xa converges toward zero.Thus, let us consider the candidate Lyapunov function as follows:

V (xa) = xTaPxa , P > 0 (43)


with

P =

(P1 00 P2

)(44)

The time derivative of (43) along the trajectories of (40) is obtained as:

V (xa) =q∑

i,j,k=1

µi(ξ)µj(ξ)µk(ξ)xTa (ΣTijkP + PΣijk)xa (45)

Therefore, we have the following stability conditions: ΣTijjP + PΣijj < 0 i, j = 1, . . . , q

(Σijk + Σikj

2)TP + P (

Σijk + Σikj

2) < 0 i, j < k s.t. µiµjµk 6= 0

(46)

which become from (42) and (44):Λ =

(Λ11 ∗Λ21 Λ22

)< 0 i, j = 1, . . . , q

Υ =

(Υ11 ∗Υ21 Υ22

)< 0 i, j < k s.t. µiµjµk 6= 0

(47)

where

Λ11 = ΓTjjP1 + P1Γjj

Λ21 = P2Φijj − (LjC1)TP1

Λ22 = (Mi − LjC1)TP2 + P2(Mi − LjC1)

Υ11 = (Γkj + Γjk)TP1 + P1(Γkj + Γjk)Υ21 = P2(Φijk + Φikj)− ((Lk + Lj)C1)

TP1

Υ22 = (2Mi − (Lk + Lj)C1)TP2 + P2(2Mi − (Lj + Lk)C1)

(48)

Now multiplying the two inequalities given in (47) on the left and right by((P−1

1 )T 00 I

)and

(P−11 00 I

)respectively and defining new variables

Y1 = P−11 , Y2 = P2, we obtain:

Θ =

(Θ11 ∗Θ21 Θ22

)< 0 i, j = 1, . . . , q

Ω =

(Ω11 ∗Ω21 Ω22

)< 0 i, j < k s.t. µiµjµk 6= 0

(49)

where

Θ11 = Y T1 ΓT

jj + ΓjjY1Θ21 = Y2ΦijjY1 − (LjC1)

T

Θ22 = (Mi − LjC1)TY2 + Y2(Mi − LjC1)

Ω11 = Y T1 (Γkj + Γjk)T + (Γkj + Γjk)Y1

Ω21 = Y2(Φijk + Φikj)Y1 − ((Lk + Lj)C1)T

Ω22 = (2Mi − (Lk + Lj)C1)TY2 + Y2(2Mi − (Lk + Lj)C1)

(50)


By considering the following:

Σ1 =

(Θ11 ∗α Θ22

)

Σ2 =

(Ω11 ∗η Ω22

)Z1 =

(0 β

)Z2 =

(0 λ

)Z =

(−I 0

)(51)

where α = Y2ΦijjY1β = LjC1

η = Y2(Φijk + Φikj)Y1λ = (Lj + Lk)C1

(52)

Matrices Θ and Ω can be rewritten again as:Θ = Σ1 + ZT

1 Z + ZTZ1

Ω = Σ2 + ZT2 Z + ZTZ2

(53)

Lemma 3.2 : Young’s inequalityFor any matrices S and T with appropriate dimensions, the following propertyholds for any invertible matrix U and scalar ε > 0:

STT + T TS ≤ εSTUS + ε−1T TU−1T (54)

Applying Lemma 1 and taking U = Y2 and ε = ε1, (53) becomes:Θ ≤ Σ1 −

(I 00 βTY2

)(−ε1Y2 0

0 −ε−11 Y2

)−1 (I 00 Y2β

)

Ω ≤ Σ2 −(I 00 λTY2

)(−ε1Y2 0

0 −ε−11 Y2

)−1 (I 00 Y2λ

) (55)

Hence, using the Schur complement [14], the inequalities Θ < 0 and Ω < 0hold if the following two matrix inequalities are satisfied.

Λ1 =

Θ11 ∗ ∗ ∗α Θ22 ∗ ∗I 0 −ε1Y2 ∗0 Y2β 0 −ε−1

1 Y2

< 0 i, j = 1, . . . , q

Λ2 =

Ω11 ∗ ∗ ∗η Ω22 ∗ ∗I 0 −ε1Y2 ∗0 Y2λ 0 −ε−1

1 Y2

< 0 i, j < k s.t. µiµjµk 6= 0

(56)


By considering again the following:

Σ3 =

Θ11 ∗ ∗ ∗0 Θ22 ∗ ∗I 0 −ε1Y2 ∗0 Y2β 0 −ε−1

1 Y2

Σ4 =

Ω11 ∗ ∗ ∗0 Ω11 ∗ ∗I 0 −ε1Y2 ∗0 Y2λ 0 −ε−1

1 Y2

Z3 =

(0 Y2 0 0

)Zx =

(Θ51 0 0 0

)Zy =

(Ω51 0 0 0

)

(57)

where Θ51 = ΦijjY1Ω51 = (Φijk + Φikj)Y1

(58)

Matrices Λ1 and Λ2 can be rewritten again as:Λ1 = Σ3 + ZxTZ3 + ZT

3 ZxΛ2 = Σ4 + ZyTZ3 + ZT

3 Zy(59)

Applying Lemma 1 and taking U = I and ε = ε2, (59) becomes:

Λ1 ≤ Σ3 −

ΘT

51 00 Y20 00 0

(−ε2I 0

0 −ε−12 I

)−1 (Θ51 0 0 00 Y2 0 0

)

Λ2 ≤ Σ4 −

ΩT

51 00 Y20 00 0

(−ε2Y2 0

0 −ε−12 Y2

)−1 (Ω51 0 0 00 Y2 0 0

) (60)

Hence, using the Schur complement [14], the inequalities Λ1 < 0 and Λ2 < 0hold if the following two matrix inequalities are satisfied.


Θ11 ∗ ∗ ∗ ∗ ∗0 Θ22 ∗ ∗ ∗ ∗I 0 −ε1Y2 ∗ ∗ ∗0 Y2β 0 −ε−1

1 Y2 ∗ ∗Θ51 0 0 0 −ε2I ∗0 Y2 0 0 0 −ε−1

2 I

< 0 i, j = 1, . . . , q

Ω11 ∗ ∗ ∗ ∗ ∗0 Ω22 ∗ ∗ ∗ ∗I 0 −ε1Y2 ∗ ∗ ∗0 Y2λ 0 −ε−1

1 Y2 ∗ ∗Ω51 0 0 0 −ε2I ∗0 Y2 0 0 0 −ε−1

2 I

< 0

i, j < k s.t. µiµjµk 6= 0

(61)

Then, from (35), (38) and the use of the changes of variables:Ui = KiY1Wi = Y2Li

(62)

we establish the LMI conditions (29) given in Theorem 3.

4 Application to the rolling disc system

In order to illustrate the efficiency of the proposed approach, let us to considera rolling disc process described by the following descriptor model given in [24]:

Ex = A(x)x+Buy = Cx

(63)

where x = (x1, x2, x3, x4)T is the state vector with x1 is the position of the

center of the disc, x2 is the translational velocity of the same point, x3 is theangular velocity of the disc, x4 is the contact force between the disc and thesurface. u is the applied input force to the disc and y = x2 is the variable ofoutput measurement.

A(x) =

0 1 0 0−k1 − k2x21

m

−bm

01

m0 1 −r 0

−k1 − k2x21m

−bm

0r2

J+

1

m

, E =

1 0 0 00 1 0 00 0 0 00 0 0 0


B =(

0 0 0−rJ

)T

, C =(

0 1 0 0)

Thus, to construct the T-S model with unmeasurable premise variable for thenonlinear descriptor systems (63), we consider the sector of nonlinearities of

the term ξ =−k1 − k2x21

m∈ [ξmin, ξmax] of the matrix A(x). Then, we can

transform the nonlinear term under the following shape:

ξ = M1ξmax +M2ξmin (64)

where M1 =

ξ − ξmin

ξmax − ξmin

M2 =ξmax − ξ

ξmax − ξmin

(65)

Hence, the global T-S fuzzy model is inferred as: Ex =2∑

i=1

µi(ξ)(Aix+Bu)

y = Cx

(66)

with

A1 =

0 1 0 0

ξmax−bm

01

m0 1 −r 0

ξmax−bm

0r2

J+

1

m

, A2 =

0 1 0 0

ξmin−bm

01

m0 1 −r 0

ξmin−bm

0r2

J+

1

m

The weighting functions are given by:

µ1(ξ) = M1

µ2(ξ) = M2(67)

The physical parameters definition and their numerical values are given intable 1. The resolution of the LMI defined in Theorem 3 with ε1 = ε2 = 0.12lead the following data:

K1 =(

9.8989 6.7540)

; K2 =(

7.8216 6.3835)

L1 =

(−2.58793.2936

); L2 =

(−2.58793.2936

)(68)

Simulation results with initial conditions:

x0 = [0.10 0.30 0.75 0.85]T , x0 = [0.14 0.30 0.75 2.24]T

are presented in Figures 1, 2, 3 and 4.


These simulation results show the performance of the proposed observer-basedcontroller designed with the parameters K1, K2, L1, L2 given by (67). Theyshow that the global asymptotic stability of the closed loop T-S system withthe control law is satisfied. Moreover, the simulation results show that theobserver gives a good estimation of unknown state variables of the rolling discprocess. On the other hand, using (19), we obtain the following control gainsgiven in algorithm (25): K11 =

(0.1116 1.0324

); K12 =

(2.5809 0.5814

)K21 =

(0.0402 0.9104

); K22 =

(2.27600.3396

) (69)

Notice that with these numerical values of K11, K12, K21 and K22, we obtainexactly the same simulation results given in Figures 1, 2, 3 and 4.

Parameter Meaning Valuek1 Spring coefficient 100 N/mk2 Spring coefficient 100 N/mm Mass 40 kgb Damper coefficient 30r Radius of the disc 0.4 mJ Inertia coefficient 3.2 kg.m2

Table 1: Model parameters of rolling disc process

Figure 1: x1 & x1 with fuzzy observer-based controller






5 Conclusion

Based on the separation approach between differential and algebraic equationsin the T-S descriptor model, a new method to design an observer-based fuzzycontroller for a class of T-S descriptor systems with unmeasurable premisevariables is proposed in this paper. The convergence of the closed-loop systemis studied by using the Lyapunov theory and the stability conditions are givenin terms of LMIs. Simulation results are given and demonstrate the good per-formance of the proposed dynamic controller.

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Received: November 27, 2016; Published: February 24, 2017

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Observer-Based Controller Design for a Class of Takagi ... · Observer-based controller design for a class of T-S descriptor systems 137 The outline of the paper is as follows. The