# Robust Stabilization for Continuous Takagi-Sugeno Fuzzy System Based on Observer Design

Authors: Yassine Manai; Mohamed Benrejeb
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

This paper investigates the influence of a new parallel distributed controller (PDC) on the stabilization region of continuous Takagi-Sugeno (T-S) fuzzy models. Using a nonquadratic Lyapunov function, a new sufficient stabilization criterion is established in terms of linear matrix inequality. The criterion examines the derivative membership function; an approach to determine state variables is given based on observer design. In addition, a stabilization condition for uncertain system is given. Finally, numeric simulation is given to validate the developed approach.


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## Body

## 1. Introduction

Fuzzy control systems have experienced a big growth of industrial applications in the recent decades, because of their reliability and effectiveness. Many researches have investigated the Takagi-Sugeno models [1–3] during the last decades. Two classes of Lyapunov functions are used to analyze these systems: quadratic Lyapunov functions and nonquadratic Lyapunov ones which are less conservative than the first class. Many researches have investigated with nonquadratic Lyapunov functions [4–10].As the information about the time derivatives of membership function is considered by the PDC fuzzy controller, it allows the introduction of slack matrices to facilitate the stability analysis. The relationship between the membership function of the fuzzy model and the fuzzy controllers is used to introduce some slack matrix variables. The boundary information of the membership functions is brought to the stability condition and thus offers some relaxed stability conditions [6]. In order to determine the state variables many approaches of observer design are given [9–11].In this paper, new stabilization conditions for Takagi-Sugeno uncertain fuzzy models based on the use of fuzzy Lyapunov function are presented. This criterion is expressed in terms of linear matrix inequalities (LMIs) which can be efficiently solved by using various convex optimization algorithms [12, 13]. The presented method is less conservative than existing results.The organization of the paper is as follows. In Section2, we present the system description and problem formulation and we give some preliminaries which are needed to derive results. Section 3 will be concerned with stabilization analysis for continuous T-S fuzzy systems by the use of new PDC controller based on derivative membership functions. An observer approach design is derived to estimate state variables. In Section 4, a new stabilization condition for uncertain system is given. Next, a new robust PDC controller design approach is presented. Illustrative examples are given in Section 5 for a comparison of previous results to demonstrate the advantage of the proposed method. Finally Section 6 makes the conclusion.Notation.
Throughout this paper, a real symmetric matrixS>0 denotes S being a positive definite matrix. The superscript “T” is used for the transpose of a matrix.


## 2. System Description and Preliminaries

Consider an uncertainT-S fuzzy continuous model for a nonlinear system as follows:
(2.1)IFz1(t)isMi1,…,zp(t)isMip,THENx˙(t)=(Ai+ΔAi)x(t)+(Bi+ΔBi)u(t)i=1,…,r,
where Mij(i=1,2,…,r,j=1,2,…,p) is the fuzzy set and r is the number of model rules, x(t)∈ℜn is the state vector, u(t)∈ℜm is the input vector, Ai∈ℜn×n, Bi∈ℜn×m are constant real matrices, and z1(t),…,zp(t) are known premise variables. ΔAi, and ΔBi are time-varying matrices representing parametric uncertainties in the plant model. These uncertainties are admissibly norm-bound and structured.The final outputs of the fuzzy systems are:(2.2)x˙(t)=∑i=1r‍hi(z(t)){(Ai+ΔAi)x(t)+(Bi+ΔBi)u(t)},
where
(2.3)z(t)=[z1(t)z2(t)⋯zp(t)],hi(z(t))=wi(z(t))∑i=1rwi(z(t)),wi(z(t))=∏j=1p‍Mij(zj(t))∀t.The termMi1(zj(t)) is the grade of membership of zj(t) in Mi1(2.4)Since{∑i=1r‍wi(z(t))≻0wi(z(t))≥0,i=1,2,…,rwehave{∑i=1r‍hi(z(t))=1,hi(z(t))≥0,i=1,2,…,r
for all t.We have the following property:(2.5)∑k=1r‍h˙k(z(t))=0.
This study investigates the PDC controller influence on the closed-loop stability region and gives robustness analysis of uncertain Takagi-Sugeno fuzzy system. Thus, we consider a PDC fuzzy controller which examines the derivative membership function and it is given by
(2.6)u(t)=-∑i=1r‍hi(z(t))Fix(t)-∑ρ=1r‍h˙ρ(z(t))(Kρ+R)x(t).
The fuzzy controller design consists to determine the local feedback gains Fi, Kρ, and R in the consequent parts. The state variables are determined by an observer and are detailed in the next section.The open-loop system is given by(2.7)x˙(t)=∑i=1r‍hi(z(t))(Ai+ΔAi)x(t).
By substituting (2.6) into (2.2), the closed-loop fuzzy system can be represented as:
(2.8)x˙(t)=∑i=1r‍∑j=1r‍hi(z(t))hj(z(t)){AΔi-BΔiFj-∑ρ=1r‍h˙ρ(z(t))BΔi(Kρ+R)}x(t),
where AΔi=Ai+ΔAi and BΔi=Bi+ΔBi.Assumption 2.1.
The time derivative of the premises membership function is upper bound such that|h˙k|≤ϕk, for k=1,…,r, where, ϕk,k=1,…,r are given positive constants.Assumption 2.2.
The matrices denote the uncertainties in the system and take the form of(2.9)ΔAi=DaiFai(t)Eai,ΔBi=DbiFbi(t)Ebi,
where Dai,Dbi,Eai, and Ebi are known constant matrices and Fai(t) and Fbi(t) are unknown matrix functions satisfying:
(2.10)FaiT(t)Fai(t)≤I,∀t,FbiT(t)Fbi(t)≤I,∀t,
where I is an appropriately dimensioned identity matrix.Lemma 2.3 (Boyd et al. Schur complement [12]).
Given constant matricesΩ1,Ω2, and Ω3 with appropriate dimensions, where Ω1=Ω1T and Ω2=Ω2T, then
(2.11)Ω1+Ω3TΩ2-1Ω3≺0
if and only if
(2.12)[Ω1Ω3T-Ω2]≻0or[-Ω2Ω3Ω1]≻0.Lemma 2.4 (Peterson and Hollot [2]).
LetQ=QT,H,E, and F(t) satisfying FT(t)F(t)≤I are appropriately dimensional matrices then the following inequality
(2.13)Q+HF(t)E+ETFT(t)HT≺0
is true, if and only if the following inequality holds for any λ≻0(2.14)Q+λ-1HHT+λETE≺0.The aim of the next section is to find conditions for the stabilization of the closed-loopT-S fuzzy system by using the Lyapunov theory.


## 3. Main Results

Consider the closed-loop system without uncertainties(3.1)x˙(t)=∑i=1r‍hi(z(t)){Aix(t)+Biu(t)}.
In order to give stability conditions, the slack matrix variables and the membership function boundary Mozelli et al. [14] are used. Consider the following null product that will serve stability analysis purposes:
(3.2)2[xT(t)M+x˙T(t)μM]×[x˙(t)-∑i=1r‍hi(z(t)){Aix(t)+Biu(t)}]=0.


### 3.1. A PDC Controller with Derivative Membership Function

By substituting (2.6) into (3.1), the closed-loop fuzzy system can be represented as:
(3.3)x˙(t)=∑i=1r‍∑j=1r‍hi(z(t))hj(z(t)){Ai-BiFj-∑m=1r‍h˙m(z(t))Bi(Km+R)}x(t).
The next Theorem gives sufficient conditions to guarantee stability of system (3.3).Theorem 3.1.
Under Assumptions2.1 and 2.2, and for μ≻0,ε≥0, the Takagi-Sugeno fuzzy system (3.1) is stabilizable with the PDC controller (2.6), with gains given by Fi=SiTH-T, Kρ=VρTH-T, and R=VTH-T, if there exist positive definite symmetric matrices Tk,k=1,2,…,r,Y, and any matrices H,Si,Vρ, and V with appropriate dimensions such that the following LMIs hold.
(3.4)Ti≻0,Ti+Y≻0(i=1,2,…,r),Λii≺0,Λ-ij≺0,(3.5)Λ-ij=Λij+Λji,
where
(3.6)Λij=[Tϕ-AiHT-HAiT+BiSjT+SjBiT+BiV-ϕT+V-ϕBiT*Ti-μ(AiHT-BiSjT-BiV-ϕT)+Hμ(HT+H)],V-ϕ=∑ρ=1r‍ϕρ(VρT+VT),T-ϕ=∑k=1r‍ϕk(Tk+Y).Proof of Theorem3.1.
Let’s consider the fuzzy weighting-dependent Lyapunov-Krasovskii functional as:(3.7)V(x(t))=∑k=1r‍hk(z(t))·Vk(x(t)),
with Vk(x(t))=xT(t)(Pk+εX)x(t), k=1,2,…,r, where Pk=PkT, X=XT, ε≥0, and (Pk+εX)≥0.
This candidate Lyapunov function satisfies(i)
V(x(t)) is C1,(ii)
V(0)=0 and V(x(t))≥0 for x(t)≠0,(iii)
∥x(t)∥→∞⇒V(x(t))→∞.
The time derivative ofV(x(t)) is given by:
(3.8)V˙(x(t))=∑k=1r‍h˙k(z(t))Vk(x(t))+∑k=1r‍hk(z(t))V˙k(x(t)).
Adding the null product, then(3.9)V˙(x(t))=∑k=1r‍h˙k(z(t))Vk(x(t))+∑k=1r‍hk(z(t))V˙k(x(t))+2[xT(t)M+x˙T(t)μM]×[x˙(t)-∑i=1r‍∑j=1r‍hi(z(t))hj(z(t)){Ai-BiFj-∑ρ=1r‍h˙ρ(z(t))Bi(Kρ+R)}x(t)].
Equation (3.9) can be rewritten as,
(3.10)V˙(x(t))=Υ1(x,z)+Υ2(x,z),
where
(3.11)Υ1(x,z)=xT(t)(∑k=1r‍h˙k(z(t))·(Pk+εX))x(t)+2[xT(t)M+x˙T(t)μM]×[∑i=1r‍∑ρ=1r‍hi(z(t))h˙ρ(z(t))Bi(Kρ+R)x(t)],Υ2(x,z)=∑i=1r‍hi(z(t)){2xT(t)(Pi+εX)x˙(t)}+2[xT(t)M+x˙T(t)μM]×[x˙(t)-∑i=1r‍∑j=1r‍hi(z(t))hj(z(t)){Ai-BiFj}x(t)].
Then, based on Assumption2.1, an upper bound of Υ1(x,z) is obtained as:
(3.12)Υ1(x,z)≤∑i=1r‍hi(z(t)){[∑ρ=1r‍ϕρBi(Kρ+R)x(t)]∑k=1r‍ϕk·x(t)T(Pk+εX)x(t)+2[xT(t)M+x˙T(t)μM]∑i=1r‍hi(z(t))×[∑ρ=1r‍ϕρBi(Kρ+R)x(t)]}.
Based on (2.5), it follows that ∑k=1rh˙k(z(t))(1-ε)X=0 where X is any symmetric matrix of proper dimension.
Suppose thatX-=∑k=1rh˙k(z(t))(1-ε)X and adding X- to (3.12), then
(3.13)Υ1(x,z)≤∑i=1r‍hi(z(t)){[∑ρ=1r‍ϕρBi(Kρ+R)x(t)]∑k=1r‍ϕk·x(t)T(Pk+X)x(t)+2[xT(t)M+x˙T(t)μM]∑i=1r‍hi(z(t))×[∑ρ=1r‍ϕρBi(Kρ+R)x(t)]}=∑i=1r‍hi(z(t)){x(t)TP-ϕx(t)+2xT(t)M·BiK-ϕx(t)+2x˙T(t)μM·BiK-ϕx(t)},
where
(3.14)P-ϕ=∑k=1r‍ϕk·(Pk+X),K-ϕ=∑ρ=1r‍ϕρ(Kρ+R).
Then,(3.15)V˙(x(t))≤∑i=1r‍∑j=1r‍hi(z(t))hj(z(t))×{x(t)TP-ϕx(t)+2xT(t)M·BiK-ϕx(t)+2x˙T(t)μM·BiK-ϕx(t)+2xT(t)(Pi+εR)x˙(t)+2xT(t)Mx˙(t)+2x˙T(t)μMx˙(t)-2xT(t)M(Ai-BiFj)x(t)-2x˙T(t)μM(Ai-BiFj)x(t)}.
Using vectorηT=[xT(t)x˙T(t)]T, (3.15) can be rewritten as
(3.16)V˙(x(t))≤∑i=1r‍∑j=1r‍hi(z(t))hj(z(t))ηTΞijη=∑i=1r‍hi2(z(t))ηTΞiiη+∑i=1r‍∑i≺j‍hi(z(t))hj(z(t))ηT(Ξij+Ξji)η,
where
(3.17)Ξij=[{P-ϕ-M(Ai-BiFj-BiK-ϕ)-(Ai-BiFj-BiK-ϕ)TMT}*(Pi+εX)-μM(Ai-BiFj-BiK-ϕ)+MTμ(M+MT)].
IfΞii≺0 and (Ξij+Ξji)≺0, then V˙(x(t))≺0 and (3.3) is stable. Pre- and postmultiplying Ξii≺0 and (Ξij+Ξji)≺0 by nonsingular matrices diag(M-1,M-1) and diag(M-T,M-T), respectively, and pre- and postmultiplying by M-1 and M-T, respectively, then we obtain
(3.18)M-1(Pi+εX)M-T≻0(i=1,2,…,r),M-1(Pi+X)M-T≻0(i=1,2,…,r),Λii≺0(i=1,2,…,r),Λ-ij≺0(i≺=1,2,…,r),
where Λ-ij=Λij+Λji, and
(3.19)Λij=[{(Ai-BiFj-BiK-ϕ)TM-1P-ϕM-T-(Ai-BiFj-BiK-ϕ)M-T-M-1(Ai-BiFj-BiK-ϕ)T}*{M-1(Pi+εX)M-T-μ(Ai-BiFj-BiK-ϕ)M-T+M-1}μ(M-T+M-1)]
for the following variables definition:
(3.20)H=M-1,Ti=H(Pi+εX)HT,Tϕ=HP-ϕHT,Sj=HFjT,Vj=HKρT,V=HRT,Y=HXHT.
If LMI in  holds then the closed-loop continuous fuzzy system (3.3) is asymptotically stable. The control gains are given by Fi=SiTH-T, Kρ=VρTH-T, and R=VTH-T. This completes the proof.Remark 3.2.
The selection ofϕk given in Assumption 2.1 is performed by using a simple procedure given in  [15].Remark 3.3.
The major contribution of the Theorem3.1 is represented by the proposed PDC controller given by (2.6). The contribution appears in the gains (Kρ+R) introduced in the controller term based on derivative membership functions. The stabilization condition proposed is less conservative than some of those in the literature, as is shown in the example below.


### 3.2. Observer Design

In order to determine state variables of the system, this section gives a solution by the means of fuzzy observer design. The following condition is to be satisfied by the observer:(3.21)x(t)-x^(t)→0ast→∞,
where x^(t) denotes the state vector estimated by a fuzzy observer. This condition guarantees that the steady-state error between x(t) and x^(t) converge to 0 and we denote this error by e(t)=x(t)-x^(t).A stabilizing observer-based controller can be formulated as follow:(3.22)x˙^(t)=∑j=1r‍hi(z(t)){Aix^(t)+Biu(t)+Lj(Cix^(t)-y(t))},y(t)=-∑i=1r‍hi(z(t))Cix^(t),
where y(t) denotes the output vector.We consider the proposed PDC controller given by (2.6):
(3.23)u(t)=-∑i=1r‍hi(z(t))Fix^(t)-∑ρ=1r‍h˙ρ(z(t))(Kρ+R)x^(t).Replacing the fuzzy controller (2.6) in fuzzy observer  we obtain the closed-loop fuzzy system as:
(3.24)x˙(t)=∑i=1r‍∑j=1r‍hi(z(t))hj(z(t)){(Ai-BiFj)-∑ρ=1r‍h˙ρ(z(t))(Kρ+R)}x(t)+∑i=1r‍∑j=1r‍hi(z(t))hj(z(t)){BiFj+∑ρ=1r‍h˙ρ(z(t))(Kρ+R)}e(t)e˙(t)=∑i=1r‍∑j=1r‍hi(z(t))hj(z(t)){Ai-LiCj}e(t).The augmented system is represented as follows:(3.25)x˙a(t)=∑i=1r‍∑j=1r‍hi(z(t))hj(z(t))G~ijxa(t)=∑j=1r‍hi(z(t))hj(z(t))G~iixa(t)+2∑i=1r‍∑i≺jr‍hi(z(t))hj(z(t)){G~ij+G~ji2}xa(t),
where
(3.26)xa(t)=[x(t)e(t)],G~ij=[Ai-BiFj-∑ρ=1r‍h˙ρBi(Kρ+R)BiFj+∑ρ=1r‍h˙ρBi(Kρ+R)0Ai-LiCj].By applying Theorem 1 in [16] in the augmented system (3.25) we derive the following Theorem.Theorem 3.4.
Under Assumptions2.1 and 2.2, and for 0≤μ≤1, the Takagi-Sugeno fuzzy system (3.1) is stable if there exist positive definite symmetric matrices Pk,k=1,2,…,r, and R, matrices F1,…,Fr such that the following LMIs holds.
(3.27)Pk+R≻0,k∈{1,…,r},Pj+μR≥0,j=1,2,…,r,Pϕ+{G~iiT(Pk+μR)+(Pk+μR)G~ii}≺0,i,k∈{1,…,r},{G~ij+G~ji2}T(Pk+μR)+(Pk+μR){G~ij+G~ji2}≺0,fori,j,k=1,2,…,rsuchthati≺j,
where
(3.28)G~ij=[Ai-BiFj-∑ρ=1r‍h˙ρBi(Kρ+R)BiFj+∑ρ=1r‍h˙ρBi(Kρ+R)0Ai-LiCj],Pϕ=∑k=1r‍ϕk(Pk+R).Proof of Theorem3.4.
The result follows immediately from the proof of  Theorem 1 in [16].


## 3.1. A PDC Controller with Derivative Membership Function

By substituting (2.6) into (3.1), the closed-loop fuzzy system can be represented as:
(3.3)x˙(t)=∑i=1r‍∑j=1r‍hi(z(t))hj(z(t)){Ai-BiFj-∑m=1r‍h˙m(z(t))Bi(Km+R)}x(t).
The next Theorem gives sufficient conditions to guarantee stability of system (3.3).Theorem 3.1.
Under Assumptions2.1 and 2.2, and for μ≻0,ε≥0, the Takagi-Sugeno fuzzy system (3.1) is stabilizable with the PDC controller (2.6), with gains given by Fi=SiTH-T, Kρ=VρTH-T, and R=VTH-T, if there exist positive definite symmetric matrices Tk,k=1,2,…,r,Y, and any matrices H,Si,Vρ, and V with appropriate dimensions such that the following LMIs hold.
(3.4)Ti≻0,Ti+Y≻0(i=1,2,…,r),Λii≺0,Λ-ij≺0,(3.5)Λ-ij=Λij+Λji,
where
(3.6)Λij=[Tϕ-AiHT-HAiT+BiSjT+SjBiT+BiV-ϕT+V-ϕBiT*Ti-μ(AiHT-BiSjT-BiV-ϕT)+Hμ(HT+H)],V-ϕ=∑ρ=1r‍ϕρ(VρT+VT),T-ϕ=∑k=1r‍ϕk(Tk+Y).Proof of Theorem3.1.
Let’s consider the fuzzy weighting-dependent Lyapunov-Krasovskii functional as:(3.7)V(x(t))=∑k=1r‍hk(z(t))·Vk(x(t)),
with Vk(x(t))=xT(t)(Pk+εX)x(t), k=1,2,…,r, where Pk=PkT, X=XT, ε≥0, and (Pk+εX)≥0.
This candidate Lyapunov function satisfies(i)
V(x(t)) is C1,(ii)
V(0)=0 and V(x(t))≥0 for x(t)≠0,(iii)
∥x(t)∥→∞⇒V(x(t))→∞.
The time derivative ofV(x(t)) is given by:
(3.8)V˙(x(t))=∑k=1r‍h˙k(z(t))Vk(x(t))+∑k=1r‍hk(z(t))V˙k(x(t)).
Adding the null product, then(3.9)V˙(x(t))=∑k=1r‍h˙k(z(t))Vk(x(t))+∑k=1r‍hk(z(t))V˙k(x(t))+2[xT(t)M+x˙T(t)μM]×[x˙(t)-∑i=1r‍∑j=1r‍hi(z(t))hj(z(t)){Ai-BiFj-∑ρ=1r‍h˙ρ(z(t))Bi(Kρ+R)}x(t)].
Equation (3.9) can be rewritten as,
(3.10)V˙(x(t))=Υ1(x,z)+Υ2(x,z),
where
(3.11)Υ1(x,z)=xT(t)(∑k=1r‍h˙k(z(t))·(Pk+εX))x(t)+2[xT(t)M+x˙T(t)μM]×[∑i=1r‍∑ρ=1r‍hi(z(t))h˙ρ(z(t))Bi(Kρ+R)x(t)],Υ2(x,z)=∑i=1r‍hi(z(t)){2xT(t)(Pi+εX)x˙(t)}+2[xT(t)M+x˙T(t)μM]×[x˙(t)-∑i=1r‍∑j=1r‍hi(z(t))hj(z(t)){Ai-BiFj}x(t)].
Then, based on Assumption2.1, an upper bound of Υ1(x,z) is obtained as:
(3.12)Υ1(x,z)≤∑i=1r‍hi(z(t)){[∑ρ=1r‍ϕρBi(Kρ+R)x(t)]∑k=1r‍ϕk·x(t)T(Pk+εX)x(t)+2[xT(t)M+x˙T(t)μM]∑i=1r‍hi(z(t))×[∑ρ=1r‍ϕρBi(Kρ+R)x(t)]}.
Based on (2.5), it follows that ∑k=1rh˙k(z(t))(1-ε)X=0 where X is any symmetric matrix of proper dimension.
Suppose thatX-=∑k=1rh˙k(z(t))(1-ε)X and adding X- to (3.12), then
(3.13)Υ1(x,z)≤∑i=1r‍hi(z(t)){[∑ρ=1r‍ϕρBi(Kρ+R)x(t)]∑k=1r‍ϕk·x(t)T(Pk+X)x(t)+2[xT(t)M+x˙T(t)μM]∑i=1r‍hi(z(t))×[∑ρ=1r‍ϕρBi(Kρ+R)x(t)]}=∑i=1r‍hi(z(t)){x(t)TP-ϕx(t)+2xT(t)M·BiK-ϕx(t)+2x˙T(t)μM·BiK-ϕx(t)},
where
(3.14)P-ϕ=∑k=1r‍ϕk·(Pk+X),K-ϕ=∑ρ=1r‍ϕρ(Kρ+R).
Then,(3.15)V˙(x(t))≤∑i=1r‍∑j=1r‍hi(z(t))hj(z(t))×{x(t)TP-ϕx(t)+2xT(t)M·BiK-ϕx(t)+2x˙T(t)μM·BiK-ϕx(t)+2xT(t)(Pi+εR)x˙(t)+2xT(t)Mx˙(t)+2x˙T(t)μMx˙(t)-2xT(t)M(Ai-BiFj)x(t)-2x˙T(t)μM(Ai-BiFj)x(t)}.
Using vectorηT=[xT(t)x˙T(t)]T, (3.15) can be rewritten as
(3.16)V˙(x(t))≤∑i=1r‍∑j=1r‍hi(z(t))hj(z(t))ηTΞijη=∑i=1r‍hi2(z(t))ηTΞiiη+∑i=1r‍∑i≺j‍hi(z(t))hj(z(t))ηT(Ξij+Ξji)η,
where
(3.17)Ξij=[{P-ϕ-M(Ai-BiFj-BiK-ϕ)-(Ai-BiFj-BiK-ϕ)TMT}*(Pi+εX)-μM(Ai-BiFj-BiK-ϕ)+MTμ(M+MT)].
IfΞii≺0 and (Ξij+Ξji)≺0, then V˙(x(t))≺0 and (3.3) is stable. Pre- and postmultiplying Ξii≺0 and (Ξij+Ξji)≺0 by nonsingular matrices diag(M-1,M-1) and diag(M-T,M-T), respectively, and pre- and postmultiplying by M-1 and M-T, respectively, then we obtain
(3.18)M-1(Pi+εX)M-T≻0(i=1,2,…,r),M-1(Pi+X)M-T≻0(i=1,2,…,r),Λii≺0(i=1,2,…,r),Λ-ij≺0(i≺=1,2,…,r),
where Λ-ij=Λij+Λji, and
(3.19)Λij=[{(Ai-BiFj-BiK-ϕ)TM-1P-ϕM-T-(Ai-BiFj-BiK-ϕ)M-T-M-1(Ai-BiFj-BiK-ϕ)T}*{M-1(Pi+εX)M-T-μ(Ai-BiFj-BiK-ϕ)M-T+M-1}μ(M-T+M-1)]
for the following variables definition:
(3.20)H=M-1,Ti=H(Pi+εX)HT,Tϕ=HP-ϕHT,Sj=HFjT,Vj=HKρT,V=HRT,Y=HXHT.
If LMI in  holds then the closed-loop continuous fuzzy system (3.3) is asymptotically stable. The control gains are given by Fi=SiTH-T, Kρ=VρTH-T, and R=VTH-T. This completes the proof.Remark 3.2.
The selection ofϕk given in Assumption 2.1 is performed by using a simple procedure given in  [15].Remark 3.3.
The major contribution of the Theorem3.1 is represented by the proposed PDC controller given by (2.6). The contribution appears in the gains (Kρ+R) introduced in the controller term based on derivative membership functions. The stabilization condition proposed is less conservative than some of those in the literature, as is shown in the example below.


## 3.2. Observer Design

In order to determine state variables of the system, this section gives a solution by the means of fuzzy observer design. The following condition is to be satisfied by the observer:(3.21)x(t)-x^(t)→0ast→∞,
where x^(t) denotes the state vector estimated by a fuzzy observer. This condition guarantees that the steady-state error between x(t) and x^(t) converge to 0 and we denote this error by e(t)=x(t)-x^(t).A stabilizing observer-based controller can be formulated as follow:(3.22)x˙^(t)=∑j=1r‍hi(z(t)){Aix^(t)+Biu(t)+Lj(Cix^(t)-y(t))},y(t)=-∑i=1r‍hi(z(t))Cix^(t),
where y(t) denotes the output vector.We consider the proposed PDC controller given by (2.6):
(3.23)u(t)=-∑i=1r‍hi(z(t))Fix^(t)-∑ρ=1r‍h˙ρ(z(t))(Kρ+R)x^(t).Replacing the fuzzy controller (2.6) in fuzzy observer  we obtain the closed-loop fuzzy system as:
(3.24)x˙(t)=∑i=1r‍∑j=1r‍hi(z(t))hj(z(t)){(Ai-BiFj)-∑ρ=1r‍h˙ρ(z(t))(Kρ+R)}x(t)+∑i=1r‍∑j=1r‍hi(z(t))hj(z(t)){BiFj+∑ρ=1r‍h˙ρ(z(t))(Kρ+R)}e(t)e˙(t)=∑i=1r‍∑j=1r‍hi(z(t))hj(z(t)){Ai-LiCj}e(t).The augmented system is represented as follows:(3.25)x˙a(t)=∑i=1r‍∑j=1r‍hi(z(t))hj(z(t))G~ijxa(t)=∑j=1r‍hi(z(t))hj(z(t))G~iixa(t)+2∑i=1r‍∑i≺jr‍hi(z(t))hj(z(t)){G~ij+G~ji2}xa(t),
where
(3.26)xa(t)=[x(t)e(t)],G~ij=[Ai-BiFj-∑ρ=1r‍h˙ρBi(Kρ+R)BiFj+∑ρ=1r‍h˙ρBi(Kρ+R)0Ai-LiCj].By applying Theorem 1 in [16] in the augmented system (3.25) we derive the following Theorem.Theorem 3.4.
Under Assumptions2.1 and 2.2, and for 0≤μ≤1, the Takagi-Sugeno fuzzy system (3.1) is stable if there exist positive definite symmetric matrices Pk,k=1,2,…,r, and R, matrices F1,…,Fr such that the following LMIs holds.
(3.27)Pk+R≻0,k∈{1,…,r},Pj+μR≥0,j=1,2,…,r,Pϕ+{G~iiT(Pk+μR)+(Pk+μR)G~ii}≺0,i,k∈{1,…,r},{G~ij+G~ji2}T(Pk+μR)+(Pk+μR){G~ij+G~ji2}≺0,fori,j,k=1,2,…,rsuchthati≺j,
where
(3.28)G~ij=[Ai-BiFj-∑ρ=1r‍h˙ρBi(Kρ+R)BiFj+∑ρ=1r‍h˙ρBi(Kρ+R)0Ai-LiCj],Pϕ=∑k=1r‍ϕk(Pk+R).Proof of Theorem3.4.
The result follows immediately from the proof of  Theorem 1 in [16].


## 4. Robust Stability Condition with PDC Controller

Consider the uncertain closed-loop system (2.8). A sufficient robust stability condition is given as follows.Theorem 4.1.
Under Assumptions2.1 and 2.2, and for μ≻0,ε≥0, the Takagi-Sugeno fuzzy system (2.2) is stabilizable with the PDC controller (2.6), with gains given by Fi=SiTH-T, Kρ=VρTH-T, and R=VTH-T, if there exist positive definite symmetric matrices Tk,k=1,2,…,r,Y, and any matrices H,Si,Vρ, and V with appropriate dimensions such that the following LMIs hold.
(4.1)Ti≻0,Ti+Y≻0(i=1,2,…,r),∑ii≺0,∑ij-≺0,(4.2)∑ij-=∑ij+∑ji,Tϕ=∑k=1r‍ϕk(Tk+Y),
where
(4.3)Σij=[Φ11Φ12MDaiMDbiEaiT-(Ebi(Fi+K-ϕ))TΦ22μMDaiμMDbi00*-λI000**-λI00***-λ-1I0****-λ-1I],
with
(4.4)Φ11=P-ϕ+[M·BiK-ϕ+K-ϕTBiTM]-[MGii+GiiTMT],Φ12=(Pi+εX)-μ(Gii-BiK-ϕ)TMT+M,Φ22=μ(M+MT),
where Gij=Ai-BiFj, Gii=Ai-BiFi, and Pϕ=∑k=1rϕk(Pk+R).Proof of Theorem4.1.
The result follows immediately from the proof of Theorem3.1 by replacing in the matrix inequality Ai with Ai+DaiF(t)Eai and Bi with Bi+DbiF(t)Ebi, we obtain the following inequality:
(4.5)V˙(x(t))≤∑i=1r‍∑j=1r‍hi(z(t))hj(z(t))×{([DaiDbi][Fai(t)00Fbi(t)][Eai-Ebi(Fj+K-ϕ)])x(t)TP-ϕx(t)+2xT(t)M·BiK-ϕx(t)+2x˙T(t)μM·BiK-ϕx(t)+2xT(t)(Pi+εR)x˙(t)+2xT(t)Mx˙(t)+2x˙T(t)μMx˙(t)-2xT(t)M(Ai-BiFj)x(t)-2xT(t)M×([DaiDbi][Fai(t)00Fbi(t)][Eai-Ebi(Fj+K-ϕ)])x(t)-2x˙T(t)μM(Ai-BiFj)x(t)-2x˙T(t)μM×([DaiDbi][Fai(t)00Fbi(t)][Eai-Ebi(Fj+K-ϕ)])x(t)}.
Using vectorηT=[xT(t)x˙T(t)]T, (4.5) can be rewritten as
(4.6)V˙(x(t))≤∑i=1r‍∑j=1r‍hi(z(t))hj(z(t))ηTΞ~ijη=∑i=1r‍hi2(z(t))ηTΞ~iiη+∑i=1r‍∑i≺j‍hi(z(t))hj(z(t))ηT(Ξ~ij+Ξ~ji)η,
where
(4.7)Ξ~ij=Ξij+[{M([DaiDbi][Fai(t)00Fbi(t)][Eai-Ebi(Fj+K-ϕ)])+([DaiDbi][Fai(t)00Fbi(t)][Eai-Ebi(Fj+K-ϕ)])TMT}*μM([DaiDbi][Fai(t)00Fbi(t)][Eai-Ebi(Fj+K-ϕ)])0],Ξ~ij=Ξij+[M[DaiDbi]μM[DaiDbi]][Fai(t)00Fbi(t)][Eai0-Ebi(Fj+K-ϕ)0]+[EaiT-(Ebi(Fj+K-ϕ))T00][Fai(t)00Fbi(t)]T[DaiTMTμDaiTMTDbiTMTμDbiTMT].
Then, based on Lemma2.4, an upper bound of Ξ~ij obtained as:
(4.8)Ξ~ij=Ξij+λ-1M[DaiDbiμDaiμDbi][DaiTμDaiTDbiTμDbiT]MT+λ[EaiT-(Ebi(Fj+K-ϕ))T00][Eai0-Ebi(Fj+K-ϕ)0]≺0
by Schur complement, we obtain,
(4.9)[Φ11Φ12MDaiMDbiEaiT-(Ebi(Fj+K-ϕ))TΦ22μMDaiμMDbi00*-λI000**-λI00***-λ-1I0****-λ-1I]≺0,
with
(4.10)Φ11=P-ϕ+[M·BiK-ϕ+K-TϕBiTM]-[M(Ai-BiFj)+(Ai-BiFj)TMT],Φ12=(Pi+εX)-μ(Ai-BiFj-BiK-ϕ)TMT+M,Φ22=μ(M+MT).
If the LMI  holds then the system (2.8) is stable. This completes the proof.The following theorem gives sufficient conditions for robust PDC controller design.Theorem 4.2.
Under Assumptions2.1 and 2.2, and for μ≻0,ε≥0, the Takagi-Sugeno robust fuzzy system (2.2) is stabilizable with the PDC controller (2.6), with gains given by Fi=S~iTH~-T, Kρ=V~ρTH~-T, and R=V~TH~-T, if there exist positive definite symmetric matrices T~k, k=1,2,…,r, Y~, and any matrices H~,S~i,V~ρ, and V~ with appropriate dimensions such that the following LMIs hold
(4.11)T~i≻0,T~i+Y~≻0(i=1,2,…,r),Λ~ii≺0,Λ~-ij≺0,(4.12)Λ~-ij=Λ~ij+Λ~ji,T~ϕ=∑k=1r‍ϕk(T~k+Y~),V-~ϕ=∑ρ=1r‍ϕρ(V~ρT+V~T),
where
(4.13)Λ~ij=[Φ~11Φ~12DaiDbiHEaiT-S~jEbiT-V-~ϕEbiTΦ~22μDaiμDbi00*-λI000**-λI00***-λ-1I0****-λ-1I],
with
(4.14)Φ~11=T~ϕ-AiH~T-H~AiT+BiS~jT+S~jBiT+BiV-~ϕT+V-~ϕBiT,Φ~12=T~i-μ(AiH~T-BiS~jT-BiV-~ϕT)+H~,Φ~22=μ(HT+H).Proof of Theorem4.2.
Consider (4.8) and pre- and postmultiplying by M-1 and M-T, respectively, then we obtain
(4.15)Λ~ij=Λij+λ-1[DaiDbiμDaiμDbi][DaiTμDaiTDbiTμDbiT]+λM-1[EaiT-(Ebi(Fj+K-ϕ))T00][Eai0-Ebi(Fj+K-ϕ)0]M-T≺0,
with Λij defined by (3.5).
By Schur complement, we obtain,(4.16)[Φ~11Φ~12DaiDbiM-1EaiT-M-1(Ebi(Fj+K-ϕ))TΦ~22μDaiμDbi00*-λI000**-λI00***-λ-1I0****-λ-1I]≺0,
with
(4.17)Φ~11=M-1P-ϕM-T-(Ai-BiFj-BiK-ϕ)M-T-M-1(Ai-BiFj-BiK-ϕ)T,Φ~12=M-1(Pi+εX)M-T-μ(Ai-BiFj-BiK-ϕ)M-T+M-1,Φ~22=μ(M-T+M-1)
for the following variables definition:
(4.18)H~=M-1,T~i=H~(Pi+εX)H~T,T~ϕ=H~P-ϕH~T,S~j=H~FjT,V~j=H~KρT,V~=H~RT,Y~=H~XH~T.
If LMI in holds then the closed-loop continuous fuzzy system (2.8) is asymptotically stable.
The control gains are given byFi=S~iTH~-T, Kρ=V~ρTH~-T, and R=V~TH~-T. This completes the proof.


## 5. Numerical Examples

In order to show the improvements of the proposed approaches over some existing results, in this section, we present a numerical example in which we present the feasible area for aT-S fuzzy system. Indeed, we compare the proposed fuzzy Lyapunov approaches (Theorem 3.4) with result provided by [17], and in [14, Theorem  6]. A second example is given to improve the given gains of robust PDC controller.Example 5.1.
Consider the following continuousT-S fuzzy system:
(5.1)x˙(t)=∑i=1r‍hi(z(t))Aix(t),
with
(5.2)r=2;A1=[3.6-1.66.2-4.3],A2=[-a-1.66.2-4.3],B1=[-0.45-3],B2=[-b-3],
where a∈[0,25], b∈[0,2], considering μ=0.04 and ϕ1,2=1.
The proposed approach (Theorem3.4) gives less conservative stabilization conditions (Figure 1) than some recent results provided by [14, 17].Figure 1
Feasible area provided by [17] (+), [14] (◯) and Theorem 3.1 (•).Example 5.2.
Consider the uncertain continuousT-S fuzzy system given by (2.8) with
(5.3)r=2;A1=[3.6-1.66.2-4.3],A2=[-a-1.66.2-4.3],B1=[-0.45-3],B2=[-b-3],Da1=Ea1=[0.3000],Db1=Eb1=[0.30],Da2=Ea2=[-0.2000.3],Db2=Eb2=[-0.20.3]
for a=1, b=0.5, considering μ=0.04, λ=0.1 and ϕ1,2=1, we find the following gains values:
(5.4)F1=[-5.03401.5230],F2=[3.7868-0.8082],K1=[-180.233465.7330],K2=[-180.949465.9073],R=[201.9216-71.7345].


## 6. Conclusion

This paper provided new conditions for the stabilization with a class of PDC controller of Takagi-Sugeno fuzzy systems in terms of a combination of the LMI approach and the use of nonquadratic Lyapunov function as fuzzy Lyapunov function. In addition, the time derivative of membership function is considered by the PDC fuzzy controller and the slack matrix variables are introduced in order to facilitate the stability analysis. An approach to design an observer is derived in order to estimate variable states. In addition, a new condition of the stabilization of uncertain system is given in this paper.The stabilization condition proposed in this paper is less conservative than some of those in the literature, which has been illustrated via examples.



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*Source: 836814-2012-10-17.xml*