# Decentralized Control of Uncertain Fuzzy Large-Scale System with Time Delay and Optimization

Authors: Hang Zhang; Xia Wang; Yi-Jun Wang; Zhao-Mei Sun
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This paper studies the decentralized stabilization problem for an uncertain fuzzy large-scale system with time delays. The considered large-scale system is composed of several T-S fuzzy subsystems. The decentralized parallel distributed compensation (PDC) fuzzy control for each subsystem is designed to stabilize the whole system. Based on Lyapunov criterion, some sufficient conditions are proposed. Moreover, the positive definite matricesPi and PDC gain Kij can be solved by linear matrix inequality (LMI) toolbox of Matlab. Then, the optimization design method for decentralized control is also considered with respect to a quadratic performance index. Finally, numerical
examples are given and compared with those of Zhang et al., 2004 to illustrate the effectiveness and less conservativeness
of our method.


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## Body

## 1. Introduction

Many real-life problems, such as power system, economic systems, societal system, and nuclear system are frequently of high dimension. Such systems are regarded as large-scale system. They consist of a number of subsystems which serve particular functions, share resources, and are governed by a set of interrelated goals and constrains [1]. Over the past decades, many methods have been to investigate the stability and stabilization of large-scale system [2–10].Fuzzy systems of Takagi-Sugeno (T-S) models [11] have become an effective method to represent complex nonlinear dynamic system by fuzzy sets and fuzzy reasoning. The method of T-S model is feasible since, in many situations, human experts can provide linguistic descriptions of local systems in terms of IF-THEN rules [12–14]. Reference [15] proposed a control concept “parallel distributed compensation” (PDC) for fuzzy controller design of T-S fuzzy model. Under some conditions, PDC can stabilize the closed-loop fuzzy system asymptotically. Linear matrix inequalities (LMIs) methods to find the common positive matrix Pi always play a key role work in PDC design [16].Now let us consider the control problem of large-scale system. Accordingly, suppose a large-scale system is composed of a number of subsystems with interconnection. Each subsystem is described by a T-S fuzzy model. That is, each subsystem dynamic is captured by a set of fuzzy implications that characterize local relations in the state space. Then, the global model of large-scale system can be achieved by smoothly connecting the local linear model in each fuzzy subspace together via the membership functions. Recently, the fuzzy decentralized control design methods and stability condition are addressed [17–24]. Wang and Luoh [17] have studied a fuzzy decentralized control design method for a fuzzy large-scale system on the assumption that all variables are available. Tseng and Chen [18] dealt with the model reference tracking control problem by using H∞ decentralized fuzzy control, which relaxed the condition that the state variables are measurable. Wang and Zhang  [19] considered the robust decentralized controller design method for nonlinear large-scale descriptor system. Hsiao et al. [20] proposed fuzzy decentralized control design methods for fuzzy large-scale system with time delays and gave the analysis of the closed-loop fuzzy large-scale system. Robust decentralized H∞ output feedback controller was designed in [21]. Tong and Zhao [22] proposed a stabilization criterion of continuous-time interconnected fuzzy systems without uncertainties. Su and Liu [23] dealt with decentralized stabilization problem for a large-scale system in which the system is composed of several T-S fuzzy subsystems with nonlinear interconnections. Wang et al. [24] addressed robust H∞ fuzzy controller design to overcome the parametric uncertainties of fuzzy large-scale systems without time delay and get H∞ performance. Zhang et al. [25] proposed the stabilization problem of fuzzy large-scale system without parametric uncertainties.However, the proposed decentralized control designs and the sufficient conditions of closed-loop system did not consider the parametric uncertainties or time delays, which is important in both theory and real-world application. In this paper, the stabilization problem of the uncertain fuzzy large-scale systems with time delays is considered. Based on decentralized control concept, we like to synthesize a PDC controller for each subsystem so that the whole system can be stabilized asymptotically. When applied to degenerated cases, (without uncertainties or time delay), the stabilization criteria are better than existing ones. Stability is one of the most important performance indecies, but it is not enough for control systems. Linear quadratic performance can reflect a lot of performance requirements, so a quadratic performance index is considered in this paper. Based on Lyapunov function stability theory, the optimization design method for decentralized control with respect to the quadratic performance index is transformed into solving linear inequality matrix by PDC controller.This paper is organized as follows. In Section2, the considered systems are stated and some preliminaries are presented.The problem of stabilization for fuzzy large-scale system with uncertainty is proposed in Section 3. Section 4 provides examples to illustrate the correctness of our theoretic results. Finally, a conclusion is given in Section 5.


## 2. System Description and Preliminaries

Consider a fuzzy large-scale systemS, with uncertainties both in isolated subsystem and in their interconnection, which consists of J interconnected subsystems Si. The ith fuzzy subsystem is described by the following equations: (2.1)ẋi(t)=∑j=1rihij(t)[(Aij+ΔAij(t))xi(t)+(Bij+ΔBij(t))ui(t)]+∑k=1J[(Cki+ΔCki(t))xk(t)+(Dki+ΔDki(t))xk(t-τki)],
where (2.2)[ΔAij(t),ΔBij(t)]=EijFij(t)[Gij,Hij],[ΔCki(t),ΔDki(t)]=E̅kiF̅ki(t)[Lki,Mki]
for for all t≥0,j=1,2,…,ri,k=1,2,…,J, xi(t) is the state vector, ui(t) is the control input, Aij,Bij,Gij,Hij,Eij,Cki,Dki,E̅ki,Mki,Lki are known constant matrices with appropriate dimensions. τki is the time delay. hij(t) is the normalized weigh in (2.6). Fij(t) and F̅ki(t) are time-varying matrices with appropriate dimensions satisfying
(2.3)Fij(t)FijT(t)≤I,F̅ki(t)F̅kiT(t)≤I.
Each isolated subsystem Si is represented by a T-S fuzzy model. The jth rule of this T-S fuzzy model is represented as follows.Rulej. If z1i(t) is Mj1i and … and zpi(t) is Mjpi, then
(2.4)ẋi(t)=(Aij+ΔAij(t))xi(t)+(Bij+ΔBij(t))ui(t),
where zi(t)=[z1i(t),z2i(t),…,zpi(t)]T,z1i,…,zpi(t) are premise variables, and Mjli(l=1,2,…,p) are fuzzy sets. By “fuzzy blending,” the final output of the ith fuzzy subsystem is described as follows: (2.5)ẋi(t)=∑j=1riωij(t)[(Aij+ΔAij(t))xi(t)+(Bij+ΔBij(t))ui(t)]∑j=1riωij(t)=∑j=1rihij(t)[(Aij+ΔAij(t))xi(t)+(Bij+ΔBij(t))ui(t)]
with
(2.6)ωij(t)=∏l=1pMjli(zli(t)),hij(t)=ωij(t)∑j=1riωij(t),
where Mjil(zil(t)) is the grade of membership of zil(t) in Mjil, and ri is the number of fuzzy rules of subsystem Si. We assumed that ωij(t)≥0 for all t, j=1,2,…,ri. Therefore
(2.7)∑j=1rihij(t)=1.Define a quadratic performance index(2.8)J=∑i=1J∫0∞(xiT(t)Qixi(t)+uiT(t)Riui(t))dt,
where Qi and Ri are positive matrices.The main propose of this paper is to synthesize a decentralized PDC fuzzy controllerui(t) for each subsystem such that the closed-loop large-scale T-S fuzzy systems (2.1) is asymptotically stable and the optimization design method for decentralized control respect to the quadratic performance index.Before starting the main results, we need the following lemmas.Lemma 2.1 (see [26]).
LetQ be any of n×n matrix; one will have for any constant k≥0; and any positive matrix S>0 that
(2.9)2xTQy≤kxTQS-1QTx+1kyTSy
for all x,y∈Rn.Lemma 2.2 (see [26]).
LetD,E be any constant matrices and FTF≤I, where a is a positive constant, all matrices with appropriate dimensions; one will have for any constant k>0 such that
(2.10)2xTDEFy≤kxTDDTx+akyTETEy,
for all x,y∈Rn.Lemma 2.3 (see [27]).
LetRi be any positive matrix; one will have for any 0≤hij(t)≤1 such that
(2.11)(∑j=1rihij(t)Kij)TRi(∑j=1rihij(t)Kij)≤∑j=1rihij(t)KijTRiKij
for all Kij∈Rn.


## 3. Stabilization and PDC Synthesis of Fuzzy Large-Scale System

In this section, the decentralized concept and PDC approach are applied to synthesize a local feedback controller for each local subsystem. Let the fuzzy controller be as the PDC form:Rulej.If z1i(t) is Mj1i and... and zpi(t) is Mjpi, then (3.1)ui(t)=-Kijxi(t),
where i=1,2,…,J,j=1,2,…,ri. The overall state feedback fuzzy control law is represented by:
(3.2)ui(t)=-∑j=1riωij(t)Kijxi(t)∑j=1riωij(t)=-∑j=1rihij(t)Kijxi(t).Substituting (3.2) into (2.1), the closed-loop fuzzy subsystem becomes (3.3)ẋi(t)=∑j=1ri∑n=1rihij(t)hin(t)[(Aij+ΔAij)-(Bij+ΔBij)Kin]xi(t)+∑k=1J[(Cki+ΔCki)xk(t)+(Dki+ΔDki)xk(t-τki)].Now, our work is to determine the local feedback gainsKij such that the whole fuzzy large-scale system (3.3) is asymptotically stable.Theorem 3.1.
The fuzzy large-scale system (2.1) can be asymptotically stabilized by the decentralized PDC fuzzy control (3.2), if there exist matrices Fij, positive definite matrices Xi to satisfy the following LMIs:
(3.4)(XiAijT+AijXi-FinTBijT-BijFinXiGijT-FinTHijTXiMiTC̅iE̅iXiGijXi-HijFin-I0000MiXi0-I000C̅iT00-I00E̅iT000-12I0Xi0000-12JI)<0,
for i=1,2,…,J, j=1,2,…,ri,n=1,2,…,ri, where MiT=[Mi1,Mi2,…,MiJ,Li1,Li2,…,LiJ]T, C̅i=[Eij,C1i,C2i,…,CJi,D1i,D2i,…,DJi],E̅i=[E̅1i,E̅2i,…,E̅Ji], Xi=Pi-1, Fin=KinPi-1.Proof.
Let the Lyapunov functional be(3.5)V(t)=∑i=1JVi(t)=∑i=1J[xiTPixi+∑k=1J∫t-τkitxkT(s)(I+MkiTMki)xk(s)ds],
where Pi>0 is to be selected. It is obviously that there exist σ1 and σ2 such that
(3.6)σ1‖xi(t)‖2≤Vi(t)≤σ2‖xi(t)‖2.
Taking the derivative of the Vi(t) along the trajectories of (3.3),
(3.7)V̇i(t)=ẋiT(t)Pixi(t)+xiT(t)PiẋiT(t)+∑k=1JxkT(t)(I+MkiTMki)xk(t)-∑k=1JxkT(t-τki)(I+MkiTMki)xk(t-τki)=∑j=1ri∑n=1rihij(t)hin(t){∑k=1JxiT(t)[(Aij+ΔAij)TPi+Pi(Aij+ΔAij)-KinT(Bij+ΔBij)TPi-Pi(Bij+ΔBij)Kin]xi(t)+2xiT(t)Pi∑k=1J(Cki+ΔCki)xk(t)+2xiT(t)Pi∑k=1J(Dki+ΔDki)xk(t-τki)+∑k=1JxkT(t)(I+MkiTMki)xk(t)-∑k=1JxkT(t-τki)(I+MkiTMki)xk(t-τki)}=∑j=1ri∑n=1rihij(t)hin(t){∑k=1JxiT(t)[Nij+ΔAijTPi+PiΔAij-KinTΔBijTPi-PiΔBijKin]xi(t)+2xiT(t)Pi∑k=1JCkixk(t)+2xiT(t)Pi∑k=1JΔCkixk(t)+2xiT(t)Pi∑k=1JDkixk(t)+2xiT(t-τki)Pi∑k=1JΔDkixk(t-τki)+∑k=1JxkT(t)(I+MkiTMki)xk(t)-∑k=1JxkT(t-τki)(I+MkiTMki)xk(t-τki)},
where Nij=AijTPi+PiAij-KinTBijTPi-PiBijKin.
Using Lemma (2.2), we have
(3.8)xiT(t)[ΔAijTPi+PiΔAij-KinTΔBijTPi-PiΔBijKin]xi(t)=xiT(t)[PiEijFij(t)(Gij-HijKin)+(Gij-HijKin)TFijT(t)EijTPi]xi(t)≤xiT(t)(Gij-HijKin)T(Gij-HijKin)xi(t)+xiT(t)PiEijEijTPixi(t),2xiT(t)Pi∑k=1JΔCkixk(t)=2∑k=1JxiT(t)PiE̅kiF̅ki(t)Lkixk(t)≤xiT(t)Pi∑k=1JE̅kiE̅kiTPixi(t)+∑k=1JxkT(t)LkiLkiTxk(t),2xiT(t)Pi∑k=1JΔDkixk(t-τki)=2∑k=1JxiT(t)PiE̅kiF̅ki(t)Mkixk(t-τki)≤xiT(t)Pi∑k=1JE̅kiE̅kiTPixi(t)+∑k=1JxkT(t-τki)MkiMkiTxk(t-τki).
Using Lemma 2.1, we get
(3.9)2xiT(t)Pi∑k=1JCkixk(t)≤xiT(t)Pi∑k=1JCkiCkiTPixi(t)+∑k=1JxkT(t)xk(t),2xiT(t)Pi∑k=1JDkixk(t-τki)≤xiT(t)Pi∑k=1JDkiDkiTPixi(t)+∑k=1JxkT(t-τki)xk(t-τki),
Noticing that the facts as follows:
(3.10)∑i=1J∑k=1JxkT(t)xk(t)=∑i=1J∑k=1JxiT(t)xi(t)=∑i=1JxiT(t)(JI)xi(t),∑i=1J∑k=1JxkT(t)LkiLkiTxk(t)=∑i=1J∑k=1JxiT(t)LikLikTxi(t),∑i=1J∑k=1JxkT(t)MkiMkiTxk(t)=∑i=1J∑k=1JxiT(t)MikMikTxi(t)
Based on (3.7)–(3.10), we have
(3.11)V̇(t)=∑i=1JV̇i(t)≤∑i=1J{∑j=1ri∑n=1rihij(t)hin(t)xiT(t)×[∑k=1JAijTPi+PiAij-KinTBijTPi-PiBijKin+(Gij-HijKin)T×(Gij-HijKin)+PiEijEijTPi+2Pi∑k=1JE̅kiE̅kiTPi+2JI+Pi∑k=1JCkiCkiTPi+Pi∑k=1JDkiDkiTPi+∑k=1JMikMikT+∑k=1JLikLikT]xi(t)}.
From Schur complement, we know
(3.12)AijTPi+PiAij-KinTBijTPi-PiBijKin+(Gij-HijKin)T(Gij-HijKin)+PiEijEijTPi+2Pi∑k=1JE̅kiE̅kiTPi+2JI+Pi∑k=1JCkiCkiTPi+Pi∑k=1JDkiDkiTPi+∑k=1JMikMikT+∑k=1JLikLikT<0
is equivalent to
(3.13)(AijTPi+PiAij-KinTBijTPi-PiBijKin+2JIGijT-KinTHijMiTCiEiGij-HijKin-I000Mi0-I00CiT00-I0EiT000-12I)<0,
where MiT=[Mi1,Mi2,…,MiJ,Li1,Li2,…,LiJ]T, Ci=Pi[Eij,C1i,C2i,…,CJi,D1i,D2i,…,DJi], Ei=Pi[E̅1i,E̅2i,…,E̅Ji].
So we haveV̇(t)<0 while xi(t)(i=1,2,…,J) are not all zero vectors. Note that the matrix inequalities in (3.13) can be transformed into certain forms of linear matrix inequalities (LMIs). Therefore, multiplying both sides of matrix inequalities (3.13) by diag⁡{Pi-1,I,I,I,I} and applying the change of variables such that Pi=Xi-1, Kin=FinXi-1,(i=1,2,…,J,n=1,2,…,ri), then (3.4) is obtained.With the similar proof of Theorem3.1, the stabilization criterion of large-scale system (2.1) without uncertainties is also discussed. The result is presented as follows.Corollary 3.2.
The fuzzy large-scale system (2.1) without uncertainties can be asymptotically stabilized by the decentralized PDC fuzzy control (3.2), if there exist matrices Fij, positive definite matrices Xi to satisfy the following LMIs:
(3.14)(XiAijT+AijXi-FinTBijT-BijFinC̅iXiC̅iT-I0Xi0-12JI)<0
for i=1,2,…,J, j=1,2,…,ri, n=1,2,…,ri, where C̅i=[C1i,C2i,…,CJi,D1i,D2i,…,DJi].Proof.
The proof is similar with that of Theorem3.1; therefore details are omitted.Remark 3.3.
This corollary is similar with Theorem 1 in [10], so the theorem in this paper is more general.Theorem 3.4.
If there exist matricesFij, positive definite matrices Xi to satisfy the following LMIs:
(3.15)(XiAijT+AijXi-FinTBijT-BijFinXiGijT-FinTHijTXiMiTC̅iE̅iXiXiFinTGijXi-HijFin-I000000MiXi0-I00000C̅iT00-I0000E̅iT000-I000Xi0000-12JI00Xi00000-Qi-10Fin000000-Ri-1)<0,
the fuzzy large-scale system (2.1) can be asymptotically stabilized by the decentralized PDC fuzzy control (3.2), and the performance index (2.8) is satisfied the following inequality:
(3.16)J≤J*=∑i=1J[xiT(0)Pixi(0)+∑k=1J∫-τki0xkT(s)(I+MkiTMki)xk(s)ds],
for i=1,2,…,J, j=1,2,…,ri, n=1,2,…,ri, where MiT=[Mi1,Mi2,…,MiJ,Li1,Li2,…, LiJ]T, C̅i=[Eij,C1i,C2i,…,CJi,D1i,D2i,…,DJi], E̅i=[E̅1i,E̅2i,…,E̅Ji], Xi=Pi-1, Fin=KinPi-1.Proof.
From the proof of Theorem3.1, we know if
(3.17)Gij=AijTPi+PiAij-KinTBijTPi-PiBijKin+(Gij-HijKin)T(Gij-HijKin)+PiEijEijTPi+2Pi∑k=1JE̅kiE̅kiTPi+2JI+Pi∑k=1JCkiCkiTPi+Pi∑k=1JDkiDkiTPi+∑k=1JMikMikT+∑k=1JLikLikT<0
then V̇(t)<0. Noticing that Qi and Ri are positive matrices, if inequality
(3.18)AijTPi+PiAij-KinTBijTPi-PiBijKin+(Gij-HijKin)T(Gij-HijKin)+PiEijEijTPi+2Pi∑k=1JE̅kiE̅kiTPi+2JI+Pi∑k=1JCkiCkiTPi+Pi∑k=1JDkiDkiTPi+∑k=1JMikMikT+∑k=1JLikLikT+Qi+KinTRiKin<0,
holds, we obtain V̇(t)<0:
(3.19)J=∑i=1J∫0∞(xiT(t)Qixi(t)+uiT(t)Riui(t))dt=∑i=1J∫0∞xiT(t)(Qi+∑n=1rihin(t)KinTRi∑n=1rihin(t)Kin)xi(t)dt.
Using Lemma 2.3, we get
(3.20)J≤∑i=1J∫0∞xiT(t)(Qi+∑n=1rihin(t)KinTRiKin)xi(t)dt.
If (3.18) holds, we have
(3.21)J≤-∑i=1J∑j=1ri∑n=1ri∫0∞hij(t)hin(t)xiT(t)Gijxi(t)dt≤-∑i=1J∫0∞V̇i(t)dt=V(0)-V(∞)=∑i=1J[xiT(0)Pixi(0)+∑k=1J∫-τki0(I+MkiTMki)ds].
Therefore, multiply both sides of (3.18) by Pi-1, and let Xi=Pi-1,Fin=KinPi-1. From Schur complement, the proof is completed.


## 4. Numerical Examples

In this section, some numerical simulations for uncertain fuzzy large-scale system will be given to illustrate the effectiveness of the proposed stabilization criteria and also compared with the existing results.Example 4.1.
Consider an uncertain FLSS time delaysS composed of three fuzzy subsystems Si,i=1,2,3 by the following equations:
(4.1)ẋ1(t)=∑j=12h1j(t)[(A1j+ΔA1j)x1(t)+(B1j+ΔB1j)u1(t)]+∑k=13[(Ck1+ΔCk1)xk(t)+(Dk1+ΔDk1)xk(t-τk1)],ẋ2(t)=∑j=12h2j(t)[(A2j+ΔA2j)x2(t)+(B2j+ΔB2j)u2(t)]+∑k=13[(Ck2+ΔCk2)xk(t)+(Dk2+ΔDk2)xk(t-τk2)],ẋ3(t)=∑j=12h3j(t)[(A3j+ΔA3j)x3(t)+(B3j+ΔB3j)u3(t)]+∑k=13[(Ck3+ΔCk3)xk(t)+(Dk3+ΔDk3)xk(t-τk3)],
in which x1T(t)=[x11,x21]T, x2T(t)=[x12,x22]T, x1T(t)=[x13,x23]T(4.2)[ΔA1j,ΔB1j]=E1jF1j[G1j,H1j],[ΔA2j,ΔB2j]=E2jF2j[G2j,H2j],[ΔA3j,ΔB3j]=E3jF3j[G3j,H3j],[ΔCk1,ΔDk1]=E̅k1F̅k1[Lk1,Mk1],[ΔCk2,ΔDk2]=E̅k2F̅k2[Lk2,Mk2],[ΔCk3,ΔDk3]=E̅k3F̅k3[Lk3,Mk3],
for i,k=1,2,3, j=1,2. 							(4.3)A11=(0.62.40.2091.9),A12=(0.62.401.9),A21=(-1-0.513),A22=(0.62.4-0.2091.9),A31=(0.61.20.2091.9),A32=(0.61.201.9),B11=(2001),B12=(1002),B21=(3103),B22=(2032),B31=(1001),B32=(1011),C11=(-0.10.100),C21=(000.10.1),C31=(-0.20.150-1),C12=(000.10.1),C22=(0000.1),C32=(-0.3-0.20.20.1),C13=(-0.1000),C23=(0.1000.1),C33=(000.10),D11=(0.10.100),D21=(0.1000.1),D31=(-1.080.360-0.72),D12=(0.10-0.10.1),D22=(00.100),D32=(0.6-0.30.10),D13=(0000),D23=(-0.1000),D33=(-0.100-0.1),E11=(-0.28-0.028-0.140.182),E12=(-0.21-0.084-1.26-0.42),E21=(-0.50.251.350.75),E22=(-0.27075.00.81),E31=(-0.280.280.140.182),E32=(0.210.841.26-0.42),E̅11=(-0.280.028-0.140.182),E̅21=(-0.21-840.0-1.26-0.42),E̅31=(0.21-0.084-1.260.42),E̅12=(-0.50.251.350.75),E̅22=(-0.2700.810.81),E̅32=(-0.2700.750.81),E̅13=(-0.280.280.140.182),E̅23=(0.210.841.26-0.42),E̅33=(-0.210.841.260.42),F11(t)=(1-sin⁡4t001-cos⁡4t),F̅11(t)=(sin⁡t200cos⁡t2),G11=H11=E11,G12=H12=E12,G21=H21=E21,G22=H22=E22,G31=H31=E31,G32=H32=E32,L11=M11=E̅11,L21=M21=E̅21,L31=M31=E̅31,L12=M12=E̅12,L22=M22=E̅22,L32=M32=E̅32,L13=M13=E̅13,L23=M23=E̅23,L33=M33=E̅33,F11(t)=F12(t)=F21(t)=F22(t)=F31(t)=F32(t),F̅11(t)=F̅21(t)=F̅31(t)=F̅12(t)=F̅22(t)=F̅32(t)=F̅13(t)=F̅23(t)=F̅33(t).Here, the T-S fuzzy models of the isolated subsystem are of the following:Subsystem 1
Rule 1.
Ifx11(t) is M111 then
(4.4)ẋ1(t)=(A11+ΔA11(t))x1(t)+(B11+ΔB11(t))u1(t).Rule 2.
Ifx11(t) is M211 then
(4.5)ẋ1(t)=(A12+ΔA12(t))x1(t)+(B12+ΔB12(t))u1(t),
and the membership functions for Rules 1 and 2 are, respectively, M111(x11(t))=1/(1+exp⁡(-3x11(t))),M211(x11(t))=1-M111(x11(t)).Subsystem 2
Rule 1.
Ifx12(t) is M112 Then
(4.6)ẋ2(t)=(A21+ΔA21(t))x2(t)+(B21+ΔB21(t))u2(t).Rule 2.
Ifx12(t) is M212 then
(4.7)ẋ2(t)=(A22+ΔA22(t))x2(t)+(B22+ΔB22(t))u2(t),
and the membership functions for Rules 1 and 2 are, respectively, M112(x12(t))=exp⁡(-2x122(t)), M212(x12(t))=1-M112(x12(t)).Subsystem 3
Rule 1.
Ifx13(t) is M113 then
(4.8)ẋ3(t)=(A31+ΔA31(t))x3(t)+(B31+ΔB31(t))u3(t).Rule 2.
Ifx13(t) is M213 then
(4.9)ẋ3(t)=(A32+ΔA32(t))x3(t)+(B32+ΔB32(t))u3(t),
and the membership functions for Rules 1 and 2 are, respectively, M113(x13(t))=1/(1+exp⁡(-1.5x13(t))), M213(x13(t))=1-M113(x13(t)).
It is noted that the above large-scale system without control(ui(t)=0) has unstable responses with initial conditions x1(t)=[-3,3]T, x2(t)=[-2,2]T, x3(t)=[1,-1]T as shown in Figure 1.The state response withui(t)=0.
(a)(b)(c)In order to stabilize the large-scale fuzzy system, three decentralized PDC fuzzy controllers are designed in the following.Fuzzy ControllerC1
Rule 1.
Ifx11(t) is M111 then
(4.10)u1(t)=-K11x1(t).Rule 2.
Ifx11(t) is M211 then
(4.11)u1(t)=-K12x1(t).Fuzzy ControllerC2
Rule 1.
Ifx12(t) is M112 then
(4.12)u2(t)=-K21x2(t).Rule 2.
Ifx12(t) is M212 then
(4.13)u2(t)=-K22x2(t).Fuzzy ControllerC3
Rule 1.
Ifx13(t) is M113 then
(4.14)u3(t)=-K31x3(t).Rule 2.
Ifx13(t) is M213 Then
(4.15)u3(t)=-K32x3(t).By using the approaches of Theorem3.1, we obtain the matrices Kij for subsystems S1,S2, and S3: (4.16)K11=(1.56810.47330.52963.0430),K12=(2.22610.26800.58011.7275),K21=(0.87170.7769-0.48561.2398),K22=(1.48310.51790.20981.7072),K31=(1.65480.72810.41222.9992),K32=(1.57690.49580.58232.0029).The complete simulation results with initial conditionsx1(t)=[-3,3]T, x2(t)=[-2,2]T, x3(t)=[1,-1]T are shown in Figure 2. It is obvious that they are stabilized asymptotically.The state response with fuzzy control by Theorem3.1.
(a)(b)(c)Example 4.2.
Consider the fuzzy large-scale systemS composed of two subsystems Si as follows [25]:
(4.17)ẋ1(t)=∑j=12h1j(t)(A1jx1(t)+B1ju1(t))+∑h=12Ch1xh(t),ẋ2(t)=∑j=12h2j(t)(A2jx2(t)+B2ju2(t))+∑h=12Ch2xh(t),
where x1(t)=[x11(t),x12(t)]T,x2(t)=[x21(t),x22(t)]T(4.18)A11=(510-4),A12=(-402-8),B11=(10),B12=(01),C11=C22=(0000),A21=(-304-2),A22=(-4-51-3),B21=(1-2),B22=(-11),C21=(5100),C12=(1010)
and the membership functions are, respectively, M111(x11(t))=exp⁡(-x112(t)), M211(x11(t))=1-M111(x11(t)), M112(x21(t))=1/(1+exp⁡(-x21(t))), M212(x21(t))=1-M112(x21(t)).
In order to stabilize the large-sale fuzzy system, three decentralized PDC fuzzy controllers are designed in the following.Fuzzy ControllerC1
Rule 1.
Ifx11(t) is M111 then
(4.19)u1(t)=-K11x1(t).Rule 2.
Ifx11(t) is M211 then
(4.20)u1(t)=-K12x1(t).Fuzzy ControllerC2
Rule 1.
Ifx21(t) is M112 then
(4.21)u2(t)=-K21x2(t).Rule 2.
Ifx21(t) is M212 then
(4.22)u2(t)=-K22x2(t).We letQ1=Q2=(5001), R1=1, R2=2. By using the approaches of Theorem 3.4, we obtain
(4.23)P1=(5.50841.63291.63291.5120),P2=(0.9692-0.0406-0.04061.0681).The control gainKij for subsystem S1,S2 is compared with [24] in Table 1. The complete simulation results with initial conditions x1(t)=[1,-1]T,x2(t)=[2,-2]T are shown in Figure 3.Table 1
Comparison of fuzzy controller, for example, with Zhang et al. [25].
IndexK11K12K21K22J*∥K∥2Zhang et al. [25][10.76024.8931][7.14241.0899][2.27020.1348][1.95270.8715]21.060614.2011This paper[9.04141.0000][2.00001.8284][1.16040.0740][0.35152.2722]12.22889.8351The state response with fuzzy control by Theorem3.4.
(a)(b)Remark 4.3.
From Figure3, we can see that the system can be stabilized through appropriate decentralized control. Obviously, in Table 1, we can see the method of this paper has smaller gain matrices and performance index, so it has better control effectiveness.


## 5. Conclusions

In this paper we explore the stabilization problems for uncertain fuzzy large-scale system with time delays. The decentralized PDC fuzzy controller has been designed under some conditions such that the whole closed-loop large-scale fuzzy system is asymptotically stable. Then, the optimization design method for decentralized control is also considered with respect to a quadratic performance index. Finally, numerical examples are provided to demonstrate the correctness and less conservativeness of the theoretical results. However, there are still some other problems to be addressed, such that time-varying delays and delay-dependent stability and stabilization of fuzzy large-scale system and the results developed in the paper can be extended to the case that the underlying systems are invovled with any switching dynamics.



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*Source: 246705-2012-03-26.xml*