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H∞ Fuzzy Control Synthesis for a Large-Scale System With a Reduced Number of LMIs

H∞Fuzzy Control Synthesis for a Large-Scale System With a Reduced Number of LMIs

Wei Chang and Wen-June Wang,Fellow,IEEE

Abstract—This paper introduces an H∞fuzzy control synthesis method for a nonlinear large-scale system with a reduced num-ber of linear matrix inequalities(LMIs).It is well known that a nonlinear large-scale system can be transformed to a Takagi–Sugeno(T–S)fuzzy system by using“sector nonlinearity”or“local approximation in fuzzy partition spaces”methods.Next,in or-der to achieve the fuzzy control design for this T–S fuzzy system, we solve the stabilization conditions that are represented by the LMIs.However,if the number of LMIs is large,the control design process may become very complicated.In this study,based on the Lyapunov method and S-procedure,several theorems are proposed for the synthesis of parallel distributed compensation(PDC)-type fuzzy control such that the nonlinear large-scale system achieves H∞control performance,and the number of LMIs to be solved is reduced explicitly.As a result,the control design process will become much easier.Furthermore,if the modeling error between the nonlinear system and T–S fuzzy system exists,the robust H∞control performance and the number reduction of LMIs are also achieved by the proposed theorem.Several examples are presented in this paper to show the number reduction effect of LMIs and the effectiveness of the proposed controller synthesis.

Index Terms—Fuzzy control,H∞control,interconnected sys-tems,large-scale systems,linear matrix inequalities(LMIs),rule reduction,S-procedure.

I.I NTRODUCTION

T HIS paper studies an H∞fuzzy control synthesis method for a large-scale system with a reduced number of LMIs.

A review of related papers,as well as the motivation and contri-bution of this paper,are given in the following three paragraphs. Finally,the organization of this study is presented.

First,there are many physical systems which consist of multi-ple subsystems and are linked via a network of interconnections. These systems are called large-scale systems and they,and espe-cially their inherent control design problems,have been consid-ered by many papers(see[1]–[16]and the references therein).In [1]–[16],the considered large-scale systems were transformed into the well-known T–S fuzzy system[17]for which the fuzzy controllers were designed.For instance,in order to deal with the time delay in data transmission within the subsystems or interconnection network,in[1],[8],[10],and[12],the sta-bilization problem of large-scale systems with time delay is Manuscript received May24,2014;accepted July9,2014.Date of publication August15,2014;date of current version July31,2015.This work was supported by the Ministry of Science and Technology(MOST)of Taiwan under contracts NSC101-2218-E-008-015and NSC99-2221-E-008-093-MY3.

The authors are with the Department of Electrical Engineering,National Central University,Jhongli32001,Taiwan(e-mail:985401013@https://www.wendangku.net/doc/5b10816615.html,.tw; wjwang@https://www.wendangku.net/doc/5b10816615.html,.tw).

Color versions of one or more of the?gures in this paper are available online at https://www.wendangku.net/doc/5b10816615.html,.

Digital Object Identi?er10.1109/TFUZZ.2014.2347995studied.Next,based on the de?ned piecewise Lyapunov func-tion,the stability of discrete-time fuzzy large-scale systems is discussed in[9].In[14],the authors derived a set of inequali-ties to be the stabilization conditions of the large-scale system. Here,one of the inequalities is a large negative-de?nite matrix, which contains the effects of all interconnection terms and all decentralized PDC[17]controller gains.Moreover,in practice, the considered systems always contain some uncertainties or disturbances;therefore,robustness or H∞performance control [18]is of signi?cant importance here and as such has been the subject of much research on the topic of large-scale systems(see [6],[9],and[16]and the references therein).

Second,it is well known that the most important and dif?cult part of designing a fuzzy controller for large-scale systems is handling the nonlinear interconnections(see[1]–[17]).Since each subsystem has several interconnections with the other sub-systems,an entire nonlinear large-scale system contains a lot of interconnections.In general,if a nonlinear large-scale system is transformed into a T–S fuzzy system,one of the two meth-ods have been used to cope with the nonlinear interconnections before designing the fuzzy controller in order to stabilize the system.The?rst method(see[1]–[5])is to set some speci?c bounded conditions,which the interconnections must satisfy. The second method(see[6]–[14])is to linearize those nonlinear interconnections using the“sector nonlinearity”method[17, p.10]or the“local approximation in fuzzy partition spaces”method[17,p.23].However,when using the?rst method,the dynamics of the interconnections must be known well in ad-vance in order to obtain a suitable bounded condition.If the dy-namics of the interconnections are not known in advance,then the interconnections’bound must be checked in the simulation process.Once the interconnections’bound does not satisfy the prespeci?ed bounded condition,the bounded condition has to be modi?ed and the controllers should be redesigned.In order to avoid these bounded condition problems,many papers(see [6]–[14]and the references therein)have treated the intercon-nections as a part of the nonlinear terms of the subsystems and have linearized them into a T–S fuzzy system[17].However,if the system consists of a large number of subsystems and each in-terconnection is transformed into a set of fuzzy rules,then this second method gives rise to the well-known“rule-explosion”problem[20,p.273].Thus,the number of LMIs[19]to be solved becomes very large and the fuzzy control design may fail if the design procedure becomes too complicated.Conse-quently,it is worth investigating a way to reduce the number of LMIs for which solutions are required when designing the control for a large-scale system through the corresponding T–S fuzzy system.

1063-6706?2014IEEE.Personal use is permitted,but republication/redistribution requires IEEE permission.

See https://www.wendangku.net/doc/5b10816615.html,/publications standards/publications/rights/index.html for more information.

Bearing the above observations in mind,this paper investi-gates the stabilization and H∞performance control of a large-scale system.Here,the“sector nonlinearity”or“local approxi-mation in fuzzy partition spaces”methods are used to transform the original system into a T–S fuzzy system,and a PDC[17] type fuzzy controller is designed.It should be emphasized that, in this study,the“rule-explosion”problem may be avoided be-cause of a special derivation,which eliminates the fuzzy rules generated by the interconnections,after which the S-procedure [19]is used to obtain the stabilization conditions.As a result, the number of LMIs[19]that need to be solved is reduced such that the controller synthesis becomes much easier.

The organization of this study is as follows.Section II presents the system description and problem formulation.In Section III, the main theorems are proposed to give the existence conditions of the controller and to show the synthesized form of the control gain.Next,in Section IV,a simple example and a theorem are given to show the reduction in the number of LMIs.In Section V, a robust control problem is considered because of the modeling error between the original nonlinear system and the T–S fuzzy system.Finally,some numerical examples and conclusions are given in Sections VI and VII,respectively.

Notation:Throughout this paper,we adopt standard notions. The notation X≥Y or X>Y means that the matrix X?Y is positive semide?nite or positive de?nite,respectively.The asterisk symbol(?)denotes the transposed elements(matrices) in symmetric positions.The notation diag(A,B)is a diagonal matrix with diagonal entries A and B.Finally,the notation mat[a b,c d]denotes the matrix[a b

c d].Here,the a,b,c,and

d ar

e scalars.

II.S YSTEM D ESCRIPTION AND P ROBLEM F ORMULATION Consider a nonlinear large-scale system,which consists of N subsystems as follows:

˙x l(t)=A l(x(t))x l(t)+B l(x(t))u l(t)+f l(x(t))+E l v l(t)

l=1,...,N(1)

where x l(t)∈ n l is the current state of the l th subsys-tem,u l(t)∈ m l is the control input,and x(t)is the state vector composed of all states x l(t).A l(x(t))∈ n l×n l and B l(x(t))∈ n l×m l,which may be nonlinear,are system and in-put matrices,respectively.The term f l(x(t))∈ n l represents the nonlinear interconnections between the l th subsystem and any other subsystems.E l∈ n l represents the disturbance gain, and v l(t)∈ denotes the unknown disturbance.

First,transforming a nonlinear system into a T–S fuzzy sys-tem is helpful to simplify the control design process for the nonlinear system[17]in general.Therefore,in this study,the large-scale system(1)is transformed to a T–S fuzzy system by using the methods of“sector nonlinearity”[17,p.10],or“local approximation in fuzzy partition spaces”[17,p.23],and then, PDC fuzzy controllers are designed to satisfy the H∞control performance such that

x T l(t)M l x l(t)dt<λ2l

v T l(t)v l(t)dt,l=1,...,N

(2) under zero initial conditions with v l(t)=0,and the stability of the large-scale system is guaranteed in the sense of Lyapunov [21]for v l(t)=0.Here,t f is the terminal time of the control, andλl is a prescribed value,which denotes the worst-case effect of v l(t)on x l(t).M l∈ n l×n l is a positive-de?nite weighting matrix.

Second,let us consider the transformation between the non-linear system and T–S fuzzy system.It should be noted that if the original system(1)consists of a large number of subsystems and all nonlinear terms of each subsystem and its interconnection are transformed into a set of fuzzy rules,then the well-known “rule-explosion”problem[20,p.273]will rise so that the con-trol design becomes much harder.In other words,the number of LMIs[19]becomes very large,such that the fuzzy control design fails.Hence,the objective of this paper is to reduce the number of LMIs and then synthesize fuzzy controllers to satisfy H∞control performance(2)for the system(1).Furthermore,a robust control problem is considered in Section V because of the modeling error between the original nonlinear system and the T–S fuzzy system.

III.R ESULTS AND C ONTROLLER S YNTHESIS

Let the individual subsystem(1)be transformed into the form of a T–S fuzzy system using the methods of“sector nonlinearity”[17,p.10]or“local approximation in fuzzy partition spaces”’[17,p.23]as follows:

Rule i:IF z l1(t)is m l i1and...and z l q(t)is m l iq,THEN

˙x l(t)=A l i x l(t)+B l i u l(t)+

m=1

F lm i x m(t)+E l v l(t)

i=1,...,r l(3) where A l i∈ n l×n l,B l i∈ n l×m l,and F lm i∈ n l×n m are con-stant matrices;r l is the total number of fuzzy rules in the l th subsystem;z l j(t),j=1,2,...,q,are the known premise vari-ables which may be the function of states and q is the total number of premise variables;and m l ij is the fuzzy membership function of z l j(t).It is known that the system(3)modeled by the“sector nonlinearity”method[17,p.10]is exactly the same as the original system(1)in some speci?c local regions for all t [17].On the other hand,if“local approximation in fuzzy parti-tion spaces”method[17,p.23]is used to model the system(1), some modeling errors between the original system(1)and the fuzzy system will be generated.In Section V,the robust control design for the fuzzy system with modeling error will be consid-ered and investigated.Before Section V,the controllers design is based on the system of(3).

Here,the PDC[17]type fuzzy controller is adopted.

Rule i:IF z l1(t)is m l i1and...and z l q(t)is m l iq,THEN

u l(t)=?K l i x l(t),i=1,...,r l(4)

CHANG AND W ANG:H∞FUZZY CONTROL SYNTHESIS FOR A LARGE-SCALE SYSTEM WITH A REDUCED NUMBER OF LMIS1199

where K l i∈ m l×n l is the control gain.The overall fuzzy con-

troller is

u l(t)=?

r l

i=1

μl i

z l(t)

K l i x l(t).(5)

Therefore,combining(1),(3),and(5),we can obtain the large-scale T–S fuzzy system with the PDC controller as follows:

˙x l(t)=

r l

i=1

μl i(z l(t))2

G l ii x l(t)

μl i(z l(t))μl j(z l(t)){ˉG l ij x l(t)}

+ˉf l(x(t))+E l v l(t)(6a)

ˉf l (x(t))=

r l

i=1

r l

j=1

μl i(z l(t))μl j(z l(t))

m=1

F lm i x m(t)

l=1,...,N(6b)

in which

ˉG l ij =(G l ij+G l ji)

2,G l ij=A l i?B l i K l j

z l(t)=

z l1(t),z l2(t),...,z l q(t)

ωi(z l(t))=

j=1

m l ij(z l j(t))

μl i(z l(t))=

ωi(z l(t))

i=1

ωi(z l(t))

,μl i(z l(t))≥0

r l

i=1

μl i(z l(t))=1.(7)

Next,based on the T–S fuzzy system(6)and controller(5), an H∞fuzzy control synthesis method for a large-scale system (1)with a reduced number of LMIs is given below.Before proceeding to the main results,the following Lemma is given. Lemma1[22]:Tchebyshev’s inequality holds for any vector X i∈ n

i=1X i

i=1

X i

≤N×

i=1

(X i)T X i.(8)

Theorem1:The nonlinear large-scale system(1)is stable in the sense of Lyapunov based on the large-scale T–S fuzzy system (6)with the PDC control(5)as v m(t)=0,and the H∞con-trol performance in(2)for the system(1)is guaranteed for the given prescribed valuesλm as v m(t)=0,if there exist positive-de?nite matrices P m=(P m)T>0,M m=(M m)T>0, matrices K m j,and scalarsτm>0,m=1,...,N such that Λm ii=Γm ii+Φm?τm S m<0,m=1,...,N

i=1,...,r m(9)Λm ij=Γm ij+Φm?τm S m<0,m=1,...,N

Γm ii=

(G m ii)T P m+P m G m ii P m

+P m E m{1/λ2m}(E m)T P m

Γm ij=

(ˉG m ij)T P m+P mˉG m ij P m

+P m E m{1/λ2m}(E m)T P m

??Φm=

M m0

S m=

???

l=1

r l

k=1

((N×r l)(F lm k)T F lm k)0

??.(11) Proof:Let the Lyapunov function be

V(x(t))=

l=1

V l(x l(t))=

l=1

x T l(t)P l x l(t)(12)

where P l=(P l)T is a positive-de?nite https://www.wendangku.net/doc/5b10816615.html,ing the in-equality X T Y+Y T X≤λX T X+(1/λ)Y T Y,the deriva-tive of the l th Lyapunov function V l(x l(t))along the trajectory of the l th subsystem is

˙V

(x l(t))≤

r l

i=1

μl i(z l(t))2{x T l(t)((G l ii)T P l+P l G l ii)x l(t)} +2

μl i(z l(t))μl j(z l(t)){x T l(t)((ˉG l ij)T P l +P lˉG l ij)x l(t)}+ˉf l(x(t))T P l x l(t)

+x T l(t)P lˉf l(x(t))+v T l(t){λ2l}v l(t)

+x T l(t)P l E l{1/λ2l}E T l P l x l(t).(13) It yields

˙V(x(t))=

l=1

˙V

(x l(t))

≤

m=1

r m

i=1

μm i(z m(t))2

x m(t)

ˉf

(x(t))

Γm ii

x m(t)

ˉf

(x(t))

m=1

μm i(z m(t))μm j(z m(t))

x m(t)

ˉf

(x(t))

Γm ij

x m(t)

ˉf

(x(t))

m=1

r m

i=1

r m

j=1

μm i(z m(t))μm j(z m(t))(v T m(t){λ2m}v m(t))

(14)

1200IEEE TRANSACTIONS ON FUZZY SYSTEMS,VOL.23,NO.4,AUGUST 2015

N m =1

r m i =1

μm i (z m (t ))2

??? x m (t )ˉf

m (x (t ))

×(Γm ii +Φm )

x m (t )ˉf

m (x (t ))

+N m =1

μm i (z m (t ))μm j (z m

(t ))×??? x m (t )ˉf m (x (t )) T (Γm

ij +Φm ) x m (t )ˉf

m (x (t )) ??????<0

(15)

holds,then (14)becomes

˙V (x (t ))<

N m =1 r m

i =1r m j =1

μm i

(z m (t ))μm j

(z m (t ))× ?x T m (t )M m x m (t )+v T m (t ) λ2m v m (t )

.(16)

Moreover,under zero-initial conditions,the above inequality implies that

t f

x T m (t )M m x m (t )dt <λ2m

t f 0

v T

m (t )v m (t )dt.(17)It can be seen that H ∞control performance (2)is achieved with a prescribed λm .Actually,if the disturbance v m (t )is zero,we have

˙V m (x m (t ))

(t )M m x m (t ).(18)

Therefore,the system is also stable in the sense of Lyapunov.Obviously,the large-scale T–S fuzzy system is stable if (15)holds or there exists a solution to satisfy Γm ii +Φm <0and Γm

ij +Φm <0in (15).However,there is a zero in the bottom-right corners of Γm ii +Φm and Γm

ij +Φm ,respectively,so that they could not be negative de?nite.Therefore,the con-sequent derivation is to continue the proof.According to (6b)and Lemma 1,we have

ˉf T l

(x (t ))ˉf l (x (t ))=??r l i =1r l j =1

μl i (z l (t ))μl j (z l (t )) N

m =1

F lm i x m (t )

??T

×??r l i =1r l j =1

μl i (z l (t ))μl j (z l (t )) N

m =1

F lm i x m (t )

??=

r l

i =1

μl i (z l (t ))

N m =1

F lm i x m (t )

r l

i =1

μl i (z l (t ))

N m =1

F lm i x m (t )

≤

r l

i =1

r l

μl i (z l

(t ))

N m =1

(F lm i x m (t ))

μl i (z l (t ))N m =1

(F lm i x m (t ))

r l i =1

r l

N m =1

μl i (z l

(t ))F lm i x m (t )

× N m =1

μl i (z l (t ))F lm i x m (t )

≤

N m =1r l i =1

(N ×r l ) μl i (z l

(t ))F lm i

x m (t ) T

× μl i (z l

(t ))F lm i

x m (t ) ≤N m =1

x T m (t ) r l

k =1

(N ×r l )(F lm k )T

F lm k

x m (t ).(19)

Next,the following equalities are obtained:

N l =1

ˉf T l

(x (t ))ˉf l (x (t ))

N l =1

m =1

x T m (t )

r l

k =1

(N ×r l )(F lm k )T

F lm k

x m (t )

N m =1

ˉf T m

(x (t ))ˉf m (x (t ))

N m =1

l =1

x T m (t )

r l

k =1

(N ×r l )(F lm k )T

F lm k

x m (t )

N m =1

r m

i =1r m j =1

μm i (z m (t ))μm j (z m

(t ))

×????? x m (t )ˉf

m (x (t )) T ????N l =1r l k =1(N ×r l )(F lm k )T F lm k

00I

??×

x m (t )ˉf m (x (t )) =N m =1

???r m i =1

μm i (z m (t ))2

??? x m (t )ˉf m (x (t )) T S m

x m (t )ˉf

m (x (t )) ??????+

N m =1

μm i (z m (t ))μm j (z m

(t ))×??? x m (t )ˉf m (x (t )) T S m x m (t )ˉf

m (x (t )) ????

?≤0.

(20)

CHANG AND W ANG:H∞FUZZY CONTROL SYNTHESIS FOR A LARGE-SCALE SYSTEM WITH A REDUCED NUMBER OF LMIS1201 With the aids of S-procedure[19],if there exists a positive

scalarτm such that matrix inequalities(9)and(10)are satis?ed,

then(15)holds.The proof is complete.

Remark1:As we can see from the proof of Theorem1,

if there is a zero in the bottom-right corners ofΓm ii+Φm

andΓm ij+Φm,thenΓm ii+Φm<0andΓm ij+Φm<0in(15)

could not be satis?ed.In order to solve this dif?culty,(19)and

(20)are constructed using the information of interconnections

f l(x(t)).Next,with the aids of(20)and S-procedure[19],both

matricesΓm ii+Φm andΓm ij+Φm are transformed into two

matrix inequalities(9)and(10),respectively,which have nega-

tive entries in the bottom-right corner so that the proof is com-

plete.However,it is dif?cult to?nd the feasible matrices P m,

M m,K m j,and scalarsτm using the MATLAB LMI-toolbox

[23]because(9)and(10)are not in LMI form.Therefore,The-

orem2is yielded.

Remark2:In the proof of Theorem1,the rules generated by

the interconnections are eliminated by a special skill presented

in(19)and(20)so that the number of LMIs can evidently be

reduced.However,the number reduction effect is not easily

found in the content of Theorems1and2.In Section IV,a

simple example will be given to explain the LMIs’number

reduction effect in Theorems1and2,and then,Theorem3(i.e.,

a modi?ed theorem of Theorem2)will be given to explicitly

show the number reduction effect.

Theorem2:The nonlinear large-scale system(1)is sta-

ble in the sense of Lyapunov based on the large-scale T–S

fuzzy system(6)with the PDC control(5)as v m(t)=0,and

the H∞control performance in(2)for the system(1)is guaran-

teed for the given prescribed valuesλm as v m(t)=0,if there

exist positive-de?nite matrices Q m=(Q m)T>0,ˉM m=

(ˉM m)T>0,vectors H m i,and scalarsξm>0such that

Θm i=ˉΩm i+L m<0,m=1,...,N,i=1,...,r m

(21)

Θm ij=ˉΩm ij+L m<0,m=1,...,N,i

(22)

where

ˉΩm i =diag(Ωm i,0,0),ˉΩm ij=diag

Ωm ij,0,0

(23)

Ωm i=(Q m(A m i)T?(H m i)T(B m i)T)

+(A m i Q m?B m i H m i)(24)

Ωm ij=1

Q m(A m i)T?(H m j)T(B m i)T

+Q m(A m j)T?(H m i)T(B m j)T

(A m i Q m?B m i H m j+A m j Q m?B m j H m i)(25)

L m=?

(1/λ2m)E m(E m)T+ˉM mξm I Q mˉF m

??ξm I0

???ξm I

(26)

ˉF

N×r1(F1m1)T,...,

N×r1(F1m r

)T,...,

N×r N(F N m

)T,...,

N×r N(F N m

r N

)T].(27)

In this case,τm=1/ξm,M m=Q?1mˉM m Q?1m,and the con-

trol gain can be obtained from K m i=H m i Q?1m.

Proof:In Theorem1,the main conditions areΛm ii<0and

Λm ij<0.Next,we need to transform the stabilization con-

ditions of Theorem1into LMI form[19].Let Q m=P?1m,

H m i=K m i Q m,andˉM m=Q m M m Q m.Perform pre-and

postmultiplication of diag(Q m,I)inΛm ii andΛm ij;then,we get

ˉΛm

Ωm i+E m{1/λ2m}E T m+ˉM m I

+τm Q m

l=1

r l

k=1

((N×r l)(F lm k)T F lm k)Q m

I?τm I

(28)

ˉΛm

Ωm ij+E m{1/λ2m}E T m+ˉM m I

+τm Q m

l=1

r l

k=1

((N×r l)(F lm k)T F lm k)Q m

I?τm I

(29)

Again,letξm=1/τm;perform pre-and postmultiplication

of diag(I,ξm I)inˉΛm ii andˉΛm ij;then,(21)and(22)are ob-

tained using the Schur-complements[19].Thus,the proof is

complete.

Remark3:It should be noted that,in Theorem2,the number of

LMIs is greatly reduced because of the matrixˉF m in L m.Here,

all interconnections of each subsystem(e.g.,the m-subsystem)

are covered by a single matrixˉF m[see(27)]because of the

special derivation in(19)and(20).Thus,the reduction of the

number of LMIs is one of the main contributions of Theorem2.

IV.R EDUCTION OF THE N UMBER OF L INEAR

M ATRIX I NEQUALITIES

It is known that the LMI is a powerful tool in the fuzzy con-

troller design.However,when using the standard fuzzy model-

ing methods,i.e.,“sector nonlinearity”or“local approximation

in fuzzy partition spaces”methods,to design fuzzy controls,the

number of LMIs that need to be solved will be large if the con-

sidered system is a large-scale system with a lot of subsystems.

Hence,the controllers designed by the LMI-toolbox[23]may

not be easily performable.Therefore,Section III has proven the

reduction of the number of LMIs such that the controller syn-

thesis becomes much easier.In the beginning of this section,a

simple example is given to show the reduction in the number

of LMIs when using Theorem2.Consequently,Theorem3is

the modi?ed result of Theorem2and is used to calculate the

reduced number of LMIs.

1202IEEE TRANSACTIONS ON FUZZY SYSTEMS,VOL.23,NO.4,AUGUST2015 TABLE I

F UZZY S YSTEM P ARAMETERS OF S UBSYSTEM1

Fuzzy Rule A1i B1i F11i F12i

i=1mat[01,

1?3]

[01]T

mat[00.1,

0.1π/20]

mat[0(π/2)3,

00.5]

i=2mat[01,

1?3]

[01]T

mat[00.1,

0.1π/20]

mat[0?(π/2)3,

00.5]

i=3mat[01,

1?3]

[01]T

mat[00.1,

?0.1π/20]

mat[0(π/2)3,

00.5]

i=4mat[01,

1?3]

[01]T

mat[00.1,

?0.1π/20]

mat[0?(π/2)3,

00.5]

i=5mat[01,

?1?3][01]T

mat[00.1,

0.1π/20]

mat[0(π/2)3,

00.5]

i=6mat[01,

?1?3][01]T

mat[00.1,

0.1π/20]

mat[0?(π/2)3,

00.5]

i=7mat[01,

?1?3][01]T

mat[00.1,

?0.1π/20]

mat[0(π/2)3,

00.5]

i=8mat[01,

?1?3][01]T

mat[00.1,

?0.1π/20]

mat[0?(π/2)3,

00.5]

Example1:A large-scale system consists of two subsystems as follows(N=2):

˙x11(t)˙x12(t)

sin(x11(t))?3

x11(t)

x12(t)

u1(t) +

00.1

0.1sin(x11(t))x21(t)0

x11(t)

x12(t)

01+(x22(t))2x12(t)

00.5

x21(t)

x22(t)

˙x21(t)˙x22(t)

(x21(t))23

x21(t)

x22(t)

u2(t) +

cos(x11(t)x22(t))0

x11(t)

x12(t)

01+(x12(t))2

x21(t)

x22(t)

.(30)

Suppose all states x11(t),x12(t),x21(t),and x22(t)are within the interval[?π/2,π/2].Set z11(t)=sin(x11(t)), z12(t)=sin(x11(t))x21(t),z13(t)=(x22(t))2x12(t),z21(t)= (x21(t))2,z22(t)=cos(x11(t)x22(t)),and z23(t)=(x12(t))2. Next,according to the sector nonlinearity method[17],we need eight fuzzy rules in each subsystem to synthesize the fuzzy controllers(the total rule number of the two subsys-tems is sixteen).The fuzzy system parameters are shown in Table I(due to space limitations,we only list the details of subsystem1).

Using Theorem2,the following16LMIs need to be consid-ered for the stabilization:

Θ1i=

Θ1i(1)ξ1I Q1ˉF1

??ξ1I0

???ξ1I

??<0

Θ2i=

Θ2i(1)ξ2I Q2ˉF2

??ξ2I0

???ξ2I

??<0

for i=1,...,8(31) where

Θ1i(1)=(Q1(A1i)T?(H1i)T(B1i)T)+(A1i Q1?B1i H1i) +(1/λ21)E1(E1)T+ˉM1

Θ2i(1)=(Q2(A2i)T?(H2i)T(B2i)T)+(A2i Q2?B2i H2i) +(1/λ22)E2(E2)T+ˉM2.

In fact,the number of above eight LMIs(i.e.,i=1, (8)

can be reduced to four LMIs in the following way.First,due to the last line in(19)and(27),the interconnection termˉF l in (31)is independent of the fuzzy rule i.Second,from Table I, the system parameters are A l1=A l2=A l3=A l4,A l5=A l6= A l7=A l8,and B l1=···=B l8.Therefore,it is seen thatΘ11=Θ12=Θ13=Θ14,Θ15=Θ16=Θ17=Θ18,Θ21=Θ22=Θ23=Θ24,andΘ25=Θ26=Θ27=Θ28.As a result,we only have to satisfy the LMIs for i=1and i=5;the16LMIs become four LMIs(i.e.,Θ11<0,Θ15<0,Θ21<0,andΘ25<0)to be solved.

Based on the explanation in Example1,Theorem2can be modi?ed as Theorem3,where the effect of number reduced of LMIs is explicated.

Theorem3:The nonlinear large-scale system(1)is stable in the sense of Lyapunov based on the large-scale T–S fuzzy system(6)with the PDC control(5)as v m(t)=0,and the H∞control performance in(2)for the system(1)is guaranteed for the given prescribed valuesλm as v m(t)=0,if the stabilization conditions

Θm i=ˉΩm i+L m<0,m=1,...,N,i=1,...,r?m

(32)Θm ij=ˉΩm ij+L m<0,m=1,...,N,i

(33) are satis?ed,where r?m is de?ned as

r?m=

2z m,in case(a)

r(A m i,B m i)|r m

i=1

,in case(b).

(34)

Here,case(a)denotes that the system is transformed to T–S fuzzy system by“sector nonlinearity”method and case (b)denotes the“local approximation in fuzzy partition spaces”transformation method.Moreover,z m is the total number of nonlinear elements in A l(x(t))and B l(x(t))of the m th

CHANG AND W ANG:H ∞FUZZY CONTROL SYNTHESIS FOR A LARGE-SCALE SYSTEM WITH A REDUCED NUMBER OF LMIS

1203

subsystem (1).r (A m i ,B m i )|r m

i =1is a number of the combina-tions of A m i and B m i ,which have the same matrices [A m i ,B m i ](see the details in Example 2).

Remark 4:The stabilization conditions (21),(22)of Theorem 2and the stabilization conditions (32),(33)of Theo-rem 3are the same.The only difference between Theorems 2

and 3is the number r m and r ?

.Based on Example 1,we have the observation r ?

m ≤r m .As a result,the number of LMIs will be reduced if the representation of Theorem 3is used.

The following example is given to show how to calculate r ?

m when a nonlinear large-scale system (1)is given.

Example 2:Consider the large-scale system (30)in Example 1.If the “sector nonlinearity”method is used (i.e.,it belongs to the case (a)of (34)),there is only one nonlinear term in A l (x (t ))and B l (x (t ))of each subsystem,then the

number z m is 1and r ?

is 2.Hence,the number of LMIs to be solved is r ?1+r ?

2=2+2=4.On the other hand,if the “local approximation in fuzzy partition spaces”method is used (i.e.,case (b)of (34)),the large-scale T–S fuzzy system becomes

Rule i :If x 11(t )is M 1i ,x 12(t )is M 2i ,x 21(t )is M 3i and x 22(t )is M 4i ,Then ˙x

l (t )=A l i x l (t )

+B l i u l (t )

2 m =1

F lm i x m (t )+E l v l (t )

i =1, (34)

for l =1,2.

(35)

There are 81+81rules in (35).Here,the matrices

A l i

and

B l i

are as follows:

A 11=···=A 1

27=

11?3

A 128=···=A 154=

010?3 A 155=···=A 1

81=

1?1?3

B 11···=B 181=

01 ,A 21=···=A 227=

011

A 228=···=

A 254=

0103

A 255

=···=A 281

0113

,B 21

=···=

B 281

(36)

and the fuzzy membership functions are shown in Fig.1.The

combinations of A l i and B l i are shown in Table II.Hence,r ?1=r A 1i ,B 1i 81i =1=3and r ?

2=r (A 2i ,B 2i ) 81i =1=2.Here,we only need r ?1+r ?

2=3+2=5LMIs to synthesize the controllers for this

system.

Fig.1.Membership functions of (35),where z (t )is x 11(t ),x 12(t ),x 21(t ),or x 22(t ).

TABLE II

C OMBINATIONS OF A l i AN

D B l

Subsystem Combinations

A l i ,

B l

A 11,

B 11 =···= A 127,B 1

27 2 A 128,B 128 =···= A 154,B 154

A 1

55,B 155 =···= A 181,B 1

1 A 2

1,B 21 =···= A 227,B 2

27 = A 255,B 255 =···= A 281,B 281

A 2

28,B 228 =···= A 254,B 2

V .R OBUST C ONTROL W ITH M ODELING E RROR

This section considers the stabilization of the T–S fuzzy sys-tem which is transformed from a nonlinear system by the “local

approximation in fuzzy partition spaces”method [17,p.23].The local approximation method will cause modeling error be-tween the original nonlinear system and the transformed T–S fuzzy system.In this section,the modeling error is considered as the uncertainty of the T–S fuzzy system and the controllers synthesis in presections still works with some modi?cation for the robust stabilizations.

The following equation shows that the modeling error appears between the original system and T–S fuzzy system,where the PDC controller (5)is used:

˙x

l (t )=A l (x (t ))x l (t )+B l (x (t ))u l (t )+f l (x (t ))+E l v l (t )=

r l i =1r l

j =1

μl i (z l (t ))μl j (z l (t ))

× A l i x l (t )?B l i K l

i x l (t )+

N m =1

F lm i x m (t )

+E l v l (t )+

A l (x (t ))x l (t )

?r l i =1r l j =1

μl i (z l (t ))μl j (z l

(t ))A l i x l (t )

+ ?B l (x (t ))K l i

x l (t )

1204IEEE TRANSACTIONS ON FUZZY SYSTEMS,VOL.23,NO.4,AUGUST 2015

r l i =1r l j =1

μl i (z l (t ))μl j (z l

(t )) ?B l i K l i x l (t )

r l i =1r l

j =1

μl i (z l (t ))μl j (z l (t ))

(A l i +ΔA l (t ))x l (t )?(B l i +ΔB l (t ))K l

i x l (t ) +r l i =1r l j =1μl i (z l (t ))μl j (z l (t )) N

m =1F lm i x m (t )

+E l v l (t )

(37)

where ΔA l (t )=

r l i =1r l

j =1

μl i (z l (t ))μl j (z l

(t )){A l (x (t ))?A l i }

ΔB l (t )=

r l i =1r l j =1

μl i (z l (t ))μl j (z l (t )){B l (x (t ))?B l i }.(38)

Suppose that there exist bounding matrices Δl a (t)and Δl b (t )such that ΔA l (t )=D l a Δl a (t )L l

a and ΔB l (t )=D l

b Δl b (t )L l b satisfy Δl a (t ) ≤εl a ,and Δl b (t ) ≤εl

b .Here,

D l a ∈ n l ×n l ,D l b ∈ n l ×n l ,L l a ∈ n l ×n l ,L l b ∈ n l ×m l

,εl a ∈ 1×1,and εl b ∈

1×1

are known constant matrices (vectors)[27].Therefore,(37)can be rewritten as

˙x

l (t )=r l

i =1

μl i (z l (t ))2 G l ii +ˉD l ˉΔl (t )ˉL l i x l (t )

μl i (z l (t ))μl j (z l

(t ))

ˉG l ij

+ˉD l ˉΔl (t )ˉL l i +ˉD l ˉΔl (t )ˉL l j 2

x l (t )

+ˉf

l (x (t ))+E l v l (t )(39a)ˉf

l (x (t ))=r l i =1r l j =1

μl i (z l (t ))μl j (z l

(t ))

N m =1

F lm i x m (t )

l =1,...,N

(39b)

in which ˉD l =[D l a D l b ],ˉΔl (t )=diag(Δl a (t ),Δl b

(t )),and ˉL l i =[L l a ?L l b K l i ]T .

Next,based on the control synthesis conditions proposed in Theorems 1and 2,below theorem will be derived to ?nd the controllers for the system (39).

Theorem 4:Suppose there are known scalars εm a and εm

b ;the nonlinear large-scale system (1)is stable in the sense of Lyapunov based on the large-scale T–S fuzzy system (39)with the PDC control (5)when v m (t )=0,and the H ∞control per-formance in (2)for the system (1)is guaranteed for the given prescribed values λm when v m (t )=0,if there exist positive-de?nite matrices Q m =(Q m )T >0,ˉM

m =(ˉM m )T >0,vectors H m i ,and scalars ξm >0such that

ˉΨm i +ˉL m i <0,m =1,...,N,i =1,...,r m (40)ˉΨm ij +ˉL m ij

<0,m =1,...,N,

(41)

where

ˉΨm i =diag (Ψm i ,0,0,0,0),ˉΨm ij

=diag(Ψm ij ,0,0,0,0)(42)Ψm i

=Ωm i

+ˉD m (ˉD m )T (43)Ψm ij =Ωm ij

+12(ˉD m (ˉD m )T +ˉD m (ˉD m )T )(44)

ˉL m i =??????

??(1 λ2m )E m (E m )T +ˉM m εm a Q m (L m a )T ??I ????

εm b (H m i )T (L m b )

Q m ˉF m ξm I 000?I 00?

?ξm I 0??

?ξm I

??????

(45)

ˉL m ij

=???????????

(1/λ2m )E m (E m )T +ˉM m

εm a Q m (L m a )

??I ???????

?εm b (H m j )T (L m b )

εm b (H m i )T (L m b )

Q m ˉF m ξm I 0000?2I 000??2I 00???ξm I 0?

?ξm I

?????????

(46)

and ˉF m is de?ned at (27).In this case,ˉD m =[D m a D m b

],τm =1/ξm ,M m =Q ?1m ˉM m Q ?1

m ,and the control gain can

be obtained from K m i =H m i Q ?1

m .

Proof:The proof is given in Appendix A. The calculation for the reduced number of LMIs is similar to Theorem 3.Therefore,(34)in Theorem 3can be modi?ed in Theorem 4but with the similar derivation in Theorem 3.

Remark 5:In order to simplify the derivation of Theorem 4,the modeling error of interconnection term f l (x (t ))is not considered in Theorem 4.In fact,this simpli?cation does not affect the LMIs reduction contribution of Theorem 4.

CHANG AND W ANG:H∞FUZZY CONTROL SYNTHESIS FOR A LARGE-SCALE SYSTEM WITH A REDUCED NUMBER OF LMIS1205

TABLE III

C OMPARISON OF C OMPUTATION L OADS B ETWEEN THE P ROPOSE

D T HEOREM3AND[14,R EMARK1]

Using Theorem3Using[14,Remark1]

Number of subsystems N N=3N N=3 Number of fuzzy rules2q×N24×3=482q×N24×3=48 Computation load(seconds)non2.3090non370.4847 Number of unknown parameters N+N+N+(2z l×N)3+3+3+(2×3)=15((1+N×r l)N×r l/2)+N+(N×r l)1176+3+48=1227 to be solved by LMIs

Number of LMIs2z l×N2×3=6((1+N×r l)N×r l/2)+11176+1=1177

Remark6:It should be emphasized that the main skill which

used to generate control process(with number-reduction ef-

fect at LMIs)at Theorems1–4is the well-known S-procedure

method[19].It is suitable to be used at many kind of sys-

tem.Therefore,the proposed number-reduction method of LMI

can be extended and applied at many kind of system and

control design requirements such as robust control,optimal

control,observer design,non-PDC type fuzzy control design

[24],and imperfect premise membership functions[25]–[26],

etc.

VI.E XAMPLES AND S IMULATIONS

In this section,two examples are given.Example3is to show

the comparison of the number of LMIs to be solved between the

proposed Theorem3and[14,Remark1].Example4is to deal

with a H∞control performance problem for a practical two-

machine interconnected system in which the modeling error is

considered.

Example3:Consider a large-scale system(47),which is trans-

formed into a large-scale T–S fuzzy system by using the“sector

nonlinearity”method[17,p.10]

˙x l(t)=A l(x(t))x l(t)+B l(x(t))u l(t)

m=1

f lm(x(t))x m(t)+E l v l(t),l=1,...,N

(47)

where

A l(x(t))=

a l×x l1(t)sin(x l2(t))?3

B l(x(t))=

0 1

f lm(x(t))=

?b l0.1

0.5cos(x lm(t))0

,E l=

a l=rand1,and

b l=rand2.

Here,rand1and rand2are two random variables with zero mean and one variance.Suppose all states x l1(t) and x l2(t)belong to the interval[?π/2,π/2].Set z l1(t)= x l1(t)sin(x l2(t)),z l(1+m)(t)=cos(x lm(t)).The simulation is performed using MATLAB7.7.0on an Intel Core i7-3770 CPU(3.4GHz)with16-GB RAM.Due to space limita-tions,the subsystems’number N=3is chosen,and a1=?1, a2=1,a3=0.51,b1=?0.61,b2=0.34,and b3=?0.21. Using Theorem3,the PDC controllers are synthesized as follows:

K11=···=K18=

4.24450.7001

×103

K19=···=K116=

4.24760.7001

×103

K21=···=K28=

4.24760.7001

×103

K29=···=K216=

4.24450.7001

×103

K31=···=K38=

4.24680.7001

×103

K39=···=K316=

4.24520.7001

×103.(48) It can be seen from Table III that the number of fuzzy rules when using Theorem3is the same as that when using[14, Remark1].However,the number of LMIs,the unknown param-eters to be solved by LMIs,and the computation load of this paper are reduced explicitly compared with the results of paper [14].Moreover,it can be seen that even a small increase of the number of subsystems N will cause a large increase in the num-ber of LMIs if we use the traditional control methods proposed in[14].This is the so-called rule-explosion problem[20,p.273] in LMIs,which always seriously increases the complexity in the control design process.However,the proposed method of this paper may avoid this problem.

Remark7:The number of LMIs and the unknown parameters of[14]in Table III are larger than those in this paper.This is not just caused by the different controller design methods but also by the stabilization stratagem that was used.In[14],in order to obtain one relaxed stabilization condition,more LMIs and unknown parameters are required.However,we have to note that the control design results of this paper can easily be extended to the stabilization stratagem used in[14],and there are still fewer LMIs required than that in[14].

Example4:A two-machine interconnected system is em-ployed to illustrate the proposed robust and H∞performance of this paper.Equation(49)represents a general type of power system[6].Here,the subsystems and interconnections are non-linear.Hence,it is a common example to be used to examine the effectiveness of the control design of large-scale system.In this example,the“local approximation in fuzzy partition spaces”method[17,p.23]is used to transform the system(50)(trans-formed from(49))into a T–S fuzzy system as(52).For the system(52),there are two subsystems,and the control design needs to solve at least18LMIs and get18different control gains

1206IEEE TRANSACTIONS ON FUZZY SYSTEMS,VOL.23,NO.4,AUGUST 2015

to stabilize the system if the traditional control design methods are used.However,for the same system (52),we only need to solve six LMIs and ?nd six control gains to stabilize the system.Furthermore,the modeling error between the original system (50)and the T–S fuzzy system (52)is also considered.There-fore,the uncertain system (53)with the modeling error is dealt with as follows.

A power system which is composed of two subsystems is considered as follows [6]:˙x l 1(t )=x l 2(t )˙x l 2(t )=?

D l M l x l 2(t )+1

M l

u l (t )+

2 m =1

(m =l )

E l E m Y lm M l

cos δ0lm ?θlm ?cos(x l 1(t )?x m 1(t )+δ0

lm ?θlm )]

+w l (t )

l =1,2

(49)

where E 1=1.017,E 2=1.005,M 1=1.03,M 2=1.25,D 1=

0.8,D 2=1.2,Y 12=Y 21=1.98,θ12=?θ21=1.5,δ0

=?δ0

21=1.2,and w l (t )=sin(t ).Then,(49)can be represented as

˙x

l (t )=A l (x (t ))x l (t )+B l (x (t ))u l (t )+

2 m =1

(m =l )

f lm (x (t ))+E l v l (t ),

l =1,2(50)

where A l (x (t ))=?????

2 m =1(m =l )

E l E m Y lm M l [?2 cos(δ0lm ?θlm )?x l 1(t )]?

D l M l ?

???B l (x (t ))=??

?01M l ?

f lm (x (t ))=?

????

E l E m Y lm

M l [sin(δ0lm ?θlm )sin(x l 1(t ))cos(x m 1(t ))]?

E l E m Y

lm M l [sin(δ0

lm ?θlm )cos(x l 1(t ))sin(x m 1(t ))]?

??????E l =

TABLE IV

F UZZY M EMBERSHIP F UNCTIONS

i 12345M 1i ?π/2?π/2?π/200M 2i ?π/20π/2?π/20

i 6789M 1i 0π/2π/2π/2M 2

i π/2

?π/2

Fig.2.

Membership functions of x 11(t )and x 21(t ).

v l (t )=w l (t )+2 m =1

(m =l )

E l E m Y lm M l cos(δ0lm ?θlm

+x l 1(t ) 2?cos(δ0

lm ?θlm )cos(x l 1(t )?x m 1(t )) .

(51)

Next,using the “local approximation in fuzzy partition spaces”method [17,p.23],the large-scale T–S fuzzy system is obtained as follows.

Rule i :If x l 1(t )is about M 1i and x m 1(t )is about M 2i ,then ˙x

l (t )=A l i x l (t )

+B l i u l (t )

2 m =1

F lm i x m (t )+E l v l (t )i =1, (9)

(52)

The premise variables x l 1(t )and x m 1(t )are given for l =1,2;m =1,2;m =l .The values of M 1i and M 2i are shown in Table IV and the fuzzy membership functions are shown in Fig.2.

Next,we consider the modeling errors as uncertainties and then the large-scale T–S fuzzy system (52)with the PDC-type fuzzy controller (5)is represented as follows:˙x

l (t )=9 i =1

μl i

z l (t )

2 G l ii +ˉD l ˉΔl (t )ˉL l i x l (t ) +2

μl i z l (t ) μl j z l (t )

CHANG AND W ANG:H ∞FUZZY CONTROL SYNTHESIS FOR A LARGE-SCALE SYSTEM WITH A REDUCED NUMBER OF LMIS

1207

ˉG l ij

+ˉD l ˉΔl (t )ˉL l i +ˉD l ˉΔl (t )ˉL l j 2

x l (t )

+ˉf

l (x (t ))+E l v l (t ).(53)

Here,ˉD l ˉΔl (t )ˉL l i and (ˉD l ˉΔl (t )ˉL l i +ˉD l ˉΔl (t )ˉL l j

)/2are uncertainties.Moreover

ˉf

l (x (t ))=9 i =19 j =1

μl i

z l (t ) μl j

z l (t )

2 m =1

F lm i

x m (t ) .(54)

ˉF 1and ˉF 2are as follows:ˉF 1=[√2×9(F 111)T ,√2×9(F 112)T ,...,√2×9(F 119)T √2×9(F 211)T ,√2×9(F 212)T ,...,√2×9(F 219)T ]ˉF 2=[√2×9(F 121)T ,√2×9(F 122)T ,...,√

2×9(F 129

)T √2×9(F 221)T ,√2×9(F 222)T ,...,√2×9(F 229)T ]

(55)

The system parameters A l i ,B l i ,ˉD l =[D l a D l b ],ˉΔl (t )=

diag (Δl a (t ),Δl b (t )),ˉL l i =[L l a ?L l b K l i ]T ,ˉL l j =[L l a ?L l b K l j ]T ,

and F lm i are given in Appendix B.

Next,by using Theorem 4,the number of LMIs can be reduced to six (i.e.,we only have to satisfy the LMI conditions for i =

1,4,7,since A l 1=A l 2=A l 3,A l 4=A l 5=A l 6,A l 7=A l

A l 9,and

B l 1=···=B l

9).

Hence,the parameters are obtained with the given pre-scribed values λl =0.1and the εl a =0.25,εl

b =0,l =1,2as follows:

Q 1= 0.1766?4.4615?4.4615115.9857 ,Q 2=

0.1766?4.4615

?4.4615115.9857 ˉM

1= 0.01430.00000.0000229.9702 ,ˉM 2= 0.0143?0.0000

?0.0000229.9702 H 11

= 0.01151.4668 ×103H 14

= 0.01101.4810 ×103H 17

= 0.01041.4952 ×103H 21

= 0.01431.7511 ×103H 24

= 0.01381.7653 ×103H 27

= 0.01321.7795 ×103ξ1=1.8063,and ξ2=1.8063.

(56)The fuzzy control gains K l i are

K 11=K 12=K 1

= 1.36490.0538

×104

K 14=K 15=K 1

= 1.36450.0538

×104

K 17=K 18=K 1

= 1.36420.0538

×104

K 21=K 22=K 2

= 1.64130.0646

×10

Fig.3.State responses of subsystem (49)with l =1

Fig.4.State responses of subsystem (49)with l =2.

K 24=K 25=K 2

= 1.64100.0646

×104

K 27=K 28=K 2

9= 1.64070.0646

×104.

(57)

Finally,the PDC-type fuzzy controllers are as follows:u l (t )=?

9 i =1

μl i K l

i x l (t )

=?(μl 1+μl 2+μl 3)(K l 1x l (t ))?(μl 4+μl 5+μl

6)(K l 4x l (t ))?(μl 7+μl 8+μl 9)(K l 7x l (t )),

l =1,2.(58)

The simulation results are shown in Figs.3–5with initial

conditions (x 11(0),x 12(0))=(88π/180,?1.5)and (x 21(0),x 22(0))=(?89π/180,0).From the simulated responses,it is seen that the controlled two-machine interconnected sys-tem (49)with the fuzzy controller (58)is stable in the sense of Lyapunov.Next,under zero initial conditions,i.e.,(x 11(0),x 12(0))=(0,0)and (x 21(0),x 22(0))=(0,0),the values of

αl = t f 0x T l (t )M l x l (t )dt/ t f 0v T

l (t )v l (t )dt ,l =1,2,are ob-tained.After 0.2s,both of the αl ,l =1,2,tend toward constants 3.3417×10?4(see Fig.6).Thus,√

αl = 3.3417×10?4=0.0183<λl =0.1.Moreover,Fig.7shows the values Δl a (t )

of two subsystems.Here, Δl

a (t ) are small than the given

values ε1a =0.25and ε2

a =0.25.

1208IEEE TRANSACTIONS ON FUZZY SYSTEMS,VOL.23,NO.4,AUGUST

2015

Fig.5.Controllers output of two

subsystems.

Fig.6.Values of αl .Here,the curves are almost overlapped for l =1,

Fig.7.

Δl a (t ) of two subsystems.

Remark 8:In Example 3,only one nonlinear term appears

in A l (x (t ))and B l (x (t ))of each subsystem.According to Theorem 3(case (a)of (34)in Theorem 3),the number z m

is 1and r ?m is 2.Hence,in Example 3,we only need r ?

1+r ?2

+r ?3=2+2+2=6LMIs to synthesize the controllers (we need at least 48LMIs if the traditional method is used).On

the other hand,in Example 4,because of A m 1=A m 2=A m

A m 4=A m 5=A m 6,A m 7=A m 8=A m 9,and

B m 1=···=B m

there are three different combinations of A m i and B m

i for the

m th subsystem.Similar to the calculation process of Theorem 3(similar result obtained from case (b)of (34)in Theorem

3),the value of r ?

is 3.Hence,in Example 4,we only need r ?1+r ?

2=3+3=6LMIs to obtain the controllers (we need at least 18LMIs if the traditional method is used).

VII.C ONCLUSION

In this paper,a novel H ∞fuzzy control synthesis method for a general type nonlinear large-scale system is proposed.Since the interconnections of each subsystem are treated separately,the number of LMIs to be solved and the computation load are reduced so that the control design becomes much easier.The proposed method is especially useful when the interconnections are nonlinear and the number of subsystems in the large-scale system is large.This is the most important contribution of this paper.Moreover,the proposed control synthesis method also works for the T–S fuzzy system with the modeling error from transformation process.Finally,we have also proposed many examples and simulations to show the effect of number reduc-tion of LMIs and effectiveness of the synthesized controller.On the other hand,in this paper,we assumed that all states can be measurable and then developed the state-feedback fuzzy con-troller design techniques to the concerned large-scale systems.However,it is possible that some parts of states are unavailable in practical applications.Thus,developing the fuzzy observers for the large-scale system will be studied in the future.Fur-thermore,to extend the result of this paper to discrete-time large-scale systems is also one of our future works.

A PPENDIX A

P ROOF OF T HEOREM 4

Let the Lyapunov function be (12).Using the inequality X T Y +Y T X ≤λX T X +(1/λ)Y T Y ,the derivative of the l th Lyapunov function V l (x l (t ))along the trajectory of the l th subsystem is

˙V l (x l (t ))≤right-hand side of (13)+

r l

i =1

μl i

(z l (t ))2{x T l

(t )(P l ˉD l (ˉD l )T P l +(ˉΔl (t )ˉL l i )T (ˉΔl (t )ˉL l i

))x l (t )}+2

μl i (z l (t ))μl j (z l (t )) x T l (t ) P l ˉD l (ˉD l )T P l

+12

(ˉΔl (t )ˉL l i )T (ˉΔl (t )ˉL l

i )+12

(ˉΔl (t )ˉL l j )T (ˉΔl (t )ˉL l

j ) x l (t )

.(A1)

Due to Δl a (t ) ≤εl a and Δl b (t ) ≤εl

b ,(A1)follows the same process as in (14)–(20).It can be seen that H ∞control performance (2)is achieved with a prescribed λl and stable in the sense of Lyapunov if disturbance v l (t )is zero.Next,the same process in Theorem 2is followed,and the obtained stabilization conditions become LMIs (see (40)and (41)).Thus,the proof is complete.

CHANG AND W ANG:H∞FUZZY CONTROL SYNTHESIS FOR A LARGE-SCALE SYSTEM WITH A REDUCED NUMBER OF LMIS1209

A PPENDIX B

For subsystem1

A11=A12=A13=

?0.7545?0.7767

A14=A15=A16=

?3.8408?0.7767

A17=A18=A19=

?6.9271?0.7767

B11=···=B19=

0.9709

D1a=√

3×

?0.77671.9648

,L1a=

√

3×

Δ1a(t)=1

i=1

μ1i

?x11(t)0

μ11+μ12+μ13

?π/20

μ17+μ18+μ19

π/20

D1b=

,L1b=

0.9709

Δ1b(t)=?

i=1

μ1i0

01?

i=1

μ1i

F111=

0.01290

,F112=

?0.36960

F113=

?0.01290

,F114=

0.02030

F115=

?0.58060

,F116=

?0.02030

F117=

0.01290

,F118=

?0.36960

F119=

?0.01290

,F121=

?0.01290

F122=

?0.02030

,F123=

?0.01290

F124=

0.36960

,F125=

0.58060

F126=

0.36960

,F127=

0.01290

F128=

0.02030

,and F129=

0.01290

.(A2)

For subsystem2

A21=A22=A23=

?0.6217?0.9600

A24=A25=A26=

?3.1648?0.9600

A27=A28=A29=

?5.7079?0.9600

B21=···=B29=

0.800

D2a=

√

3×

?0.961.6190

,L2a=

√

3×

Δ2a(t)=

i=1

μ2i

?x21(t)0

+(μ21+μ22+μ23)

?π/20

μ27+μ28+μ29

π/20

D2b=

,L2b=

0.800

Δ2b(t)=

i=1

μ2i0

01?

i=1

μ2i

F211=

0.01060

,F212=

0.01670

F213=

0.01060

,F214=

?0.30460

F215=

?0.47840

,F216=

?0.30460

F217=

?0.01060

,F218=

?0.01670

F219=

?0.01060

,F221=

?0.01060

F222=

0.30460

,F223=

0.01060

F224=

?0.01670

,F225=

0.47840

1210IEEE TRANSACTIONS ON FUZZY SYSTEMS,VOL.23,NO.4,AUGUST2015

F226=

0.01670

,F227=

?0.01060

F228=

0.30460

,and F229=

0.01060

.(A3)

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1999.

Wei Chang received the B.S.and M.S.degrees from

the Department of Marine Engineering,National Tai-

wan Ocean University,Keelung,Taiwan,in2003and

2005,respectively.He received the Ph.D.degree from

the Department of Electrical Engineering,National

Central University,Jhong-li,Taiwan,in2014.

His research interests include nonlinear control,

fuzzy control,and large-scale system

analysis.

Wen-June Wang(F’08)was born in Hsin-Chu,

Taiwan,in1957.He received the B.S.degree in con-

trol engineering from National Chiao-Tung Univer-

sity,Hsin-Chu,in1980;the M.S.degree in electri-

cal engineering from Tatung University,Taipei,Tai-

wan,in1984;and the Ph.D.degree from the Institute

of Electronics,National Chiao-Tung University,in

1987.

He is currently the Chair Professor with the De-

partment of Electrical Engineering and serves as the

Dean of College of Electrical Engineering and Com-puter Science,National Central University,Chung-Li,Taiwan.His research interests include the areas of fuzzy theory and systems,robust control,neural networks,and pattern recognition.He has published more than140journal pa-pers and150conference papers.

Dr.Wang received the Distinguished Research Award from the National Sci-ence Council of Taiwan in1999,2001,and2003,respectively.He is a Member of the editorial board of numerous journals,including the International Journal of Electrical Engineers,the IEEE T RANSACTIONS ON F UZZY S YSTEMS,and the IEEE T RANSACTIONS ON C YBERNETICS.He is also the Editor-in-Chief of the International Journal of Fuzzy Systems.