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International Journal of Fuzzy Logic and Intelligent Systems, vol.12, no. 3, September 2012, pp. 187-192

http://dx.doi.org/10.5391/IJFIS.2012.12.3.187 pISSN 1598-2645 eISSN 2093-744X

Controller Design for Continuous-Time Takagi-Sugeno Fuzzy Systems with

Fuzzy Lyapunov Functions : LMI Approach

Ho Jun Kim1,Young Hoon Joo2, and Jin Bae Park1

1 Department of Electrical and Electronic Eng., Yonsei Univ., Seoul, 120-749 Korea2 Department of Control and Robotics Eng., Kunsan National Univ. Kunsan, Chonbuk 573-701, Korea

Abstract

This paper is concerned with stabilization problem of continuous-time Takagi-Sugeno fuzzy systems. To do this, the

stabilization problem is investigated based on the new fuzzy Lyapunov functions (NFLFs). The NFLFs depend on not

only the fuzzy weighting functions but also their first-time derivatives. The stabilization conditions are derived in terms of

linear matrix inequalities (LMIs) which can be solved easily by the Matlab LMI Toolbox. Simulation examples are givento illustrate the effectiveness of this method.

Key Words: Fuzzy Lyapunov functions (FLFs); Takagi-Sugeno fuzzy systems; Linear matrix inequalities (LMIs);

Stabilization

1. Introduction

During the last two decades, Takagi-Sugeno (T-S) fuzzy

system is an interesting issue because of its capability to

represent nonlinear systems effectively. Nonlinear systems

can be represented as an average weighted sum of linear

systems by the T-S fuzzy systems. It is very powerful

capability because the nonlinear systems can be partially

treated by linear control method. The Lyapunov method

is main approach to determine stability and solve stabiliza-

tion problem in the T-S fuzzy systems. In [1], common

quadratic Lyapunov function is used for solving stability

and stabilization problem. However it has result in very

conservative conditions because only one matrix should

satisfy all subsystems of the T-S fuzzy systems.

There have been a lot of efforts to resolve this prob-

lem. In [2], interactions among all fuzzy subsystems of

T-S fuzzy systems are considered. In [3], [4], membership

functions are considered in the Lyapunov method. In [5],

[6], non-parallel distributed compensation (NPDC), whichis the fuzzy controller dose not share the same fuzzy sets

with the fuzzy systems, is used, and in [7], [8], the Lya-

punov functions are changed for presenting relaxed stabil-

ity and stabilization conditions. Especially, changing Lya-

punov functions is one of the main issues to reduce con-

Manuscript received Jun. 20, 2012; revised Sep. 21, 2012; accepted Sep.

24. 2012.Corresponding Author : [email protected]

Acknowledgment: This work was supported by the Human Resources

Development program(No. 20124010203240) of the Korea Institute of

Energy Technology Evaluation and Planning(KETEP) grant funded by the

Korea government Ministry of Knowledge Economy.

cThe Korean Institute of Intelligent Systems. All rights reserved.

servativeness. In [7], fuzzy Lyapunov functions (FLFs) are

presented at the first time. The FLFs are consist of com-

mon Lyapunov functions whose number is the number of

fuzzy rules. There have been a lot of efforts to improve

the performance of FLFs since those are presented. Es-

pecially, continuous-time fuzzy systems (CFSs) are harder

to analyze their stability and stabilization than those of thediscrete-time fuzzy systems (DFSs) because FLFs of CFSs

depend on the time-derivative of the membership functions.

In [9], the systematic approach to analyze stability and sta-

bilization of CFSs based on FLFs is proposed by introduc-

ing some null terms. In [8], The new fuzzy Lyapunov func-

tions (NFLFs) are proposed which depend on not only the

fuzzy weighting functions but also their first-order time-

derivatives. Although this method is very useful for stabil-

ity analysis of CFSs, stabilization problem is open issue.

The stabilization conditions based on FLFs are derived in

terms of bilinear matrix inequalities (BMIs) in the many

cases which are harder to be solved than the LMIs. Fur-thermore, the stabilization conditions based on NFLFs are

not yet discussed.

In this paper, stabilization analysis of CFSs based on

NFLFs are considered. NFLFs depend on time-derivative

of membership functions as well as membership functions.

Two assumptions, which are upper bounds of membership

functions and those of first time-derivatives, are needed for

formula development. Stabilization conditions are derived

in terms of LMIs which are easily solved by Matlab LMI

Toolbox [10]. Finally, numerical example is given to illus-

trate the effectiveness of the proposed method.

Notations: Rn := n-dimensional real space. Rmn :=

Set of all realm byn matrices.AT := Transpose of matrix

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International Journal of Fuzzy Logic and Intelligent Systems, vol.12, no. 3, September 2012

A. P 0 (resp. P 0) := Positive (resp. negative)-definite symmetric matrix. * := The transposed element in

the symmetric position. hi := hi(z(t)), x := x(t), u :=

u(t).

2. Preliminaries

Consider a continuous nonlinear system

x= f(x) + g(x)u (1)

wherex Rn is the state vector,u Rm is the input statevector, and f(x), g(x)are smooth nonlinear functions. Thei-th T-S fuzzy rule for the (1) is represented as

Plant Rulei:Ifz1(t)isF

i1

and z2(t)isFi2

and .. . zp(t)isFip

Then x(t) =Aix(t) + Biu(t),

(2)

wherei = 1, 2, . . . , ris the rule number of the fuzzy rules,Fij is the fuzzy sets where j = 1, 2, . . . , p, zj(t) is thepremise variables, Ai R

nn, and Bi Rnm are the

i-th local system matrices. (2) is described as followingequation by using singleton fuzzifier, product inference en-

gine, and center-average defuzzifier.

x=r

i=1

hi(Aix + Biu) (3)

where hi is the normalized membership function of i-thrule and satisfying following properties.

hi 0,r

i=1

hi= 1 (i= 1, 2, . . . , r) (4)

Thei-th rule of the fuzzy controller for the nonlinear sys-tem based on parallel distributed compensation (PDC) is

described as

Controller Rulei:

Ifz1(t)isFi1 and z2(t)isFi2 and .. . zp(t)isFip

Thenu = Kix,

(5)

Using the same fuzzifier, inference engine, and defuzzifier

of the plant, (5) is described as:

u= r

i=1

hiKix (6)

and substituting (6) into (3), the final expression is repre-

sented as

x=

ri=1

rj=1

hihj(Ai BiKj)x (7)

In this paper, stabilization analysis is considered based

on the NFLF which is described as :

V(x) =

ri=1

hixTPix +

rk=1

hkxTPkx (8)

Notice that Lyapunov functions depend on membership

function and its first time-derivatives. However, member-

ship function and its time-derivatives are not easy to be han-

dled in terms of LMIs because of their nonlinearity. By this

reason, two assumptions, upper bound of first and second

time-derivatives, are needed for smooth formula develop-

ment.

|hk| 1k, |hl| 2l, (9)

where1k and2l are positive real number and followingproperties are derived from the (4)

rk=1

hk = 0,r

l=1

hl = 0 (10)

3. Controller Design

3.1 Stability Analysis

In this section, sufficient stability conditions of the T-Sfuzzy system (3) which were proposed in [8] is introduced

. Also the NFLF is adopted, and two assumptions (9) are

considered. In this section, only the unforced ,that is, u =0, system of (7) is considered, so it is presented as

x=r

i=1

hiAi (11)

Lemma 3.1. [8] T-S fuzzy system (11) is asymp-

totically stable where (9) is considered, i ,k,l ={1, 2, . . . , r}, j > i and if there exist symmet-

ric matrices Pi, Pk, X , Y 11, Y22, Z11, Z22, any matricesM1i, M2i, N1k, N2k, L1l, L2l,, Y21, Z21, and satisfying

following inequalities.

Pk+ X 0, (12)

r

k=1

1k(Pk+ X) + Pi 0, (13)

1ik := (14)Pk N1kAi A

Ti N

T1k+ Y11

Pk+ NT1k N2kAi+ Y21 N2k+ N

T2k+ Y22

0, (15)

2il :=

Pl L1lAi ATi LT1l+ Z11 LT1l L2lAi+ Z21 L2l+ L

T2l+ Z22

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0, (16)

3ij+ 3

ji 0 (17)

where

3ij :=

M1jAi A

Ti M

T1j

Pi+ MT1j M2jAi M2j + MT2j

+r

k=1

1k1

ik+r

l=1

2l2

il

3.2 Stabilization analysis

The stability conditions with NFLFs are well discussed

in the previous section. However, the stabilization condi-

tions based on the the NFLF is not tackled at all because

of its difficulties to be derived in terms of LMIs. In this

section, stabilization conditions based on the NFLF areconsidered. The concept of parallel distributed compensa-

tion (PDC) is employed to design a fuzzy controller which

shares the same fuzzy rules and fuzzy sets with the plant of

the T-S fuzzy system. Two assumptions (9) are considered

and following null terms are added.

2

xTM+ xT1M

x r

i=1

rj=1

hihj(Ai BiKj)

= 0

(18)

2xT

r

k=1

hkM+ xT

r

k=1

hk2kM

x r

i=1

rj=1

hihj(Ai BiKj)

= 0 (19)

2

xTr

l=1

hlM+ xT

rl=1

hl3lM

x r

i=1

rj=1

hihj(Ai BiKj)

= 0 (20)

where1, 2k,and3l are proper scalar values.

Theorem 3.2. T-S fuzzy system (3) is stabilized with the

fuzzy controller (6) gains given by Ki = BiR where(9) is considered, and if there exist symmetric matri-

ces Pi, Pi, Ti, Tk, X, Y11,Y22, Z11, Z22, ij any matrices

R,Y21, Z21,and satisfying following inequalities.

Pk+ X 0, k= 1, 2, . . . , r (21)

r

k=1

1k(Pk+ X) + Pi 0, i, k= 1, 2, . . . , r (22)

1ijk :=

ijk + Y11 ijk + Y21 2k(R + R

T) + Y22

0, i, j, k= 1, 2, . . . , r (23)

2ijl :=

ijl + Z11 ijl + Z21 3l(R + R

T) + Z22

0, i, j, l= 1, 2, . . . , r (24)

3ii+ ii 0 i= 1, 2, . . . , r (25)

3

ij+

3

ji + 2ij 0, 1 i < j r (26)

11 12 1r12 22 2r

.... . .

...

1r 2r rr

0 (27)

where

3ij := ij+r

k=1

1k1

ijk +r

l=1

2l2

ijl

ijk := Tk (AiRT BiSj) (AiR

T BiSj)T

ijk := Tk+ R 2k(AiRT BiSj)

ijl := Tl (AiRT BiSj) (AiR

T BiSj)T

ijl :=R 3l(AiRT BiSj)

ij :=(AiR

T BiSj) (AiRT BiSj)

T Ti+ R 1(AiR

T BiSj) 1(R + RT)

Proof. The (8) should be demonstrated to show that V(x)is a Lyapunov candidate function. Let us consider (9), (21),

and (22). First,V(x)> 0 should be verified.

V(x) =

rk=1

hkxTPkx +

ri=1

hixTPix

=xT rk=1

hkPk+r

i=1

hiPi+r

k=1

hkX

x

=xT rk=1

hk(Pk+ X) +r

i=1

hiPi

x

r

i=1

hixT

rk=1

1k(Pk+ X) + Pi

x

>0.

From now on V(x)< 0 should be demonstrated.

V(x) =r

l=1

hlxTPlx +

rk=1

hkxTPkx

+ 2r

k=1

hkxTPkx + 2

ri=1

hixTPix

Let us consider null terms (18) - (20).

V(x) =r

l=1

hlxTPlx +

rk=1

hkxTPkx

+ 2r

k=1

hkxTPkx + 2

ri=1

hixTPix

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+ 2

xTM+ xT1M

x r

i=1

r

j=1

hihj(Ai BiKj)+ 2

xT

rk=1

hkM+ xT

rk=1

hk2kM

x r

i=1

rj=1

hihj(Ai BiKj)

+ 2

xTr

l=1

hlM+ xT

rl=1

hl3lM

x r

i=1

rj=1

hihj(Ai BiKj)

It is equivalent with the next equation.

ri=1

rj=1

rk=1

rl=1

hihjhkhl

xx

T

Pl+ Pk 3M(Ai BiFj) 3(Ai BiFj)

TMT

Pk+ Pi+ 3MT (1+ 2k+ 3l)M(Ai BiFj)

(1+ 2k+ 3l)(M+ MT)

xx

=r

i=1

r

j=1

hihj x

xT

M(Ai BiFj) (Ai BiFj)

TMPi+ M

T 1M(Ai BiFj)

1(M+ M

T)

xx

+r

i=1

rj=1

rk=1

hihjhk

xx

T

Pk M(Ai BiFj) (Ai BiFj)

TMT + Y11Pk+ M

T 2kM(Ai BiFj) + Y21

2k(M+ MT) + Y22 x

x

+r

i=1

rj=1

rl=1

hihj hl

xx

T

Pl M(Ai BiFj) (Ai BiFj)

TMT + Z11MT 3lM(Ai BiFj) + Z21

3l(M+ M

T) + Z22

xx

The above equation should be derived in terms of LMI.

Define the following transformation matrix.

M1 I

I M1

Take a congruence transformation to represent in terms of

LMI and consider (9).

ri=1

rj=1

hihj

xx

Tij

xx

+r

i=1

rj=1

rk=1

hihjhk

xx

Tijk + Y11 ijk + Y21 2k(R + R

T) + Y22

xx

+

ri=1

rj=1

rl=1

hihj hl

x

xT

ijl + Z11

ijl + Z21 3l(R + RT

) + Z22

x

x

=r

i=1

rj=1

hihj

xx

T ij+

rk=1

hk1

ijk +r

l=1

hl2

ijl

xx

r

i=1

rj=1

hihj

xx

T ij+

rk=1

1k1

ijk +r

l=1

2l2

ijl

xx

=

xx

T r

i=1

h2i 3

ii+r

i=1

rj>i

(3ij+ 3

ji) x

x

xx

T ri=1

h2i ii+r

i=1

rj>i

2ij x

x

=

h1zh2z

...

hrz

T

11 12 1r12 22 2r

.... . .

...

1r 2r rr

h1zh2z

...

hrz

< 0

wherezT

=

xT xT

Therefore, if (27) is satisfied, Vis negative definite and theequilibrium point of the fuzzy system is stabilized with the

fuzzy controller (6).

Remark 3.3. The contribution of this paper is that the sta-

bilization conditions are derived in terms of LMIs based on

the new fuzzy Lyapunov function.

Remark 3.4. Theorem 1 can induce conservative re-

sults according to changes in the value of the parameters

1, 2k,and3l. There are no ways to choose these scalar

vector systematically but just do trial and error for set theseparameters. The main factor of the conservativeness may

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Controller Design for Continuous-Time Takagi-Sugeno Fuzzy Systems with Fuzzy Lyapunov Functions : LMI Approach

1.5 1 0.5 0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x2

(t)

GradeofMembership

Figure 1: Membership functions of the fuzzy model. (dot-

ted line : h1, dashed line : h2)

be the common matrixMbetween the (18) - (20). There-fore, it is an open issue to remove the common matrix for

solving the problem.

4. Computer Simulations

To show the effectiveness of the proposed method, nu-

merical example is given and performed with the Matlab

LMI Toolbox. Consider the fuzzy system (3) with

A1 =

2 101 0

, A2 =

16 101 0.65

,

B1 =

10

, B2 =

30

where 1k = 2k = 0.8, 1 = 2k = 3l = 1, andx(0) = [ 1 1 ]T.The membership functions are

h1 =1 + sin x2(t)

2 , h2 =

1 sin x2(t)

2

, andx2(t) [/2/2]. The Fig. 1 shows the member-ship functions of the fuzzy model. The feedback gains are

obtained in the Theorem 1 as

K1 = [ 6.9924 0.8427 ], K2 = [ 7.0212 1.0479 ] (28)

The Fig. 2 and Fig. 3 show the state trajectory of the

each state variable x1and x2with controller feedback gains(28). All states of the system go to zero, and the fuzzy

system (3) is stabilized with the fuzzy controller (6).

0 2 4 6 8 102

1.5

1

0.5

0

0.5

1

x1

(t)

time (sec)

Figure 2: The system state response ofx1(t)with the fuzzycontroller

0 2 4 6 8 100.2

0

0.2

0.4

0.6

0.8

1

1.2

x2

(t)

time (sec)

Figure 3: The system state response ofx2(t)with the fuzzycontroller

5. Conclusion

In this paper, the stabilization conditions with NFLFs are

given. The NFLFs depend on not only the fuzzy weighting

functions but also their first-time derivatives. The stabi-

lization conditions are derived in terms of LMIs and easily

solved by the LMI Toolbox. There are open issues to solve

the common matrix problem which induce the conserva-

tiveness. Simulation example is given to show the effec-

tiveness of this method.

References

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Ho Jun Kim

Student of the Yonsei University

Research Area: Fuzzy Control

E-mail : [email protected]

Young Hoon Joo

Professor of the Kunsan National University

Research Area: Fuzzy, Nonlinear Systems


Jin Bae Park

Professor of the Yonsei University

Research Area: Nonlinear System, Filter, Fuzzy


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