Design of interval type 2 t s fuzzy logic control systems

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http://www.sj-ce.org

Scientific Journal of Control Engineering October 2014, Volume 4, Issue 5, PP.122-137

Design of Interval Type-2 T-S Fuzzy Logic

Control Systems Li Li

1#, Yijun Du

2, Yimin Li

2

1. School of Computer Science Telecommunication Engineering, Jiangsu University, Zhenjiang Jiangsu 212013, China

2. Faculty of Science, Jiangsu University, Zhenjiang Jiangsu 212013, China

#Email: [email protected]

Abstract

In this paper, an interval type-2 T-S fuzzy logic control systems (IT2T-S FLCS) is designed. The proposed IT2T-S FLCS is a

combination of IT2 fuzzy logic system (FLS) and T-S FLCS, and also inherits the benefits of these two methods. Furthermore,

Krasovskii’s method is utilized to testify the sufficient condition for the asymptotic stability of IT2T-S FLCS, which requires the

calculation of the Jacobian matrix. Finally, the simulation results show that the IT2T-S FLCS achieves the best tracking

performance in comparison with the type-1 (T1) T-S FLCS and the proposed method can handle unpredicted internal disturbance

and data uncertainties well.

Keywords: Type-2 Fuzzy Sets; Stability; Jacobian Matrix

1 INTRODUCTION

Since Takagi and Sugeno [1] proposed the T-S fuzzy model in 1985, the T-S fuzzy system has emerged as one of the

most active and fruitful areas of fuzzy control. The T-S FLS [1, 2] was proposed in an effort to develop a systematic

approach to generate fuzzy rules from a given input-output data set. This model consists of rules with fuzzy anteced-

ents and a mathematical function in the consequent part. By using this modeling approach, a complex non-linear

system can be represented by a set of fuzzy rules of which the consequent parts are linear state equations. The

complex non-linear plant can then be described as a weighted sum of these linear state equations. This T-S fuzzy

model is widely accepted as a powerful modeling tool. Their applications to various kinds of non-linear systems can

be found in [3-5].Quite often, the knowledge used to construct rules in T-S FLS is uncertain. This uncertainty leads

to rules having uncertain antecedents and/or consequents, which in turn translates into uncertain antecedent and/or

consequent membership functions. When the measured data is less and the model is inexact, we can use Type-2

fuzzy sets[6]. Such type-2 fuzzy sets whose membership grades themselves are type-1 fuzzy sets; they are very

useful in circumstances where it is difficult to determine an exact membership function for a fuzzy set, they are

useful for incorporating linguistic uncertainties. The type-2 FLS has been successfully applied to sliding-mode

controller designs, fault tolerant systems design, robust adaptive interval type-2 fuzzy tracking control of

multivariable nonlinear systems and impulsive control of nonlinear systems[7-10].An indirecta daptive interval type-

2 fuzzy control is proposed in[11,12]. Moreover,direct and indirect adaptive interval type-2 fuzzy controlis

developedin[13,14] for a multi-input/multi-output(MIMO) nonlinear system. Type -2 fuzzy systems have shown a

great potential in various modeling as well as control application [15]. A. Abbadi[16] propose an interval type-2

fuzzy controller that has the ability to enhance the transient stability and achieve voltage regulation simultaneously

for multimachine power systems. The design of this controller involves the direct feedback linearization technique.

Hence, in this paper, we study the design of the IT2T-S FLCS.The proposed IT2T-S FLCS is a combination of IT2

FLS and T-S FLCS, and also inherits the benefits of these two methods. It was believed that IT2T-S FLS have the

potential to be used in control and other areas where a T1T-S model may be unable to perform well.

This paper is organized as follows: In Section 2, design procedure of the IT2T-S FLCS is addressed in detail. In

Section 3, the proposed Krasovskii’s method is utilized to testify the sufficient condition for the asymptotic stability

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of IT2T-S FLCS. In Section 4, developed expressions are obtained deriving the IT2T-S fuzzy system model with

regard to every coordinate of the state space. In Section 5, developed expressions in the previous section are used to

implement an algorithm. In Section 6, the proposed methodology is illustrated by an example. The conclusions are

drawn in Section 7.

2 ANALYSIS AND DESIGN OF IT2 FUZZY CONTROLLER ON T-S MODEL

An easy way to comply with the journal paper formatting requirements is to use this document as a template and

simply type your text into it.

2.1 RULE BASE

lR : If 1x is

1

l

iA and 2x is

2

l

iA … and nx is l

niA and1u is

1

l

iB and 2u is

2

l

iB … and mu is l

miB

Then 0 1 1 1 1( , )l l l l l l l

i i i i ni n i mi mx g X U a a x a x b u b u (1)

Where 1, ,l M is the number of rules, l

kiA are the IT2 fuzzy sets defined in a universe of discourse for the state

variablekx , 1, ,k n and l

ix is the first-order differential equation i , l

jiB is the IT2 fuzzy set defined in a universe

of discourse for the j th control signal ju , 1, ,j m , l

kia and l

jib represent the respective constant coefficients for

the state variable kx and for the j th signal of control of the differential equation i that represents the process. The

proposed IT2T-S FLCS for the case when antecedents are IT2 and consequents are crisp numbers.

2.2 FUZZY INFERENCE ENGINE

The inference engine combines all the fired rules and gives a non-linear mapping from the input IT2 fuzzy sets to the

output IT2 fuzzy sets. Input 1 2 n( , , , )T nX x x x R denotes the state vector, and

1 2U ( , , , ) m

mu u u R denotes

the input control vector. This rule represents an IT2 relation between the input space1 1n mX X U U , and

the output space X , of the IT2 FLS.

The firing strength of the l th rule is

11 1 1 1

1 1 1 1

1 1

1 1

( ) ( )

( ) ( )

ln n n n

l lm m m m

l

i x X x X n nX XA A

u U u U m mU UB B

w x x x x

u u u u

(2)

Here we focus on a major simplification of (2) as a result of singleton fuzzification, hence the fuzzy set X or U is

such that it is has a membership grade 1 corresponding to X X or U U and has zero membership grade for all

other inputs; therefore, (2) reduces to

1 1

1 11 1

l l l l l ln m k j

n ml

i n m k jA A B B A Bk jw x x u u x u

(3)

where ( 1,2, )kx k n and ( 1,2, )ju j m denote the location of the singleton.

lR is described by the membership function ( )l

l

iRx

1 1

( )l l l li k j

n ml l

i i k jR X A Bk jx x x u

(4)

where now each l

iX is a crisp value l

ix , so (4) reduces to

1 1

( )l l lk j

n ml

i k jR A Bk jx x u

(5)

Then output fuzzy set ( )R ix for a IT2 FLS is

1 1 1 1

( ) ( )l l lk j

M M n ml

R i i k jR A Bl l k jx x x u

(6)

where denotes the product operation.

Let COG ( ( )iRx ) and COG ( ( )R ix ) denote the centroid of the lower membership function (LMF) and upper

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membership function (UMF), respectively, of ( )R ix . Then, we can define the output of an IT2 FLSix , as

ix =w COG ( ( )iRx )+(1-w) COG( ( )R ix ) (7)

where weight w∈[0, 1] can be tuned during a design procedure [7].

So the output of an IT2T-S FLCSix is shown as

1 1

1 1

( , ) ( , )(1 )

M Ml l l l

i i i il l

i M Ml l

i il l

w g X U w g X Ux w w

w w

(8)

1 1

( ) ( )l lk j

n ml

i k jA Bk j

w x u

(9a)

1 1

( ) ( ))l lk j

n ml

i k jA Bk j

w x u

(9b)

1 1

( ) ( )l lk j

n ml

i k jA Bk j

w x u

,ll l

ii iw w w

(10)

Where w can be tuned during a design procedure. ( )k

l

kAx and ( )

j

l

jBu are the membership functions defined in

their corresponding universe of discourse X and U .

Take the consequent of the rule (1) into (8), which can be written as

0 1 1 1 11 1 1 1

2 21 1 1

M M M Ml l l ll l l l

i i i ii i ni n il l l l

i M l M Ml ll li i ii mi ml l l

w a w a x w a x w b uwx

w w b u w b u

+ 0 1 1 1 11 1 1 1

2 21 1 1

1M M M Ml l l l l l l l

i i i i i ni n i il l l l

M M Ml l l l li i i i mi ml l l

w a w a x w a x w b uw

w w b u w b u

(11)

This is

0 1 1 2 2 1 1 2 2

t t t t t t t

i i i i ni n i i mi mx a a x a x a x b u b u b u

0 1 1

n mt t t

i ki k ji jk ja a x b u

(12)

Where0

t

ia , t

kia and t

jib are variable coefficients computed as

0 01 1

0

1 1

1

M Ml l l l

i i i it l l

i M Ml l

i il l

w a w aa w w

w w

(13a)

1 1

1 1

1

M Ml l l l

i ki i kit l l

ki M Ml l

i il l

w a w aa w w

w w

1, ,k n (13b)

1 1

1 1

1

M Ml l l l

i ji i jit l l

ji M Ml l

i il l

w b w bb w w

w w

1, ,j m (14)

Up to here, the IT2 fuzzy plant model is obtained from the development of the methodology. Next, we will develop

the IT2 fuzzy controller model.

The process of the fuzzy controller design that stabilizes the plant is considered independently from the process of

the plant identification. Therefore, it does not require the model plant and the controller have the same number of

rules.

Considering those mentioned above, the achievement of the IT2 fuzzy plant model proceeds in a similar way. The

IT2 fuzzy controller can be represented by the following group of rules:

rR : If 1x is 1

r

jC and 2x is 2

r

jC … and nx is r

njC Then 0 1 1

r r r r

j j j nj nu c c x c x (15)

Where r

kjC represents the IT2 fuzzy sets of 1, ,k n state variables, with 1, ,j m control actions and

1, ,r N rules.

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Control signals can be expressed as follows in a similar way shown in (8)-(12):

0 1 11 1 1

1

N N Nr r rr r r

j j jj j j nj nN r r rr

jr

wu w c w c x w c x

w

+

0 1 11 1 1

1

1 N N Nr r r r r r

j j j j j nj nN r r rr

jr

ww c w c x w c x

w

(16)

Or, in a more simplified way:

0 1 1 2 2

t t t t

j j j j nj nu c c x c x c x = 0 1

nt t

j kj kkc c x

(17)

where the variable coefficients are given by

0 01 1

0

1 1

1

N Nr r r r

j j j jt r rj N Nr r

j jr r

w c w cc w w

w w

(18a)

1 1

1 1

1

N Nr r r r

j kj j kjt r rkj N Nr r

j jr r

w c w cc w w

w w

1, ,k n (18b)

and the lower and upper matching degree of the rule r of the controller, respectively by

1

( )rk

nr

j kCk

w x

(19a)

1

( )rk

nr

j kCk

w x

(19b)

Replacing expression (17) into (12), the following IT2T-S closed loop output function can be written as:

0 01 1 1

m n mt t t t t t

i i ji j ki ji kj kj k jx a b c a b c x

, , 1, ,i k n and 1, ,j m (20)

Expression (20) is an equivalent mathematical closed loop model of IT2T-S FLCS. In order to simplify Eq. (20), the

state vector will be extended in a coordinate, that is

0 1

1, ,

T

nx x x xX

(21)

Replacing the extended state vector into (20) the simplified following expression is obtained

0 1( )

n mt t t

i ki ji kj kk jx a b c x

0,1, ,k n , 1, ,i n and 1, ,j m (22)

3 STABILITY VERIFICATION OF THE IT2T-S FLCS

The stability of nonlinear systems has to be investigated without making linear approaches. In order to do this,

several approaches can be adopted based on Lyapunov’s second method, for instance, Krasovskii’s theorem [17, 19].

Krasovskii’s theorem for global asymptotic stability offers sufficient conditions for nonlinear systems. An

equilibrium state of a nonlinear system can be stable even when the conditions specified in this theorem are not

satisfied; however, the synthesis of systems that complete this theorem assures that they are asymptotically stable

globally.

Krasovskii’s theorem demonstrates that given a system X = ( )X , if the system’s Jacobian matrix ( ) ( )TJ X J X is

negative definite in a region around the equilibrium state eX , this is asymptotically stable in its proximities.

Moreover, if ( ) ( )TX X when X , the asymptotical stability is global. The Lyapunov function for this

system is ( )TX ( )X .

In section 4, we present a general algorithm to compute the Jacobian matrix of a closed loop IT2T-S FLCS. The

algorithm has been improved by a simplification of the equations based on the state vector extension. The algorithm

is independent of the type of membership function used for building the models of the plant and the controller.

Different types of membership functions in the same model can also be combined in the proposed algorithm.

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4 COMPUTATION OF THE JACOBIAN MATRIX OF A CLOSED LOOP IT2T-S FLCS

Let the IT2T-S control system modeled by Eq. (22), which can be represented in the following generic way:

1 1 1 2( , , , )nx x x x

2 2 1 2( , , , )nx x x x

....

1 2( , , , )n n nx x x x (23)

Let 1 2( , , , )e e e e T

nX x x x be an equilibrium state of system (20). Whereas the function is continuous over the range,

a Taylor series expansion about the point eX may be used:

1 2

1 1

1

( , , , )

e

e e i n

i i

X

x x xdx x x x

dt x

+…+ 1 2( , , , )

e

e i n

n n

n X

x x xx x

x

+ ( )e

iR X (24)

When the nonlinear term ( )e

iR X fulfills (25), being iK a sufficiently small constant, then the Jacobian matrix of the

system determines the properties of stability of the equilibrium state eX :

2 2

1 1( )e e e

i i n nR X K x x x x

(25)

The Jacobian matrix is calculated applying the following expressions:

1 1 2 1 1 2

1

1 2 1 2

1

( , , , ) ( , , , )

( )

( , , , ) ( , , , )

e e

e

e e

n n

nX X

iq X

n n n n

nX X

x x x x x x

x x

J X

x x x x x x

x x

(26)

In the previous expression it is needed to calculate the partial derivative from Eq. (22) with regard to each one of the

state variables. This equation can be divided in two addends:

0 1( )

n mt t t

i ki ji kj kk jx a b c x

=

0 0 1

n n mt t t

ki k ji kj kk k ja x b c x

(27)

The partial derivatives will be obtained in a generic form for the 1, ,q n state variables and the 1, ,i n first

degree differentials equations.

4.1 DERIVATIVE OF THE FIRST ADDEND

The first partial derivative to calculate is from the first adding of Eq. (27):

0

0 0

n t t tki k n nk ki k tki

k qik kq q q

a x a x ax a

x x x

(28)

Bearing in mind (13b):

1 1 1 1

1 1 1 1

1

1

M M M Ml ll l l l l l

i iki i ki ki i kil l l l

M M M Ml ll lti ii il l l lki

q q q q

w a w a w a w aw w

w w w waw w

x x x x

(29)

Calculate the first adding of Eq. (29):

1 1 1

1 11

2

1

M M Ml l ll li ki i ikil M Ml ll l l

M l i i kil li q ql

M lq

il

w a w a ww w a

w x xw w

xw

(30)

while 1

1

M l ll

i ki Ml li

kilq q

w a wa

x x

(31)

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and 1

1

M ll

i Ml i

lq q

w w

x x

(32)

Replacing now the previous two expressions in (30) and simplifying, it results in

1

1 1

1

2

1( )

M l ll

i ki M Ml p l piM l i ki kil p

i ql

M lq il

w a ww a a

w xw w

x w

(33)

calculate the second adding of Eq. (29) is similar to calculate the first adding of Eq. (29):

1

1 1

1

2

1

1 1( )

M l ll

i ki M Ml p l piM i ki kil l p

i ql

M lq il

w a ww a a

w xw w

x w

(34)

Replacing (33), (34) expressions in (29) , it results in

1 1 1 1

2 2

1 1

1( ) ( )

l lM M M Mp l p p l pi i

i ki ki i ki kil p l ptq qki

M Ml lq i il l

www a a w a a

x xaw w

x w w

(35)

Finally, Eq. (28) can be written as

0

1 1 1 1

2 20

1 1

1

n t

ki kk

q


i ki ki i ki kil p l pn q q t

k qik M Ml l

i il l

a x

x

www a a w a a

x xw w x a

w w

(36)

Notice that to evaluate expression (36) it is necessary to calculate previously the derivative of the matching degree of

the rules of the plant, given from Eq. (9a), (9b). This point is solved in Section 4.4.

4.2 DERIVATIVE OF THE SECOND ADDEND

To calculate the derivative of the second addend, the following expression must be solved:

0 1 1

0 1

n m mt t t t

ji kj k ji kjn mk j j t t

k ji qjk jq q

b c x b cx b c

x x

(37)

Where

1

1

m t tt t

ji kj mj ji kjt t

kj jijq q q

b c b cc b

x x x

(38)

Given the similarity of the expressions for the calculation of t

kia and t

jib , it is easy to deduce that

1 1 1 1

2 2

1 1

1( ) ( )


i ji ji i ji jit l p l pq qji

M Ml lq i il l

www b b w b b

x xbw w

x w w

(39)

The derivative of the term corresponding to the fuzzy controller will be:

1 1 1 1

1 1 1 1

1

1

N N N Nr rr r r r r r

j jkj j kj kj j kjr r r r

N N N Nr rr rtj jj jkj r r r r

q q q q

w c w c w c w cw w

w w w wcw w

x x x x

(40)

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calculate the first adding of Eq. (40):

1 1 1

1 1

1

2

1

N N Nr r rr rj kj j jkjr N Nr rr r r

N r j j kjr rj q qr

N rq

jr

w c w c ww w c

w x xw w

xw

(41)

while 1

1

N r r rj kj Nr j r

kjrq q

w c wc

x x

(42)

and 1

1

N rr

j Nr j

rq q

w w

x x

(43)

Replacing now the previous two expressions in (41) and simplifying, it results in

1

1 1

1

2

1( )

N r r rj kj N N sr j r s

N j kj kjr r sqjr

N rq jr

w c ww c c

xww w

x w

(44)

calculate the second adding of Eq. (40) is similar to calculate the first adding of Eq. (40):

1

1 1

1

2

1

(1 ) (1 )( )

N r r rj kj N Nr j s r s

N j kj kjr r sj qr

N rq jr

w c ww c c

w xw w

x w

(45)

Finally, Eq. (40) can be written as

1 1 1 1

2 2

1 1

1( ) ( )

r rN N N Nsj r s j s r s

j kj kj j kj kjr st r sq qkj

N Nr rq j jr r

w ww c c w c c

x xcw w

x w w

(46)

Notice that to evaluate expression (46) it is necessary to calculate previously the derivative of the matching degree of

the rules of the controller, given from Eq. (19a), (19b). This point is solved in Section 4.3.

As soon as intermediate expressions (39) and (46) are calculated, they can be replaced in (38):

1

1 1 1 1

1 2 2

1 1

1( ) ( )

m t t

ji kjj

q


i ji ji i ji jil p l pm q q t

kjM Mj l l

i il l

b c

x

www b b w b b

x xw w c

w w

1 1 1 1

2 2

1 1

1( ) ( )


j kj kj j kj kjr s r sq q t

jiN Nr r

j jr r

w ww c c w c c

x xw w b

w w

(47)

Thus, the derivative of the second addend given by Eq. (37) is

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0 1

1 1 1 1

0 1 2 2

1 1

1( ) ( )

n m t t

ji kj kk j

q


i ji ji i ji jil p l pn m q q t

kjM Mk j l l

i il l

b c x

x

www b b w b b

x xw w c

w w

1 1 1 1

12 2

1 1

1( ) ( )


j kj kj j kj kjr s r sq mq t t t

ji k ji qjN N jr r

j jr r

w ww c c w c c

x xw w b x b c

w w

(48)

4.3 DERIVATIVE OF THE MATCHING DEGREE OF THE RULES OF THE CONTROLLER

Given expression (19a) that defines the lower matching degree of the rules of the controller, the partial derivative of

this expression regarding to every state variable is obtained from

11

1 1 11

( )( )( ) ( ) ( )

rrn

r r rk k k

rn n nnCCj

k k kC C Ck k kq q q qk k n

xxwx x x

x x x x

(49)

Bearing in mind that

( )0i

j

f x

x

i j (50)

Eq. (49) can be written as

1

( )( )

rq

rk

rnqCj

kCkq q k q

xwx

x x

(51a)

Similar of the expression for the calculation, it is easy to deduce that

1

( )( )

rq

rk

rnqCj

kCk

q q k q

xwx

x x

(51b)

4.4 DERIVATIVE OF THE MATCHING DEGREE OF THE RULES OF THE PLANT

Given expression (9a) that defines the lower matching degree of the rules of the plant, the partial derivative of this

expression with regard to every state variable is obtained from

1 1

( )l lk j

l lln m

l li ii

i ik jA Bk jq q q q

w x w uwx u w u w x

x x x x

(52)

( 1

( )lk

nl

i kAk

w x x

, 1

( )lj

ml

i jBj

w u u

)

where

1

1

1 1 11

( )( )( ) ( ) ( )

llm

l l lj j j

lm m mmBBi

j j jB B Bj j jq q q qj j m

uuw uu u u

x x x x

(53)

then 1 1

( )( )( )

lv

lj

lmvm Bi

jv Bjq q j v

uw uu

x x

(54)

Now, bearing in mind (50):

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1 1

( )( ) ( )

lq

l lk k

ln nqAi

k kA Ak kq q q k q

xw xx x

x x x

(55)

Replacing Eqs. (54) and (55) in (52):

11 1 1 1

( ) ( )( ) ( ) ( ) ( )

l lq v

l l l lj k k j

lm n n mq vA m Bi

j k k jvB A A Bj k k jq q qk q j v

x uwu x x u

x x x

(56a)

Similarity of the expression for the calculation, it is easy to deduce that

11 1 1 1

( ) ( )( ) ( ) ( ) ( )

l lq v

l l l lj k k j

l m n n mq vA m Bi

j k k jB A A Bvj k k jq q qk q j v

x uwu x x u

x x x

(56b)

Finally, the calculation of

l

i

q

w u

x

implies the derivative of the control vector. Writing again Eq. (17) with the

extended state vector showed in (21):

0

n t

j kj kku c x

(57)

Then 0

0

n tt

kj k nkj kjt

qj kkq q q

c xu cc x

x x x

(58)

Replacing now

t

kj

q

c

x

from (46):

0

1 1 1 1

0 2 2

1 1

1( ) ( )

n t

kj kkj

q q


j kj kj j kj kjr s r sqn qt

qj kN Nk r r

j jr r

c xu

x x

w ww c c w c c

x xc w w x

w w

(59)

4.5 JACOBIAN MATRIX

Every element of the Jacobian matrix, Eq. (26), is given by the derivative of the expression (27). Two addends of

this expression have been calculated in Sections 4.1 and 4.2, and they are given by Eqs. (36) and (48), therefore

1

1 1 1 1

0 2 2

1 1

( )

1( ) ( )

mi t t t

iq qi ji qjjq


i ki ki i ki kil p l pn q q

kM Mk l l

i il l

xJ x a b c

x

www a a w a a

x xw w x

w w

1 1 1 1

1 2 2

1 1

1( ) ( )


i ji ji i ji jil p l pm q q t

kjM Mj l l

i il l

www b b w b b

x xw w c

w w

1 1 1 1

2 2

1 1

1( ) ( )


j kj kj j kj kjr s r sq q t

ji kN Nr r

j jr r

w ww c c w c c

x xw w b x

w w

(60)

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Notice that Eqs. (51a), (51b)and (56a), (56b)and therefore Eq. (60), are independent on the type of membership

function that is chosen to fulfill the fuzzy plant and controller models, so, it is possible to evaluate the Jacobian

matrix for any membership function. Moreover, it is possible to combine different membership functions in a same

model.

5 ALGORITHMS

Let IT2T-S FLCS defined by (1) and (15), whose fuzzy equivalent, considering IT2T-S consequent which is given

by (12), and the controller, considering equally IT2T-S consequent which is given by (17); the IT2T-S fuzzy closed

loop model is given by (22). An equilibrium state is given for the programming of this model.

Given a concrete state 1, ,T

nX x x , the steps to compute the Jacobian matrix of the closed loop system are

shown as follows:

(1). Calculate the matching degree of the rules of the fuzzy controller by (19a), (19b).

(2). Compute the variable coefficients of the fuzzy controller by (18b), 0,1, ,k n .

(3). Find the value of each of the signals generated by the fuzzy controller by (17).

(4). Calculate the matching degree of the rules of the fuzzy plant model by (9a), (9b).

(5). For the i th equation of the process and the q th state variable:

5.1. Calculate the variable coefficient of the plant model by (18b) and by (14) with 0,1, ,k n .

5.2. Compute the second addend of Eq. (60).

5.3. Calculate the partial derivative of the matching degree of the rules of the fuzzy controller model by (51a),

(51b).

5.4. Calculate the partial derivative of the matching degree of the rules of the fuzzy process model by (56a), (56b).

Eq. (59) is used to complete this derivative.

5.5. Compute third addend of (60).

5.6. Calculate the corresponding term iqJ of the Jacobian matrix.

(6). Repeat step 5 until all terms of the Jacobian matrix evaluated in the state X are computed. (see Fig.1)

As we reach this point, a general method is available for the calculation of the Jacobian matrix at a point of a fuzzy

control system identified on an input/output data basis.

6 SIMULATION AND EXAMPLE

Let the following plant given by its internal representation be:

1 1 2 1

2

2 1 2 2

2

2

x x x u

x x x u

(61)

This plant has an equilibrium point at the origin of the state space. Linearizing the nonlinear model given by (61)

around this point, the following system is obtained:

1 1 2 1

2 1 2

2x x x u

x x u

(62)

whose dynamic matrix is 1 2

1 0A

Eigenvalues of the matrix A are: 1 1 and 2 2 indicate that the system (61) is unstable, and the origins of the

state space act as a saddle point for this system.

For the implementation of this example, the plant dynamics is supposed to be unknown, so that Eq.(61) is only used

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to obtain input/output data of the plant. T1T-S model of the plant has been obtained by the methodology based on

descending gradient developed in [17-18]. (see Appendix) The input/output data of the plant has been obtained by a

random number generator in the interval 1 2, 10,10x x [17].

FIG.1. ALGORITHM TO COMPUTE EQ. (60)

Now, once the plant dynamics is known, it is possible to design T1T-S model of controller that stabilizes the system

around the equilibrium point in the origin of the state space. (see Appendix)

We illustrate the IT2T-S FLS by extending the T1T-S FLS. Assume that uncertainties exist in the antecedents of

each rule, and extend the T1T-S FLS to a IT2T-S FLS by spreading antecedent MF stand deviation values ±a% (see

Appendix).

The Jacobian matrix of the IT2T-S fuzzy system evaluated in the equilibrium state by the “FuzJac’’ function is

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-7.4678 11.1494

-74.4091 -47.7924J

To determine the control system stability by the application of Krasovskii’s theorem, it is obvious that

( ) ( )TJ X J X is negative definite in an environment of the equilibrium state.

Fig.2 and Fig.3 show the comparison of the states for the T1 and IT2 T-S fuzzy system. The fuzzy closed loop

control systems are asymptotically stable in the origin of the state space. Furthermore, they show that the IT2 T-S

fuzzy system is better than the T1 T-S fuzzy system, respectively.

FIG.2. THE COMPARISON OF 1x t FOR THE T1 AND IT2 T-S FUZZY SYSTEM

FIG.3. THE COMPARISON OF 2x t FOR THE T1 AND IT2 FUZZY T-S SYSTEM

In order to show that the IT2 T-S fuzzy system can handle the measurement uncertainties, training data is corrupted

by a random noise. From Fig.4 to Fig.7, figures show the responses with disturbance a = 0.06 for the T1 and IT2

fuzzy T-S system, respectively. When the IT2 T-S fuzzy system is influenced by random disturbance, the effect is

very small, and the output trajectories of 1x t and 2x t have smaller amplitude than the trajectories without

random disturbance. The simulation results show that the IT2 T-S fuzzy system can handle unpredicted disturbance

and data uncertainties very well.

FIG.4. THE 1x t WITH DISTURBANCE FOR THE T1 FUZZY T-S SYSTEM

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FIG.5. THE 1x t WITH DISTURBANCE FOR THE IT2 FUZZY T-S SYSTEM

FIG.6. THE 2x t WITH DISTURBANCE FOR THE T1 FUZZY T-S SYSTEM

FIG.7. THE 2x t WITH DISTURBANCE FOR THE IT2 FUZZY T-S SYSTEM

7 CONCLUSIONS

In this paper, we present a design of IT2T-S FLCS in details. Furthermore, Krasovskii’s method is introduced to

testify the sufficient condition for the asymptotic stability of IT2T-S FLCS. Later, the Jacobian matrix of a closed

loop fuzzy system is computed and an algorithm to solve the Jacobian matrix is proposed. Finally, the simulation

results show that the IT2T-S FLCS achieves the best tracking performance in comparison with the T1T-S FLCS.

Moreover, the IT2T-S FLCS can handle unpredicted internal disturbance and data uncertainties well.

ACKNOWLEDGMENT

The work was supported by the National Natural Science Foundation of China (11072090). The work was supported

by the Opening Project of Guangxi Key Laboratory of Automobile Components and Vehicle Technology, Guangxi

University of Science and Technology (2012KFMS12).

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[2] M. Sugeno, G. Kang. Structure Identification of Fuzzy Model. Fuzzy Sets and Systems, 1988; 28(1): 15-33

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trailer. IEEE Transaction on Fuzzy Systems, 1994, 2 (2): 119-134

[4] K. Tanaka, T. Ikeda, H.O. Wang. Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: quadratic

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[5] H.O. Wang, K. Tanaka, M.F. Griffin. An approach to fuzzy control of nonlinear systems: stability and design issues, IEEE

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[6] J.M. Mendel. On answering the question “Where do I start in order to solve a new problem involving interval type-2 fuzzy sets?”

Information Sciences, 2009; 179(19): 3418-3431

[7] Ming-Ying Hsiao, Tzuu-Hseng S. Li, J.-Z. Lee, etc, Design of interval type-2 fuzzy sliding-mode controller, Information Sciences,

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[8] Ondrej Linda, Milos Manic, Interval Type-2 fuzzy voter design for fault tolerant systems,Information Sciences, 2011, 181(14):

2933-2950

[9] Tsung-Chih Lin, Observer-based robust adaptive interval type-2 fuzzy tracking control of multivariable nonlinear systems,

Engineering Applications of Artificial Intelligence, 2010, 23(3): 386-399

[10] Yimin Li, Yijun Du, Indirect adaptive fuzzy observer and controller design based on interval type-2 T-S fuzzy model,Applied

Mathematical Modelling, 2012, 36(4): 1558-1569

[11] Kheireddine, C., Lamir, S., Mouna, G., Khier, B., 2007. Indirect adaptive interval type-2 fuzzy control for nonlinear systems.

International Journal of Modeling, Identification and Control 2, 106-119

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36(6): 2710-2723

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multimachine power systems ,International Journal of Electrical Power & Energy Systems, 2013, 45(1): 456-467

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APPENDIX

The T1T-S fuzzy model of the plant:

IF 1x is GAUSSMF(−13.9,6.5) and 2x is GAUSSMF(−13.5,5.1) and 1u is GAUSSMF(−13.6,5.4) and 2u is GAUSSMF(−13.1,5.3)

THEN 1 1 2 1 216.7 1.93 1.26 0.114 0.235x x x u u

IF 1x is GAUSSMF(−6.02,5.1) and 2x is GAUSSMF(−5.03,4.1) and 1u is GAUSSMF(−6.85,5.03) and 2u is GAUSSMF(−5.24,5)

THEN 1 1 2 1 213.5 1.13 1.01 1.57 1.13x x x u u

IF 1x is GAUSSMF(0.68,4.1) and 2x is GAUSSMF(2.67,4.5) and 1u is GAUSSMF(1.62,4.7) and 2u is GAUSSMF(1.59,4.9) THEN

1 1 2 1 211.5 0.617 1.36 0.423 0.498x x x u u

IF 1x is GAUSSMF(8.54,5.5) and 2x is GAUSSMF(9.69,5.4) and 1u is GAUSSMF(9.24,5.2) and 2u is GAUSSMF(8.27,5.3) THEN

1 1 2 1 218.5 2.2 2.05 0.917 0.894x x x u u

http://www.sciencedirect.com/science/article/pii/S002002550700494X



http://www.sciencedirect.com/science/article/pii/S0307904X11005725

http://www.sciencedirect.com/science/article/pii/S0307904X1100610X




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IF 1x is GAUSSMF(−18.5,4.9) and

2x is GAUSSMF(−5.4,3.3) and 1u is GAUSSMF(−16.7,5.4) and

2u is GAUSSMF(−19.8,4.8)

THEN 2 1 2 1 25.91 003 130 68 222 244x e x x u u


2x is GAUSSMF(−9.74,4.1) and 1u is GAUSSMF(−6.48,5.2) and

2u is GAUSSMF(−6.78,5.6)

THEN 2 1 2 1 23.04 003 10.8 52.7 15.3 10.3x e x x u u

IF 1x is GAUSSMF(1.28,6.2) and

2x is GAUSSMF(3.07,5.2) and 1u is GAUSSMF(1.37,5.6) and

2u is GAUSSMF(1.37,5.2) THEN

2 1 2 1 26.36 003 110 370 39.2 121x e x x u u


2x is GAUSSMF(18.4,5.1) and 1u is GAUSSMF(11.9,5.8) and

2u is GAUSSMF(13.6,5.6) THEN

2 1 2 1 25.76 003 88 515 19.2 62.4x e x x u u

A possible T1 controller fuzzy rules could be:


2x is GAUSSMF(2.72,6.07)

THEN 1 10.0002 5.69u x


2x is GAUSSMF(−13.01,6.2)

THEN 1 1 20.00047 3.01 0.00385u x x

IF 1x is GAUSSMF(4.04, 5.5) and

2x is GAUSSMF(13.12, 5.3)

THEN 1 1 20.00044 2.62 0.00155u x x

IF 1x is GAUSSMF(−6.01, 6.1) and

2x is GAUSSMF(−9.72,6.2)

THEN 2 1 20.0004 1.83 8.42u x x



THEN 2 1 21.32 9.16u x x



THEN 2 1 20.000504 2.06 2.81u x x

The IT2T-S fuzzy model of the plant is the following:

IF 1x is GAUSSMF(−13.9,6.5(1±a%)) and 2x is GAUSSMF(−13.5,5.1(1±a%)) and 1u is GAUSSMF(−13.6,5.4(1±a%)) and 2u is

GAUSSMF(−13.1,5.3(1±a%))

THEN 1 1 2 1 216.7 1.93 1.26 0.114 0.235x x x u u


GAUSSMF(−5.24,5(1±a%))

THEN 1 1 2 1 213.5 1.13 1.01 1.57 1.13x x x u u

IF 1x is GAUSSMF(0.68,4.1(1±a%)) and 2x is GAUSSMF(2.67,4.5(1±a%)) and 1u is GAUSSMF(1.62, 4.7(1±a%)) and 2u is

GAUSSMF(1.59,4.9(1±a%))

THEN 1 1 2 1 211.5 0.617 1.36 0.423 0.498x x x u u

IF 1x is GAUSSMF(8.54,5.5(1±a%)) and 2x is GAUSSMF(9.69,5.4(1±a%)) and 1u is GAUSSMF(9.24, 5.2(1±a%)) and 2u is

GAUSSMF(8.27,5.3(1±a%))

THEN 1 1 2 1 218.5 2.2 2.05 0.917 0.894x x x u u


GAUSSMF(−19.8,4.8(1±a%))

THEN 2 1 2 1 25.91 003 130 68 222 244x e x x u u


GAUSSMF(−6.78,5.6(1±a%))

THEN 2 1 2 1 23.04 003 10.8 52.7 15.3 10.3x e x x u u

IF 1x is GAUSSMF(1.28,6.2(1±a%)) and 2x is GAUSSMF(3.07,5.2(1±a%)) and 1u is GAUSSMF(1.37,5.6(1±a%)) and 2u is

GAUSSMF(1.37,5.2(1±a%))

THEN 2 1 2 1 26.36 003 110 370 39.2 121x e x x u u

IF 1x is GAUSSMF(13.6,5.4(1±a%)) and 2x is GAUSSMF(18.4,5.1(1±a%)) and 1u is GAUSSMF(11.9,5.8(1±a%)) and 2u is

GAUSSMF(13.6,5.6(1±a%))

THEN 2 1 2 1 25.76 003 88 515 19.2 62.4x e x x u u

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A possible IT2T-S controller could be:

IF 1x is GAUSSMF(−8.34,4.4(1±a%)) and

2x is GAUSSMF(2.72,6.07(1±a%)) THEN 1 10.0002 5.69u x

IF 1x is GAUSSMF(14.62,5.6(1±a%)) and

2x is GAUSSMF(−13.01,6.2(1±a%)) THEN 1 1 20.00047 3.01 0.00385u x x


2x is GAUSSMF(13.12,5.3(1±a%)) THEN 1 1 20.00044 2.62 0.00155u x x

IF 1x is GAUSSMF(−6.01,6.1(1±a%)) and

2x is GAUSSMF(−9.72,6.2(1±a%)) THEN 2 1 20.0004 1.83 8.42u x x


2x is GAUSSMF(1.6,6.54(1±a%)) THEN 2 1 21.32 9.16u x x


2x is GAUSSMF(14.8,7.8(1±a%)) THEN 2 1 20.000504 2.06 2.81u x x

AUTHORS 1Li Li, born in Pucheng, Shanxi province, China, in Sep. 1964.

She graduated from Jiangsu University with a doctorate’s degree

in Agricultural Electrification and Automation in 2009.

She is Associate Professor working in Faculty of Computer

Science and Telecommunication Engineering, Jiangsu

University, Zhenjiang, China. She is currently engaged in the

following research fields: (1) Fuzzy Fault Detection and

Diagnosis;(2) Simulation and Intelligent Control;(3) Intelligent

Algorithm and Optimization.

2Yijun Du is a Graduate Student in Faculty of Science, Jiangsu

University, Zhenjiang, China.

3Yimin Li, born in Luoyang, Henan province, China, in 1963.

He obtained his doctorate in Control Theory and Control

Engineering from Nanjing University of Aeronautics and

Astronautics in 2005.

He is a Professor and Postgraduate Supervisor in Faculty of

Science, Jiangsu University, Zhenjiang, China. He is currently

engaged in the following research fields: (1) Modeling of

complex ecosystem and study of features of bio-system; (2) The

fuzzy control theory based on biological features; (3) The theory

and application of Bionics intelligent control; (4) Type-2 fuzzy

control methods.

Design of interval type 2 t s fuzzy logic control systems

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