Design of interval type 2 t s fuzzy logic control systems

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 Li1#, 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: llym@ujs.edu.cn

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 antecedents 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 type2 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 - 122 http://www.sj-ce.org

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 l R l : If x1 is A1li and x2 is A2l i … and xn is Anil and u1 is B1li and u2 is B2l i … and um is Bmi

Then xil  gil ( X ,U )  a0l i  a1li x1 

 anil xn  b1li u1 

l  bmi um

(1)

Where l  1, , M is the number of rules, Akil are the IT2 fuzzy sets defined in a universe of discourse for the state variable xk , k  1, , n and xil is the first-order differential equation i , B lji is the IT2 fuzzy set defined in a universe of discourse for the j th control signal u j , j  1, , m , akil and blji represent the respective constant coefficients for the state variable xk 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 X  ( x1 , x2 ,, xn )T  Rn denotes the state vector, and U  (u1 , u2 , , um )  Rm denotes the input control vector. This rule represents an IT2 relation between the input space X1   X n  U1  U m , and the output space X , of the IT2 FLS. The firing strength of the l th rule is   xn X n  X n ( xn )   A1  xn  n  

wil   x1X1  X1 ( x1 )   Al  x1     1 

(2)  u1U1 U1 (u1 )   Bl  u1     um U m U m (um )   Bl  um   1  m   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   Al  xn   Bl  u1  

wil   Al  x1   1

n

1

 Bl  um     Al  xk   Bl  u j  n

m

k 1

m

k

j 1

(3)

j

where xk (k  1, 2,n) and u j ( j  1, 2,m) denote the location of the singleton.

R l is described by the membership function Rl ( xil )

R ( xil )   X  xil      A  xk   B  u j  j 1  k 1  n

l

m

l i

l k

(4)

l j

where now each X il is a crisp value xil , so (4) reduces to

R ( xil )    A  xk   B  u j  n

l

m

l k

k 1

j 1

(5)

l j

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

R ( xi )   R ( xil )      A  xk   B  u j   l 1 l 1 k 1 j 1 M

M

l

n



m

l k

l j



(6)

where  denotes the product operation.

Let COG (  R ( xi ) ) and COG ( R ( xi ) ) denote the centroid of the lower membership function (LMF) and upper - 123 http://www.sj-ce.org

membership function (UMF), respectively, of R ( xi ) . Then, we can define the output of an IT2 FLS xi , as xi =w COG (  R ( xi ) )+(1-w) COG( R ( xi ) )

(7)

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

So the output of an IT2T-S FLCS xi is shown as xi

 w

M

n

  (1  w)

l

l 1

wi gil ( X ,U )



M

l

w l 1 i

M l 1

wil gil ( X ,U )



(8)

M

wl l 1 i

m

wi    Al ( xk )   Bl (u j ) l

k 1

j 1

k

n

(9a)

j

m

wil    Al ( xk )  Bl (u j )) k 1

j 1

k

n

(9b)

j

m

l wil   wi , wil 

wil    Al ( xk )  Bl (u j ) k 1

j 1

k

j

(10)

Where w can be tuned during a design procedure.  Al ( xk ) and  Bl j (u j ) are the membership functions defined in k their corresponding universe of discourse X and U .

Take the consequent of the rule (1) into (8), which can be written as  M wli a0l i   M wli a1li x1    M wli anil xn   M wli b1li u1  l 1 l 1 l 1 l 1  xi  M l  M M l l l l   w  l 1 i  l 1 wi b2i u2    l 1 wi bmium  w

1  w  l 1 wi a0i   l 1 wi a1i x1  M

+



M

l 1

l

M

l

M wil   wil b2l i u2  l 1 

  l 1 wil anil xn   l 1 wil b1li u1   M l    l 1 wil bmi um  l

M

l

M

(11)

This is xi  a0t i  a1ti x1  a2t i x2 

 anit xn  b1ti u1  b2t i u2 

t  bmi um

 a0t i   k 1 akit xk   j 1 btji u j n

m

(12)

Where a0t i , akit and btji are variable coefficients computed as t 0i

a

akit

b

t ji

 w a  1  w  w a w  w  w  w a  1  w  w a w  w  w  w b  1  w  w b w  w  w M l l l 1 i 0i M l l 1 i

M l l l 1 i 0i M l l 1 i

(13a)

M l l l 1 i ki M l l 1 i

M l l l 1 i ki M l l 1 i

k  1,

,n

(13b)

M

M l l l 1 i ji M l l 1 i

j  1,

,m

(14)

l 1 M

l 1

l i

l ji

l i

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: R r : If x1 is C1rj and x2 is C2r j … and xn is Cnjr Then u rj  c0r j  c1r j x1 

Where Ckjr represents the IT2 fuzzy sets of k  1, r  1, , N rules.

 cnjr xn

, n state variables, with j  1,

- 124 http://www.sj-ce.org

(15)

, m control actions and

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

w

 N wrj c0r j   N wrj c1r j x1  r 1 r 1  r 1 w 

  r 1 w j cnjr xn  + 

 N wrj c0r j   N wrj c1r j x1  r 1 r 1  r 1 w 

  r 1 wrj cnjr xn  

N

N

r j

1  w N

r

N

r j

(16)

Or, in a more simplified way:  cnjt xn = c0t j   k 1 ckjt xk n

u j  c0t j  c1t j x1  c2t j x2 

(17)

where the variable coefficients are given by

 w c  1  w  w c  w  w  w c  1  w  w c w  w  w N

c0t j  w

r 1

N

ckjt

N

r r j 0j

r 1 N

r j

r 1

r 1 N

r j

r 1

(18a)

r j

r 1

N

r r j kj

r 1 N

r r j 0j

r 1 N

r r j kj r j

k  1,

(18b)

,n

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

w j    C r ( xk ) r

k 1

(19a)

k

n

wrj   C r ( xk ) k 1

(19b)

k

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





xi  a0t i   j 1 btji c0t j   k 1 akit   j 1 btji ckjt xk , i, k  1, m

n

m

, n and j  1,

,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 1  T x      x0 , x1 , xn  X   Replacing the extended state vector into (20) the simplified following expression is obtained

xi   k 0 (akit   j 1 btji ckjt )xk n

m

k  0,1,

, n , i  1,

, n and j  1,

,m

(21)

(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 J ( X )  J ( X )T is negative definite in a region around the equilibrium state X e , this is asymptotically stable in its proximities.

Moreover, if  ( X )T  ( X )   when X   , the asymptotical stability is global. The Lyapunov function for this system is  ( X )T  ( 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: x1  1 ( x1 , x2 ,

, xn )

x2  2 ( x1 , x2 , , xn ) .... xn  n ( x1 , x2 , , xn )

(23)

Let X  ( 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 X e may be used: e

e 1

e 2

e T n

 ( x , x , d xi  xie    x1  x1e  i 1 2  dt x1

+…+  xn  xne 

, xn ) Xe

i ( x1 , x2 , xn

, xn )

+ Ri ( X e )

(24)

Xe

When the nonlinear term Ri ( X e ) fulfills (25), being K i a sufficiently small constant, then the Jacobian matrix of the system determines the properties of stability of the equilibrium state X e : 2 2 Ri ( X e )  Ki  x1  x1e     xn  xne   (25)   The Jacobian matrix is calculated applying the following expressions:  1 ( x1 , x2 , , xn )  x1  Xe      n ( x1 , x2 , , xn )   x1 Xe 

  Xe   (26) J iq ( X ) e X   n ( x1 , x2 , , xn )    xn Xe  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: 1 ( x1 , x2 , xn

, xn )

xi   k 0 (akit   j 1 btji ckjt )xk =  k 0 akit xk   k 0  j 1 btji ckjt xk n

m

n

n

The partial derivatives will be obtained in a generic form for the q  1, degree differentials equations.

m

(27)

, n state variables and the i  1,

, n first

4.1 DERIVATIVE OF THE FIRST ADDEND The first partial derivative to calculate is from the first adding of Eq. (27): 



n k 0

akit xk

xq





  akit xk 

n

xq

k 0

t  n  a   k 0  ki xk   aqit  x   q 

(28)

Bearing in mind (13b): M   M wli akil  wil akil   w l M1 l  1  w l M1 l   l 1 wi  l 1 wi akit   xq xq

   M wli akil    l M1 l     w   l 1 wi xq

   M wil akil    l M1 l     1  w   l 1 wi   xq

   

(29)

Calculate the first adding of Eq. (29):   M wli akil   l M1 l   wi l 1 w  xq

    w



l

wi akil



xq



while

M l 1



M l 1

l

wi akil

xq



M l 1



w  l i



M l 1



l i

w

 

xq 2

w  l 1 x i akil q

- 126 http://www.sj-ce.org

M l 1

l

wi





M l 1

l

wi akil

(30)

l

M

(31)



and



M

l

l 1

xq

wi



w  l 1 x i q l

M

(32)

Replacing now the previous two expressions in (30) and simplifying, it results in   M wli akil  l  M M   wi p   l M1 l  wi  akil  akip      l 1 p 1    wi   l 1 w  xq  w  M l 2 xq ( l 1 wi )

(33)

calculate the second adding of Eq. (29) is similar to calculate the first adding of Eq. (29):   M wil akil  l  M M  wi   l M1 l    l 1  p 1  x wip  akil  akip     wi   q  1  w  l 1   1  w M l 2 xq ( l 1 wi )

(34)

Replacing (33), (34) expressions in (29) , it results in l l   M M   wi M M  wi p l p w a  a wip  akil  akip            i ki ki l  1 p  1 l  1 p  1  x   x  akit  q   1 w  q  w   M M l 2 l 2 xq ( l 1 wi ) ( l 1 wi )

(35)

Finally, Eq. (28) can be written as 



n k 0

akit xk



xq l l     M M  w M M  wi p p l p (36)   l 1  p 1  i wi  akil  akip   w a  a     i  ki ki   l 1  p 1      x  x  n   q   1 w  q   x  at   k  0  w    k  qi 2 2 M M l l  w w     i i l 1 l 1       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: n m m    k 0  j 1 btji ckjt xk  j 1 btji ckjt  m t t n    k 0 xk   j 1 b ji cqj   xq xq     Where









m

btji ckjt







(37)

 btji t ckjt t   j 1  x ckj  x bji  xq q  q  t t Given the similarity of the expressions for the calculation of aki and b ji , it is easy to deduce that l l   M M   wi M M  wi p l p w b  b wip  blji  b jip          i ji ji  t l 1  p 1  l 1  p 1   b ji  xq   1 w  xq  w   M M l 2 l 2 xq ( l 1 wi ) ( l 1 wi ) j 1

m

(38)

(39)

The derivative of the term corresponding to the fuzzy controller will be: N   N wrj ckjr  wrj ckjr   w r N1 r  1  w r N1 r  ckjt  r 1 w j  r 1 w j   xq xq

   N wrj ckjr    r N1 r     w   r 1 w j xq

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   N wrj ckjr    r N1 r     1  w   r 1 w j   xq

   

(40)

calculate the first adding of Eq. (40):   N wrj ckjr   r N1 r   wj r 1 w  xq

    w 

while





N

w j ckjr

xq



and



r

w j ckjr



xq

r

r 1

N r 1

N

r

r 1

xq

wj







N

w  r 1 r j

 



w j

N





r

wj r 1



N r 1

r

wj



xq



N r 1

r

w j ckjr

2

(41)

r

N r 1

xq

ckjr

w j

(42)

r

N r 1

xq

(43)

Replacing now the previous two expressions in (41) and simplifying, it results in   N wrj ckjr   r N1 r   wj r 1 w  xq

r   N N  w j s  w j  ckjr  ckjs      r  1 s  1   xq  w   N r 2 ( r 1 w j )

(44)

calculate the second adding of Eq. (40) is similar to calculate the first adding of Eq. (40):   N wrj ckjr   r N1 r   wj r 1 (1  w)  xq

 r  N N  w j  wsj  ckjr  ckjs      r  1 s  1   x    (1  w)  q  N r 2 ( r 1 w j )

(45)

Finally, Eq. (40) can be written as r   wrj s r   N N  w j s w c  c   wsj  ckjr  ckjs       j kj kj   r 1  s 1  t r  1 s  1    ckj  xq   1 w  xq  w   N N r 2 r 2 xq ( r 1 w j ) ( r 1 w j ) N

N

(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): 



m j 1

btji ckjt



xq l l    M M  w M M  wi p p l p   l 1  p 1  i wi  blji  b jip   w b  b        i ji ji l 1 p 1   x  xq m  q      ct   j 1  w  1  w   kj M M l 2 l 2 ( l 1 wi ) ( l 1 wi )       r r     N N  w j N N  w j s s r s   r 1  s 1  w j  ckjr  ckjs   w c  c     r 1  s 1  x j kj kj     xq      q  bt  w  1  w   ji  N N r 2 r 2 ( r 1 w j ) ( r 1 w j )          Thus, the derivative of the second addend given by Eq. (37) is

- 128 http://www.sj-ce.org

(47)





n k 0



m j 1

btji ckjt xk



xq

  k 0 n

l l     M M  w M M  wi p    l 1  p 1  i wi  blji  b jip   wip  blji  b jip      l 1  p 1      m   xq   1 w  xq  ct w    j 1   kj M M l 2 l 2 ( l 1 wi ) ( l 1 wi )         

r r      N N  w j N N  w j s s r s   r 1  s 1  w j  ckjr  ckjs   w c  c     r 1  s 1  x j kj kj      xq      q  bt x   m bt c t  w  1  w   ji  k   j 1 ji qj N N r 2 r 2 ( r 1 w j ) ( r 1 w j )            

(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 w j r



xq

  C r ( x1 ) n   n 1  C r ( xk )     r ( xk )   k xq  1 k xq kk 11 Ck 



  C r ( xn ) n

xq

n

  C r ( xk )

k 1 k n

(49)

k

Bearing in mind that f ( xi ) 0 x j

i  j

(50)

Eq. (49) can be written as w j r

xq



  C r ( xq ) q

xq

n

  C r ( xk )

k 1 k q

(51a)

k

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



C r ( xq ) q

xq

n

 C r ( xk )

k 1 k q

(51b)

k

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 m w  x w u   wi   n l   Al ( xk )   Bl  u j    wli  u  i  wi  x  i  k k j j 1 xq xq  1 xq xq  l

l

n

l

(52)

m

( wi  x     Al ( xk ) , wi  u     Bl (u j ) ) l

k 1

l

k

j 1

j

where  wi  u  l

xq

then



  Bl (u1 ) m  m 1  Bl (u j )     l (u j )   xq  j 1 j xq jj 11 B j 



l    Bl (uv ) m   wi (u ) m v   v 1    Bl (u j )  j j 1  xq  xq j v  

Now, bearing in mind (50): - 129 http://www.sj-ce.org

  Bl (um ) m

xq

m

  Bl (u j )

j 1 j m

(53)

j

(54)

 wi  x  l

xq



  Al ( xq ) n   n q  Al ( xk )     l ( xk )  k k  1 xq  xq kk 1q Ak 

(55)

Replacing Eqs. (54) and (55) in (52): l    Bl (uv ) m  n n  wi   Aql ( xq ) m m v (56a)    Bl (u j )   Al ( xk )    Al ( xk ) v 1    Bl (u j )  j k k j j 1 k 1 k 1 j 1  xq  xq xq k q j  v   Similarity of the expression for the calculation, it is easy to deduce that   Bl (uv ) m  n n wil  Aql ( xq ) m m v (56b)   Bl (u j )   Al ( xk )    Al ( xk ) v 1    Bl (u j )  j k k j 1 k 1 k 1  xq jj 1v j  xq xq k q   l  wi  u  Finally, the calculation of implies the derivative of the control vector. Writing again Eq. (17) with the xq extended state vector showed in (21): u j   k 0 ckjt xk n

Then Replacing now

ckjt xq

u j xq



u j xq







n

t k  0 kj k

c x

xq

(57)

 c

t qj

t  n  ckj   k 0  xk   x   q 

(58)

from (46): 



n

t k  0 kj k

c x



xq

r r     N N  w j N N  w j s s r s   r 1  s 1  w j  ckjr  ckjs   w c  c     r 1  s 1  x j kj kj     xq  n     q  x t  cqj   k  0  w  1  w   k N N r 2 r 2 ( r 1 w j ) ( r 1 w j )         

(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   x  m J iq ( x)  i  aqit   j 1 btji cqjt xq l l    M M  w M M  wi p   l 1  p 1  i wi  akil  akip   wip  akil  akip       l  1 p  1  x   x  n   q   1 w  q  x   k  0  w    k M M l 2 l 2 ( l 1 wi ) ( l 1 wi )       l l    M M  w M M  wi p p l p   l 1  p 1  i wi  blji  b jip   w b  b        i ji ji l 1 p 1   x  xq m  q      ct   j 1  w  1  w   kj M M l 2 l 2 ( l 1 wi ) ( l 1 wi )       r r      N N  w j N N  w j s s r s   r 1  s 1     w j  ckjr  ckjs   w c  c      j kj kj r 1 s 1   xq       x  q    1 w   bt x   w    ji  k  N N r 2 r 2 ( r 1 w j ) ( r 1 w j )            

- 130 http://www.sj-ce.org

(60)

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 X   x1 , shown as follows:

, xn  , the steps to compute the Jacobian matrix of the closed loop system are T

(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), k  0,1,

,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 k  0,1,

,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 J iq 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: x1  x1  2 x2  u1 x2  x1  2 x22  u2

(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: x1  x1  2 x2  u1 x2  x1  u2

(62)

1 2  A  1 0  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.

whose dynamic matrix is

For the implementation of this example, the plant dynamics is supposed to be unknown, so that Eq.(61) is only used - 131 http://www.sj-ce.org

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  x1 , x2    10,10 [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 - 132 http://www.sj-ce.org

 -7.4678 11.1494  J   -74.4091 -47.7924 To determine the control system stability by the application of Krasovskii’s theorem, it is obvious that J ( X )  J ( X )T 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 x1  t  FOR THE T1 AND IT2 T-S FUZZY SYSTEM

FIG.3. THE COMPARISON OF x2  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 x1  t  and x2  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 x1  t  WITH DISTURBANCE FOR THE T1 FUZZY T-S SYSTEM - 133 http://www.sj-ce.org

FIG.5. THE x1  t  WITH DISTURBANCE FOR THE IT2 FUZZY T-S SYSTEM

FIG.6. THE x2  t  WITH DISTURBANCE FOR THE T1 FUZZY T-S SYSTEM

FIG.7. THE x2  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).

- 134 http://www.sj-ce.org

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APPENDIX The T1T-S fuzzy model of the plant: IF x1 is GAUSSMF(−13.9,6.5) and x2 is GAUSSMF(−13.5,5.1) and u1 is GAUSSMF(−13.6,5.4) and u2 is GAUSSMF(−13.1,5.3) THEN x1  16.7  1.93x1  1.26 x2  0.114u1  0.235u 2 IF x1 is GAUSSMF(−6.02,5.1) and x2 is GAUSSMF(−5.03,4.1) and u1 is GAUSSMF(−6.85,5.03) and u2 is GAUSSMF(−5.24,5) THEN x1  13.5  1.13x1  1.01x2  1.57u1 1.13u 2 IF x1 is GAUSSMF(0.68,4.1) and x2 is GAUSSMF(2.67,4.5) and u1 is GAUSSMF(1.62,4.7) and u2 is GAUSSMF(1.59,4.9) THEN

x1  11.5  0.617 x1  1.36 x2  0.423u1  0.498u 2 IF x1 is GAUSSMF(8.54,5.5) and x2 is GAUSSMF(9.69,5.4) and u1 is GAUSSMF(9.24,5.2) and u2 is GAUSSMF(8.27,5.3) THEN x1  18.5  2.2x1  2.05x2  0.917u1  0.894u 2 - 135 http://www.sj-ce.org

IF x1 is GAUSSMF(−18.5,4.9) and x2 is GAUSSMF(−5.4,3.3) and u1 is GAUSSMF(−16.7,5.4) and u2 is GAUSSMF(−19.8,4.8) THEN x2  5.91e  003  130 x1  68x2  222u1  244u 2 IF x1 is GAUSSMF(−6.82,6.9) and x2 is GAUSSMF(−9.74,4.1) and u1 is GAUSSMF(−6.48,5.2) and u2 is GAUSSMF(−6.78,5.6) THEN x2  3.04e  003  10.8x1  52.7 x2  15.3u1  10.3u 2 IF x1 is GAUSSMF(1.28,6.2) and x2 is GAUSSMF(3.07,5.2) and u1 is GAUSSMF(1.37,5.6) and u2 is GAUSSMF(1.37,5.2) THEN

x2  6.36e  003  110 x1  370 x2  39.2u1  121u 2 IF x1 is GAUSSMF(13.6,5.4) and x2 is GAUSSMF(18.4,5.1) and u1 is GAUSSMF(11.9,5.8) and u2 is GAUSSMF(13.6,5.6) THEN x2  5.76e  003  88x1  515x2  19.2u1  62.4u 2 A possible T1 controller fuzzy rules could be: IF x1 is GAUSSMF(−8.34,4.4) and x2 is GAUSSMF(2.72,6.07) THEN u1  0.0002  5.69 x1 IF x1 is GAUSSMF(14.62,5.6) and x2 is GAUSSMF(−13.01,6.2) THEN u1  0.00047  3.01x1  0.00385x2 IF x1 is GAUSSMF(4.04, 5.5) and x2 is GAUSSMF(13.12, 5.3) THEN u1  0.00044  2.62x1  0.00155x2 IF x1 is GAUSSMF(−6.01, 6.1) and x2 is GAUSSMF(−9.72,6.2) THEN u2  0.0004  1.83x1  8.42 x2 IF x1 is GAUSSMF(3.86,6.07) and x2 is GAUSSMF(1.6,6.54) THEN u2  1.32 x1  9.16 x2 IF x1 is GAUSSMF(14.2,5.2) and x2 is GAUSSMF(14.8,7.8) THEN u2  0.000504  2.06x1  2.81x2 The IT2T-S fuzzy model of the plant is the following: IF x1 is GAUSSMF(−13.9,6.5(1±a%)) and x2 is GAUSSMF(−13.5,5.1(1±a%)) and u1 is GAUSSMF(−13.6,5.4(1±a%)) and u2 is GAUSSMF(−13.1,5.3(1±a%)) THEN x1  16.7  1.93x1  1.26 x2  0.114u1  0.235u 2 IF x1 is GAUSSMF(−6.02,5.1(1±a%)) and x2 is GAUSSMF(−5.03,4.1(1±a%)) and u1 is GAUSSMF(−6.85,5.03(1±a%)) and u2 is GAUSSMF(−5.24,5(1±a%)) THEN x1  13.5  1.13x1  1.01x2  1.57u1 1.13u 2 IF x1 is GAUSSMF(0.68,4.1(1±a%)) and x2 is GAUSSMF(2.67,4.5(1±a%)) and u1 is GAUSSMF(1.62, 4.7(1±a%)) and u2 is GAUSSMF(1.59,4.9(1±a%)) THEN x1  11.5  0.617 x1  1.36 x2  0.423u1  0.498u 2 IF x1 is GAUSSMF(8.54,5.5(1±a%)) and x2 is GAUSSMF(9.69,5.4(1±a%)) and u1 is GAUSSMF(9.24, 5.2(1±a%)) and u2 is GAUSSMF(8.27,5.3(1±a%)) THEN x1  18.5  2.2 x1  2.05x2  0.917u1  0.894u 2 IF x1 is GAUSSMF(−18.5,4.9(1±a%)) and x2 is GAUSSMF(−5.4,3.3(1±a%)) and u1 is GAUSSMF(−16.7,5.4(1±a%)) and u2 is GAUSSMF(−19.8,4.8(1±a%)) THEN x2  5.91e  003  130 x1  68x2  222u1  244u 2 IF x1 is GAUSSMF(−6.82,6.9(1±a%)) and x2 is GAUSSMF(−9.74,4.1(1±a%)) and u1 is GAUSSMF(−6.48,5.2(1±a%)) and u2 is GAUSSMF(−6.78,5.6(1±a%)) THEN x2  3.04e  003  10.8x1  52.7 x2  15.3u1  10.3u 2 IF x1 is GAUSSMF(1.28,6.2(1±a%)) and x2 is GAUSSMF(3.07,5.2(1±a%)) and u1 is GAUSSMF(1.37,5.6(1±a%)) and u2 is GAUSSMF(1.37,5.2(1±a%)) THEN x2  6.36e  003  110 x1  370 x2  39.2u1  121u 2 IF x1 is GAUSSMF(13.6,5.4(1±a%)) and x2 is GAUSSMF(18.4,5.1(1±a%)) and u1 is GAUSSMF(11.9,5.8(1±a%)) and u2 is GAUSSMF(13.6,5.6(1±a%)) THEN x2  5.76e  003  88x1  515x2  19.2u1  62.4u 2 - 136 http://www.sj-ce.org

A possible IT2T-S controller could be: IF x1 is GAUSSMF(−8.34,4.4(1±a%)) and x2 is GAUSSMF(2.72,6.07(1±a%)) THEN u1  0.0002  5.69 x1 IF x1 is GAUSSMF(14.62,5.6(1±a%)) and x2 is GAUSSMF(−13.01,6.2(1±a%)) THEN u1  0.00047  3.01x1  0.00385x2 IF x1 is GAUSSMF(4.04,5.5(1±a%)) and x2 is GAUSSMF(13.12,5.3(1±a%)) THEN u1  0.00044  2.62x1  0.00155x2 IF x1 is GAUSSMF(−6.01,6.1(1±a%)) and x2 is GAUSSMF(−9.72,6.2(1±a%)) THEN u2  0.0004  1.83x1  8.42 x2 IF x1 is GAUSSMF(3.86,6.07(1±a%)) and x2 is GAUSSMF(1.6,6.54(1±a%)) THEN u2  1.32 x1  9.16 x2 IF x1 is GAUSSMF(14.2,5.2(1±a%)) and x2 is GAUSSMF(14.8,7.8(1±a%)) THEN u2  0.000504  2.06x1  2.81x2

AUTHORS 1

3

She graduated from Jiangsu University with a doctorate’s degree

He obtained his doctorate in Control Theory and Control

in Agricultural Electrification and Automation in 2009.

Engineering from Nanjing University of Aeronautics and

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

She is Associate Professor working in Faculty of Computer Science

and

Telecommunication

Engineering,

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

Astronautics in 2005.

Jiangsu

He is a Professor and Postgraduate Supervisor in Faculty of

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

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

following research fields: (1) Fuzzy Fault Detection and

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

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

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

Algorithm and Optimization.

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

2

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

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

University, Zhenjiang, China.

control methods.

- 137 http://www.sj-ce.org