j ij





ij ij 2

ij

achieve, the output of three fuzzy neural  networks is used to approximate f1, h1, f2, g2, h2 and f3,  g3,  h3 to realize adaptive sliding mode control。 Fuzzy neural network (FNN) is pided into four levels: input layer, membership layer, rule layer and output layer。 The network structure is shown in Fig。 2。

Because g1 (g1=1/M) is constant, the outputs of    the

O(2)    f (2) (I (2) )  exp(I (2) )

j j j j

where cij and σij are respectively represented as the i-th variable’s j-th node Gaussian function’s central  value and the standard deviation, both of them are adjustable parameters。

3)Rule layer, that is, the multiplication operation on

~ ~

first fuzzy neural network are

f1  and h1。

fuzzy rules。

In the network, the fuzzy space of each variable is pided into five fuzzy sets {NM, NS, Z, PS, PM} and

twenty-five rules altogether as follows:

w(3)  1

ij

 2 2

l l 

I (3) 

w(3) x(3) 

(3)

i

j 1,

, 25

(11)

Rl: if  x1  A1     and  x1  A2   then  y l

 i1

i1

l (3) (3)     (3) (3)

where  x1  e ,

l

x2  eare  the system input variables;   A1

Oj

f j

(I j

) I j

and  A2    are fuzzy sets of x1  and x2, respectively; y is   the

systematic output variable。 The FNN inference process is described as follows。

where  x(3)  is the i-th input variable of the third layer。

4)Output layer, that is, output variable clarity。

J。 Cent。 South Univ。 (2012) 19: 2774−2781 2777

2

(4) (4)

   M

(2)

(O(1) aij )

I j





 Vxi

i1

(12)

aij  

(aij

2 j   

)

(bij )2

(19)

O(4)  f (4) (I (4) ) I (4)

(O(1)  a )2

j i j j

b   M

2 (2)  j ij

(20)

where V are  w(4) ,

(4)

ij

u(4) ,  respectively, which are the

ij (b ) j (b )3