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
i1
i1
l (3) (3) (3) (3)
where x1 e ,
l
x2 eare 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
i1
(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
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