cos x

sin x

cos x

 5 4   6   x 5  x

 5 5  D x ;

x1   x2

x x M 2

x M l   4

3 3 3

2776

g

cos x5  ; h

cos x5 sin x5 。

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

3  

Mx3

Mx3

Thus, the bridge crane system was pided  into three coupled subsystems: the positioning subsystem, lifting rope subsystem and anti-swing subsystem。

The control object of the bridge crane was to move the trolley to its destination and complement anti-swing of the load at the same time when the  system model exists uncertainty and disturbance (for example, winds and different payloads)。 In order to decouple the system, four sliding mode functions were defined for the three subsystems with sliding surface, sx=cxe1+e2, sθ=cθe5+e6, sl=cle3+e4, s=αsx+sθ。

Take the index reaching law as

s1 sgn(s) k1s

sl   2 sgn(sl )  k2 sl

(5)

(6)

Fig。 2 Fuzzy neural network structure

where cx, cl, cθ, α, ε1, k1, ε2 and k2 are positive numbers。 Then, according to Eqs。 (4)−(6), the following equations can be introduced。

u1  [h2 f1 (h1  h3 ) f2 h2 f3 h2cxe2 

(h3 h1 )(cle4 lsgn(s ) k s ) 

1)

Input layer

Each node in this layer is connected with the input vector。 The error and its change rate are the network inputs, which is corresponding to the j-th node in the input−output and can be expressed as

(1)

 h2 xd   c e6 h2   1h2 sgn(s)  k1h2 s]/

wij    1



(g2h3 g1h2 g3h2 g2h1 )

(7)

I (1)  w(1) x(1)  x(1)

(9)



 

j ij i i

(1) (1)      (1) (1)

u   cl e4 f2 ld 2 sgn(sl ) k2 sl  g2u1

h2

(8)

Oj

f j

(I j

) I j

where   (1) (1)

x1       e(t) , x2

e(t)。

However, f1, h1, f2, g2, h2, f3, g3, and h3  are generally

unknown in the actual system, therefore, the control law is difficult to implement。

4 Fuzzy neural network

2)

Membership layer

In this layer, input variables are defuzzied, Gaussian membership function is chosen。

w(2)  1

4。1Fuzzy neural network structure

ij



I (2)  w(2) (c

,2 ) 

(I (2) cij )2

(10)

As  the  actual  system  control  law  is  difficult   to