connection weights between the rule of the first j and the output node, all of them are adjustable parameters。

4。2Weight adjustment

The weight choice had a tremendous influence on system performances, if the weight is inappropriate, the convergence speed of neural networks would be reduced。 This article carried on the training based on the gradient descent law to the network weight。

Define objective function as follows:

As the control laws Eqs。 (7)−(8) contain sgn(s), it makes the system to produce high-frequency chattering easily。 Therefore, the saturation function is used instead of the sign function to smooth control signal。 Then, the control law can be rewritten as

u1  [hf(hh) fhfhc e 

(hh)(c e  ls at(s ) k s ) 

hxc  e hhs at(s) k hs]/

   

M  1 

(d (4) O(4) )2

(13)

(g2 h3    g1h2   g3h2    g2 h1 )

cl e4  fls at(s ) k  s  gu

(21)

 j j

j

u2  

 (22)

2

where d (4)  is the desired output of the network, while

(4)

5Particle swarm optimization

Oj is the actual output of the network。

Suppose that the learning rate of V, aij  and bij  are η1,

η2 and η3, respectively, the adjustment values are

5。1 Principle of particle swarm optimization

As seen in Eqs。 (21)−(22), the controller parameters

V  M

c , c , c , α, ε , k ,    ε

and k

have a direct impact on  the

 1   (I (4) )

x l θ

1 1      2 2



a    

j

 M

(14)

system control law。 The greater the values of cx, cl, cθ,  α,

ε1,  k1, ε2  and k2,  the faster  the  system approached    the

 ij



2

(aij )

sliding surface。 However, too large values would make the   control   excessive   and   may   cause   the    system



b   

M

 ij



3

(bij )

chattering, which can affect the dynamic performances of approaching process。 Otherwise, the values are   smaller,

where  V  are  w(4) ,

v(4)  and u(4) ,  respectively。 Suppose

although  the  system chattering  is  weakened,  the speed

w(4) ,

(4)