10    10   10   10

c3  10    10   10   10,

b  2。8   2。8   2。8   2。8T  ,

b   36   36   36   36T ,

2

b  3   3   3   3T  。

3

(n) 1。2 

n Gmax

0。5

(25)

According  to  the  above  method,  the    simulation

results are shown in Figs。 3−13。 Figures 3−5 describe the output of the fuzzy neural network, Fig。 6 describes    the

where Gmax is the maximum calculated cutoff generation。

5。2 Design steps

Step 1: Initialize a group of random particles (e。g。 group size N, random position, velocity, and initial vector)。

Step 2: Evaluate the fitness value of each particle according to the objection function J and fitness function F as follows:

progress of parameter optimization with the PSO,  where

cx=2。658  1,  cl=0。218  8,  cθ=1。265  8,  α=3。824  5,     ε1=

0。836 0, ε2=0。754 9, k1=2。367 2, and k2=2。622 4。

Figures 7−9 describe the change of system variables using FNNSMC and SMC, the maximum swing angle of FNNSMC is ±0。1 rad, the maximum swing angle  of SMC is ±0。12 rad, the rapidity of FNNSMC than SMC。 Figures 10−13 describe the change curves with 0。5 step disturbance within the 13 s。 It can be seen from Figs。 10−

13, the anti-swing capability of this method is stronger

min

J 1 eT e (26)

2

than conventional sliding mode control and the  method of   Ref。   [10]。   In   the   presence   of   disturbances, the

1

Ffit   J

(27)

maximum swing angle of it is only ±0。1 rad, but the maximum swing angle ±0。13 rad of SMC and is 0。25 rad

Step 3: For each particle, by comparing the inpidual fitness value at present and the best position pbest itself in the past, the best position pbest is updated if the present value is better than the past。

Step 4: For each particle, by comparing the inpidual fitness value and the best position gbest of the group with those in the past, the global best position gbest is updated if the present value is better than the global optimal position。

in Ref。 [10]。

As can be seen from the simulation results, the PSO can search the most excellent value fast in the solution space (the three generations)。 The simulation shows that the proposed control method guarantees anti-swing control and accurate tracking control of trolley when the system model exists uncertainties。 And the sliding function can reach rapidly to the sliding mode surface, which improves the system robustness。

J。 Cent。 South Univ。 (2012) 19: 2774−2781 2779

Fig。 3 Output of first fuzzy neural network: (a)  ~ ;  (b) ~

f1 h1

Fig。    4    Output    of    second    fuzzy    neural    network:  (a)

Fig。 5 Output of third fuzzy neural network: (a)

~

;  (b)  g ;

~ ; (b) g~  ; (c) ~