j  j j j

as the distance between the position of the best    particle

(3) O(3)

k

(17)

in the swarm pg,d(n) and xi,d(n)。

vi,d (n 1) vi,d (n) 1r1 ( pi,d (n) xi,d (n)) 

Then, the revised three-parameters of V, aij  and    bij

are as follows:

2r2 ( p

g ,d

(n) xi,d

(n))

vi,d (n 1)  vmax

if vi,d  vmax

V M (4) O(3)

(18)

v   (n 1) v

if v v

(23)

V j j

  i,d min

i,d min

2778

where ω is the inertia weight; η1 and η2 are the acceleration constants, namely cognitive and social parameters, respectively; and r1 and r2 are two random values in the range of [0, 1]。 The above deterministic and probabilistic parameters reflect the effects of the inpidual memory and swarm influence on the particle positions。 The position of particle i, xi,d(n) is iteratively updated as

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

Step 5: Update the velocity and position of particles according to Eqs。 (23)−(24)。

Step 6: Return to Step 2 if the termination condition is not met。 The termination condition is generally the perfect fitness or the maximum calculated cutoff generation。

6Simulation research

xi,d (n 1) xi,d (n) vi,d (n 1)

(24)

To verify the effectiveness of the proposed   method

The optimal solutions can, thus, be acquired by choosing the best particles in a D-dimensional space, where D is the number of variables。 From Eqs。 (23)− (24), it can be observed that the collective intelligence was the distinguishing property of the PSO method。

The optimization progress for parameters cx, cl, cθ, α, ε1, k1, ε2 and k2 in the control law is initialized with a group  of  random  particles  N。  Throughout  the process,

(FNNSMC), a bridge crane system [10]  is introduced into the simulation, M=1 kg, m=0。25 kg, Dx=0。15 N/(s·m), Dl=0。1 N/(s·m), g=9。8 m/s2。 The desired position of trolley is 0。7 m, ld is π-type function, as shown in Fig。 9, the lifting-rope length from 0。7 m to 0。4 m to 0。7 m。  The

initial weights of three RBF networks are 0。001, the central values and widths of twelve RBF neurons in hidden layer are taken as follows:

each particle i monitors three values: its current position (Xi), the best position in previous cycles (Pi) and  its flying velocity (Vi)。

The operator ω played the role of balancing the global search and the local search。 In order to improve the convergence performance of PSO algorithm to assure the initial global search and the subsequent  local research,    a    time-varying    inertia    weight    ω(n)   is

formulated, which is the function of iteration n。

10    10   10   10

c1  10    10   10   10,

95    95    95  95

c2  95   95   95   95,