3。5。The discrete-time model of finite-time particle swarm optimization

In this subsection, we discretize the system (14) to derive the discrete-time model of the FPSO algorithm by employing (7)。 The motivation to consider the discrete-time version is that the PSO algorithm is initially proposed to cope with the opti- mization problems in the discrete-time form。 Hence, the discrete-time model is described by

8 y1 ðk þ DkÞ¼ y1ðkÞþ y2 ðk þ DkÞDk

>< a

y2 ðk þ DkÞ¼ y2ðkÞ— cDk/l ðkÞ— bDksigð/lðkÞÞ

ð26Þ

>: /l ðkÞ¼ ð1 — xÞy2ðkÞþ ay1 ðkÞ

Introducing

( y1 ðk þ DkÞ¼ xiðk þ DkÞ— pi ðk þ DkÞ

pi ðkþDkÞ—pi ðkÞ

ð27Þ

y2 ðk þ DkÞ¼ vi ðk þ DkÞ— Dk

we obtain the discrete-time model of the FPSO   algorithm。

In the light of the form of the position and the velocity, the discrete-time model of the FPSO algorithm is given by

( pi ðkþDkÞ—2pi ðkÞþpi ðk—DkÞ a

pi ðkþDkÞ—2pi ðkÞþpi ðk—DkÞ

Remark 8。  From (28), one can see that p€iðtÞ ’ Dk2 。 However, it is not easy to calculate piðk þ DkÞ。 Hence, we

will give a method to calculate piðk þ DkÞ in the following  simulations。

To illustrate the characteristics of the discrete-time FPSO algorithm, we set pi ðkÞ ¼ 0, and use the PSO algorithm and the GPSO algorithm as comparison examples whose results were also presented in [15]。 Figs。 6–8 show the corresponding re- sults。 The magnitude of position oscillation denotes the exploration capability of particles while the number of sampling

points refers to the exploitation capability of particles。 From three figures, one can see that the discrete-time FPSO algorithm provides a flexible mechanism to tradeoff the exploration capability and the exploitation capability of the particle swarm。

Fig。 7。  The  convergence  curves  of  the  states  in  the  GPSO  algorithm  (10)  (x ¼ 0:8;  a ¼ 2:2;  b ¼ 0;  c ¼ 1;  Dk ¼ 0:5;  xi ð0Þ ¼ 5,  and  vi ð0Þ ¼ —9)。

Remark 9。 For each algorithm, the trajectories of all particles are the same due to the fact that pi ðkÞ ¼ 0 and other parameters are constant。 One reason is that the movement of the particle from the same initial position to the same equilibrium position can be easily shown in the same environment。 The other reason is that the characteristics of the discrete-time FPSO algorithm can also be easily   illustrated。

For the discrete-time model of the FPSO algorithm, a convergence condition is given in the following theorem。

Theorem 2。  Consider the discrete-time model of  the FPSO  algorithm (28)  with ðx; a; c; aÞ 2 Xd   given  by

( 2 4 — 2cDkð1 — xÞ )

Xd  ¼    ðx; a; cÞj1 — cDk < x < 1; 0 < a <