In the basic equations of PSO the j coefficient is usually interpreted as a social/cognitive one, or a confidence coefficient. And it is then quite " normal " it can be modified at each time step. Now, if we are looking for a deterministic version, it is interesting to find an interpretation in which j can be easily understood as a constant. The model is here a classical oscillating unit mass (the particle) attached to a spring.
The coordinate of the mass is x,
the fixation point is on p. Then, at each time t, the acceleration
at time is .
Let be the velocity of the
particle at time t. We have immediately
and . So, finally, for discrete
time steps (), we have the following
system
Equ. 1

and we retrieve easily the basic simplified
PSO equations
The only difference is that j is now seen as the ridigity coefficient of a virtual spring between the particule and its objective. Another point of view could be to see the particle as more or less " intelligent " and applying the following rules :
" The more I am far ahead the objective, the more I have to speed up, the more I am beyond it, the more I have to slow down".
But let us continue with our mechanical
analogy. We now add the usual damping part depending on the velocity (the
particle is getting " tired "), and we define the pulsation .
We obtain then :
Equ. 3

the Equ. 2 become
Equ. 4

and we find the well known solution
Equ. 5

where c_{1} and c_{2} are depending
on the initial conditions. It is yet another particular case of the general
five parameters model, near of we have called Constriction Type 1’’. We
have indeed here :
Equ. 6

with
Equ. 7

Maurice.Clerc@WriteMe.com 1998/11/23