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
 

Equ. 2

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 c₁ and c₂ 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