Swarm Intelligence Techniques Applied to Nonlinear Systems State Estimation - Advances in Heuristic Signal Processing and Applications - page 227

the population. This state is shown by p gbest . The logic of PSOPF is to change the
velocity of each particle toward the local best p pbest and the global best p gbest at
each time step. It will be discussed in the next section.

10.4.1.3 Velocity and Position Update Rules

In PSOPF, the velocity and position of each particle is updated continuously, as in
PSO. The new velocity of particles is calculated using p pbest and p gbest as follows:

r 1 p pbest − x k + r 2 p gbest − x k v k =

(10.11)

where r 1 and r 2 are positive random numbers with Gaussian probability distribution,
i.e., abs [ N ( 0 , 1 ) ] . The position vector will simply be updated as follows:

x j
k +

x j v j

1 = k + k .

(10.12)

In PSOPF, the velocity may become very large and the performance may be
degraded. So, the velocity should be limited to an interval

[− v max , v max ] .

10.4.1.4 Weight Computation and Normalization

After the termination of the inner loop, each particle is weighted according to
Eq. ( 10.5 ). In this equation, the most popular suboptimal choice of the proposal
density is the transitional prior as follows:
q x k x k − 1 , z k = p x k x k − 1 . (10.13)

Therefore, the weight of particle j at time k will be assigned recursively as fol-
lows:

w k − 1 p z k x k . w k =

(10.14)

Finally, the weighted particles are normalized according to Eq. ( 10.6 ).

10.4.1.5 Stopping Condition

PSOPF has two loops, each with its own specific stopping condition. The inner loop
stops when the cost of the global best estimation ( p gbest ) reaches a certain threshold
( ε ). The outer loop terminates when the measurements are ended.