Swarm Intelligence Techniques Applied to Nonlinear Systems State Estimation - Advances in Heuristic Signal Processing and Applications - page 230

10.4.2.3 Update Pheromone Distribution

When ant i selects ant j , τ ij (t) is updated as follows:

τ ij (t + 1 ) = ( 1 − ρ)τ ij (t) + Δτ ij (t) (10.17)

where 0 <ρ 1 is the pheromone evaporation rate and Δτ is a constant value that

simulates the pheromone deposition over the visited links. Also, when ant j is not
chosen by ant i , τ ij (t) is evaporated as follows:

≤

τ ij (t + 1 ) = ( 1 − ρ)τ ij (t). (10.18)

The initial pheromone distribution, τ ij ( 0 ) , has been proposed to be a function of
ants weight [ 15 ].

10.4.2.4 Stopping Condition

PF ACO has two loops, each with its own specific stopping conditions. The inner loop
stops when the distance between ants i and j becomes less than a certain threshold:

| 1

k

w j

ε j = c | r −¯ (10.19)

w k is the normalized weight of ant j , r is a normal random number, and c is
a constant value; or the number of iterations exceeds a maximum value. The other
loop is terminated when the measurements are finished.

where

10.4.3 Particle Filter with Ant Colony for Continuous Domains

The fundamental idea in continuous ant colony algorithms is to define a contin-
uous pheromone model [ 30 ]. The Continuous Ant Colony System (CACS) uti-
lizes a Gaussian PDF to model pheromone distribution over the continuous search
space [ 30 ]. The Ant Colony Optimization for continuous domains (ACO R )[ 31 , 32 ]
utilizes a weighted sum of several Gaussian PDF instead of a single one. The Parti-
cle Filter with Ant Colony for Continuous Domains incorporates ACO R into PF to
optimize the sampling process of PF. Figure 10.9 shows the general iterative struc-
ture of this method.
The Particle Filter with Ant Colony for Continuous Domains has two loops. The
outer loop iterates every time a new measurement is entered. Here, the distribution
of samples is propagated. The inner loop iterates to find the best estimation. In this
loop, the output, estimated using each sample, is made. The estimated outputs are
compared with the real measurement and the cost of each sample is evaluated. In this
algorithm, the propagated samples and the corresponding cost functions are stored
in an external archive to represent the pheromone, as in ACO R . The Gaussian PDFs