Swarm Intelligence Techniques Applied to Nonlinear Systems State Estimation - Advances in Heuristic Signal Processing and Applications - page 228

10.4.2 Ant Colony Optimization Assisted Particle Filter ( PF ACO )

The Ant Colony Optimization (ACO) is a stochastic optimization algorithm based
on the swarm intelligence of ant colonies. Each ant navigates from the nest to the
food sources to find a solution and then communicates with other ants by leaving a
pheromone trail within the environment. PF ACO incorporates ACO into PF to opti-
mize the sampling step of GPF. Figure 10.8 shows the general iterative structure of
PF ACO .
PF ACO has two loops. The main outer loop iterates every time a new measure-
ment is entered. At first, the initial distribution of ants is propagated. Then, the
outputs are estimated by the ants, and consequently each ant is weighted. The inner
loop iterates to find the best estimates of the states, corresponding to the entered
measurement. In this loop, the threshold parameter, explained in Sect. 10.4.2.4 ,is
computed for each ant. The threshold is used to define a neighborhood around the
ant. The next movement of each ant is selected based on a probability function such
that it coincides with one of its elite (low cost function) neighbors. The probabil-
ity function is utilized to model the pheromone distribution over the discrete search
space. Each ant is assigned a weight which is proportional to its cost. Moreover,
ants use their experience to update the pheromone distribution. The inner loop is
stopped when the estimation error reaches the predefined threshold. Then, ants are
weighted again, normalized, and resampled. Finally, the current state is estimated.
In the following subsections, these steps are discussed in detail.

10.4.2.1 Computation of Probability Function

Each ant chooses its direction using a probability function, defined on the basis of
the quality of other ants within the neighborhood. The probability that the ant i
selects ant j is expressed as follows:

α β

[ τ ij (t) ] [ η ij (t) ]

s ∈ N(i) [

p ij (t) = (10.15)

α β

τ is (t) ] [ η is (t) ]

where N(i) is the set of all ants which are in the neighborhood of ant i , τ ij (t) is the
pheromone density of the link between ants i and j , as introduced in Sect. 10.4.2.3 ,
and η is a heuristic function, defined as:

1
d ij

η ij (t) = (10.16)

where d ij is the distance between ants i and j .If p ij = 1, then ant i moves toward

ant j . When convergence occurs, it means that most of the ants have moved to a
high likelihood region. The parameters α and β determine the relative influence of
pheromone trails and the heuristic information, respectively.