Swarm Intelligence Techniques Applied to Nonlinear Systems State Estimation - Advances in Heuristic Signal Processing and Applications - page 225

Filter (PF ACO )[ 15 ], Particle Filter with Ant Colony for Continuous Domains [ 17 ],
and Continuous Ant Colony Filter (CACF) [ 8 ]. A detailed description of the swarm
filters is discussed in the following subsections.

10.4.1 Particle Swarm Optimized Particle Filter (PSOPF)

The Particle Swarm Optimization (PSO) is a robust stochastic optimization tech-
nique based on the movement and intelligence of swarms. It was developed in 1995
by James Kennedy and Russell Eberhart [ 29 ]. Individuals interact with each other
while they are learning from the swarm experiences and gradually move towards
the goal. PSOPF merges PSO into PF to optimize the sampling step of GPF. Fig-
ure 10.7 shows the general iterative structure of PSOPF. A high level description of
the sequential steps is shown in this figure.
PSOPF has two loops. The main outer loop iterates every time a new measure-
ment is entered. The inner loop iterates to find the best estimates of the current
states, corresponding to the entered measurement. At first, the inner loop propagates
the initial distribution of particles. Then the output, estimated using each particle, is
made. The estimated outputs are compared with the real measurement and the cost
of each particle is evaluated using a Gaussian function. Particles use local and global
experiences to update their position and velocity in the state space. The inner loop
is terminated after the cost function reaches a certain threshold. Then, particles are
weighted and normalized. To eliminate the particles with small weights and repli-
cate the ones with large weights, a resampling step is executed. Finally, the current
state is estimated. In the following subsections, these steps are discussed in detail.

10.4.1.1 Computation of Cost Function

The cost function of each particle is evaluated using a Gaussian distribution of the
difference between the estimated output, z k , and the real measurement, z k . There-
fore, the cost assigned to particle j at time k , is calculated as follows:

exp −

k

2 z k −

k T R − k z k −

1

f j z j z j

k =

(10.10)

where R k is the observation covariance.

10.4.1.2 Local Best and Global Best Update

PSOPF utilizes a set of moving particles to perform an intelligent search in the state
space, looking for the best estimate. Each particle keeps track of its coordinates