Advanced Metaheuristics-Based Approach for Fuzzy Control Systems Tuning - Complex System Modelling and Control Through Intelligent Soft Computations - page 622

By the law of motion, the acceleration of the agent i at time k, and in dth
direction, is given as follows:

F i ; d
k
M i k

a i ; d
k

¼ ð Þ

12

where M i k is the inertial mass of ith agent in the search space with dimension d.
Hence, the position and the velocity of an agent are updated respectively by the
mean of equations of movement given as follows:

x i ; d x i ; k þ v i ; d

k þ 1 k þ 1 ¼

ð 13 Þ

v i ; d
k

rand i v i ; d
k

a i ; d
k

¼ þ ð 14 Þ

þ 1

where rand i is a uniform random number in the interval [0, 1], used to give a
randomized characteristic to the search.
To control the search accuracy, the gravitational constant G k , is initialized at the
beginning and will be reduced with time. In this study, we use an exponentially
decreasing of this algorithm parameter, as follows:

G 0 e g

k
kmax

G k ¼ ð 15 Þ

where G 0 is the initial value of G k , g is a control parameter to set, and k max is the

total number of iterations.
In GSA, gravitational and inertia masses are calculated by the fitness evaluation.

A heavier mass means a more ef cient agent. Better agents have higher attractions

and walk more slowly.
As given in Rashedi et al. ( 2009 ), the values of masses are calculated using the

fitness function and gravitational and inertial masses are updated by the following
equations:

M pi
k

M ai
k

M i k ¼ M k ¼ ¼ ð 16 Þ

m i k
P j ¼ 1 m k

M k ¼ ð 17 Þ

fit k

worst k
best k

m i k ¼ ð 18 Þ

worst k

where fit k

represents the fitness value of the agent i at iteration k, and, worst k and

best k are de ned, for a minimization problem, as follows: