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values of several parameters with beneficial effects on algorithms' convergence speed
and search accuracy.
Building upon our recent results (Precup et al. 2011a ; David et al. 2012 ; Precup
et al. 2012b ; Precup et al. 2013a ), this chapter discusses two adaptive GSAs on the
basis of considering either fixed or adaptive lengths of the three stages in the stan-
dard GSA (Rashedi et al. 2009 ; Al-Hadithi et al. 2012 ). The adaptive lengths are
calculated on the basis of two fuzzy logic blocks with an input variable which com-
putes the minimum and maximum Popov sums, and the convergence is guaranteed
in an elegant manner by Popov's hyperstability theory. Our adaptive evolutionary
optimization algorithms solve a class of optimization problems which aim the min-
imization of objective functions expressed as the sum of absolute control error plus
squared output sensitivity function. Since the variables in the optimization problems
are the parameters of simple Takagi-Sugeno proportional-integral fuzzy controllers
(TS PI-FLCs), the adaptive GSA-based solving of the optimization problems yields
TS PI-FLCs with a reduced process parametric sensitivity. Experimental results are
included in the context of a case study focused on the optimal tuning of TS PI-FLCs
for the angular position control of a laboratory nonlinear direct current (DC) servo
system. The approaches presented here are different to another other fuzzy adaptation
approaches embedded inGSAs characterized by fuzzy blocks designed for adaptively
controling one parameter (Zahiri 2012 ) or for adaptively controling the gravitational
constant and the number of effective agents (Saeidi-Khabisi and Rashedi 2012 ). The
chapter also discusses the extension of these adaptation approaches such that to result
in adaptive CSS algorithms.
The chapter is organized as follows. The problem setting is presented in Sect. 8.2 .
The processmodels, the optimization problem, the TS PI-FLC structure and its design
approach are discussed. The adaptive GSAs are given in Sect. 8.3 . The extension to
adaptive CSSs is treated in Sect. 8.4 . The case study presented in Sect. 8.5 includes a
set of experimental results concerning the GSA-based optimal tuning of TS PI-FLCs
for the angular position control of a laboratory nonlinear direct current servo system.
The conclusions are highlighted in Sect. 8.6 .
8.2 Problem Setting
The state-space model of the process that characterizes a class of second-order servo
systems with dead zone and saturation nonlinearity is
0
,
if
|
u
(
t
) |≤
u a ,
m
(
t
) =
k u , m [
(
)
(
(
)) ] ,
if u a < |
(
) | <
u b ,
u
t
u a sgn
u
t
u
t
k u , m (
u b
u a )
sgn
(
u
(
t
)),
if
|
u
(
t
) |≤
u b ,
˙
01
0
x P 1 (
0
k P 1 /
m
1
0
d
(8.1)
x P 1 (
)
)
t
t
=
+
(
) +
(
),
t
t
x P 2 (
˙
t
)
1
/
T
x P 2 (
t
)
T
x P 1 (
t
)
y
(
t
) =[
10
]
,
x P 2 (
t
)
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