Page 172 - Mathematical Techniques of Fractional Order Systems

160 Mathematical Techniques of Fractional Order Systems

is that any control scheme that is performing a parameter estimation, which
is simultaneously being used in the controller parameter updating, leads to
the situation of two IOEM with a constraint on their true parameters. Thus,
AL taking into account this parameter linkage are called for. This is pre-
cisely the problem solved in Duarte and Narendra (1988a, 1996a) in a gen-
eral manner that leads to better solutions than those where the parameters of
the controller and the identifier are independently updated. It is worth men-
tioning that the so-called combined direct and indirect adaptive control intro-
duced by Duarte & Narendra (e.g., combined model reference adaptive
control) (Duarte and Narendra, 1987a,b, 1989a,b) and the dynamical indirect
adaptive control (e.g., dynamical indirect MRAC) (Duarte and Narendra,
1988b, 1996b) belong to this class of problems.
On the other hand, a great deal of attention has been paid on the study of
the FO systems (FOS) where FO operators (integrals and/or derivatives)
rather than IO operators are used either in the plant description (modeling)
and/or in the controller design. This fact has allowed to expand the control-
lers/observers designs to include FO operators providing additional degrees
of freedom in the design. For example, the IOPID controller contains three
design parameters (proportional, integral, and derivative constants ) whereas
the FOPID contains five parameters (proportional, integral, and derivative
constants; plus the integral and derivative orders) augmenting the search
space for tuning the controller allowing to obtain better solutions according
to prespecified performance indexes. As an example, in Vale ´rio and Da
Costa (2006) is presented a Ziegler Nichols type of method to tune FOPID
controllers, whereas in Aguila-Camacho and Duarte-Mermoud (2013)an
application of FOPID to an automatic voltage regulator (AVR) is discussed.
The AVR is a controller whose main purpose is to maintain the voltage level
in an electric generator by adjusting the generator exciter voltage. In that
work a genetic optimization algorithm was used to tune a FOPID. Several
other applications of FO controllers and observers have been lately reported.
It is worth mentioning the work by (Kavuran et al., 2016) where an experi-
mental test platform of a low-cost coaxial rotor was developed and a model
reference adaptive controller with FO AL was explored and implemented in
MATLAB/Simulink. In Huan et al. (2016) a method is proposed to solve all
possible FOPD controllers to stabilize a given system. Robust stability with
phase margin and gain margin is considered and the parameters regions are
described on the delay-parameter plane. The rather complex dynamics of
lithium-ion battery is described in Takamatsu and Ohmori (2015) as a frac-
tional order (FO) system and then the Kreisselmeier-type of adaptive
observer is used to estimate the state and parameters battery. The results
obtained by numerical simulations indicate that the proposed model repre-
sents adequately the system behavior.