Page 555 - Mathematical Techniques of Fractional Order Systems

534 Mathematical Techniques of Fractional Order Systems

The nonlinear equations of the PMSM are generally described as follows
18.3.
q
D XðtÞ 5 fðX t ; X 0 ; θÞ ð18:3Þ
t
where X denotes the state vector and X 0 is the initial state vector.
T
θ 5 ðθ 1 ; θ 2 ; ...; θ d Þ is a set of original parameters. q is the noninteger deriva-
tive orders.
Similarly, the nonlinear equation of the PMSM with the estimated para-
meters can be written as follows 18.4
q ^ ^ ^
D XðtÞ 5 fðX t ; X 0 ; θÞ ð18:4Þ
t
^ ^
^
^
^
T
where X is the estimated state vector, θ 5 ðθ 1 ; θ 2 ; .. .; θ d Þ denotes a set of
the extracted parameters and q is the noninteger derivative orders.
The proposed nature-inspired optimization algorithms search for the
global unknown parameters of the models based on a fitness function which
is the Mean Square Error (MSE) between the known and the estimated sys-
tem as in 18.5.
k
1 X
^
MSE 5 jXðiÞ 2 XðiÞj 2 ð18:5Þ
k
i51
where K is the number of samples.
18.4 PROPOSED CHAOTIC OPTIMIZATION TECHNIQUES
OVERVIEW

Developing the meta-heuristic algorithms by mixing the chaos maps with
them to tune some of their parameters is becoming a new research trend
(Gandomi and Yang, 2014). In this chapter, 10 different chaos maps are inte-
grated with the Grey Wolf Optimizer (GWO) and Grasshopper Optimization
algorithm (GOA) to improve the performance of the original algorithms.

18.4.1 Chaos Maps

Almost all the meta-heuristic optimization algorithms are random-based techni-
ques. This randomization is satisfied by using uniform or Gaussian distribution.
Nowadays, a recent approach has been introduced to substitute this randomness
with chaos maps to capitalize from the better properties of the chaos maps ran-
domization (Alatas, 2010a, Rezaee Jordehi, 2015; Gandomi and Yang, 2014).
In this approach, integrating the properties of chaos with the original algorithms
assists the standard algorithms to converge for the global solution rapidly and
accurately, especially in the case of multimodal functions problems (Gandomi
and Yang, 2014; Saremi et al., 2014; Rezaee Jordehi, 2015; Emary and
Zawbaa, 2016). In this manuscript, 10 different one-dimensional chaos maps