Page 632 - Mathematical Techniques of Fractional Order Systems
P. 632
Enhanced Fractional Order Chapter | 20 603
l
where v 5 L n μ l is the true value of the l th implication and μ l is
i51 F ðx i Þ F ðx i Þ
i
i
the membership function value of the fuzzy variable x i (Lin et al., 2004;
Hartley et al., 1995)
Eq. (20.14) can be rewritten as:
T
yXðÞ 5 θ ξ XðÞ ð20:15Þ
l
where θ 5 θ θ ... θ T M is an adjustable parameter vector and
T
T T
l
1 2
M
T
1
2
X
ξ XðÞ 5 ½ξ XðÞ; ξ XðÞ; ... ; ξ ðÞ is a fuzzy basis function vector defined
as:
l
v 1X T
l
ξ XðÞ 5 P M l
l51 v
th
l
When the inputs are fed into the T S, the true value v of the l implica-
tion is computed. Applying the common defuzzification strategy, the output
expressed as Eq. (20.14) is pumped out.
Based on the universal approximation theorem (Wang and Mendel, 1992;
Wang, 1994; Castro, 1995), the above fuzzy logic system is capable of uni-
formly approximating any well-defined nonlinear function over a compact
set Uc to any degree of accuracy. Also, it is straightforward to show that a
multioutput system can always be approximated by a group of single-output
approximation systems.
20.4 FUZZY ADAPTIVE ROBUST H N CONTROL: SLIDING
MODE APPROACH (VSC)
Consider a fractional order SISO nonlinear dynamic system of the form (Lin
and Kuo, 2011, 2012; Khettab et al., 2017c):
8 ðq 1 Þ
x
> 1 5 x 2
>
>
>
> ^
>
>
<
x ðq n21 Þ 5 x n ð20:16Þ
n21
>
>
> ðq n Þ
> x 5 fðX; tÞ 1 gðX; tÞu 1 dðtÞ
> n
>
>
:
y 5 x 1
where
T T n
X 5 x 1 ; x 2 ; ...; x n 5 x; x ; x ð2qÞ ; ...; x ððn21ÞqÞ AR is the system’s
ðqÞ
½
state vector, uAR is the control input, and yAR is the output, with the initial
conditions:
uð0Þ 5 0 and yð0Þ 5 0.
The initial conditions are set to zero to avoid unrobustness for Nussbaum
type adaptive controller as proved by Georgiou and Smith (1997),
