Page 635 - Mathematical Techniques of Fractional Order Systems

606 Mathematical Techniques of Fractional Order Systems

e ðnqÞ 1 k n e ðn21Þq 1 ... 1 k 1 e 5 0 ð20:27Þ
which is the main objective of control lim t-N eðtÞ 5 0.
In the reaching phase we get sðX; tÞ 6¼ 0, and a switching-type control u sw
must be added in order satisfy the sufficient condition

sðX; tÞ_ sðX; tÞ #2 η sðX; tÞ ; η . 0 which implies the global control.

Therefore the global sliding mode control law is given by:
1 X n21
u 5 k i e ðiqÞ 2 fðX; tÞ 1 y ðnqÞ 2 u a 2 KsgnðsÞ ð20:28Þ

d
gðX; tÞ i51
We can show by taking a Lyapunov function candidate defined as
1
2
V 5 s ðeÞ ð20:29Þ
2
And differentiating Eq. (20.29) with respect to time to the fractional order
ðqÞ
q, V ðtÞ along the system trajectory (Aguila-Camacho et al., 2014), we obtain
!
n21
X
V ðqÞ 5 ss ðqÞ 5 s k i e ðiqÞ 1 e ðnqÞ
i51
!
n21
X
52 s k i e ðiqÞ 2 y ðnqÞ 1 y ðnqÞ
d
i51 ð20:30Þ
!
n21
X
52 s k i e ðiqÞ 1 fðX; tÞ 1 gðX; tÞu eq 2 y ðnqÞ
d
i51

#2 η sðX; tÞ

Hence, the sliding mode control u guarantees the sliding condition of
sðX; tÞ_ sðX; tÞ.
However, the functions fðX; tÞ and gðX; tÞ are usually unknown in practice
and it is difficult to apply the control law Eq. (20.28) for an unknown nonlin-
ear plant. Moreover, the chattering problem appears when adding the switch-
ing control term u sw 5 K sgnðsÞ:
To deal with these problems, we consider the adaptive sliding mode con-
trol scheme using a fuzzy logic system and the saturation function to avoid
chattering problem.
The Chattering phenomenon can be reduced by replacing the function
sign by a function of adequate saturation which filters the high frequencies.
The modified resulting control law uðtÞ, which includes a fuzzy system to
approximate the unknown functions fðX; tÞ and gðX; tÞ and a saturation func-
tion that attenuates the chattering and improves performance, is as follows:

1 h T i
u 5 2 f X θ f 1 y ð d nqÞ 1 k e 2 u a 2 satðsðX; tÞÞ ð20:31Þ
gðXjθ Þ
g