Page 302 - Mathematical Techniques of Fractional Order Systems

Sliding Mode Stabilization and Synchronization Chapter | 10 291

so for the terminal sliding mode controller design consider the following
fractional order dynamical system:
α
D xðtÞ 5 fðxðtÞÞ 1 UðtÞ ð10:9Þ
n
n
where xðtÞAR is the state variable, fðxðtÞÞAR is a nonlinear vector, and
n
UðtÞAR is the system input. So, consider the following vector:
0 1
f 1 ðxðtÞÞ
fðxðtÞÞ 5 @ ^ A ð10:10Þ
f n ðxðtÞÞ
so the terminal sliding mode controller for the stabilization of system (10.6)
and (10.8) is obtained by the following theorem (Aghababa, 2015;
Komurcugil, 2012):

The following terminal sliding mode control law
Theorem 1:
ρ
u i ðtÞ 5 k i D 12α ðx i ðtÞ 1 signðx i ðtÞÞjx i ðtÞj Þ 2 f i ðxðtÞÞ 2 s i stabilizes the system
(10.9) in its equilibrium point where k i is the sliding mode gain and s i is the
sliding mode surface defined later for i 5 1; :::; n.

Proof: Consider the following sliding mode surface:
ρ
s i ðtÞ 5 D α21 x i ðtÞ 1 k i D 2α ðx i ðtÞ 1 signðx i ðtÞÞjx i ðtÞj Þ ð10:11Þ
and the following Lyapunov function (Aghababa, 2015):
n
X
VðtÞ 5 :SðtÞ: 5 js i ðtÞj ð10:12Þ
1
i51
so deriving (10.12) yields:
n
_
X
VðtÞ 5 signðs i ðtÞÞ_ s i ð10:13Þ
i51
and implementing the fractional order calculus properties explained in
Section 10.2, the following result is obtained:
n
ρ
X
_
VðtÞ 5 signðs i ðtÞÞ½f i ðxðtÞÞ 1 u i ðtÞ 1 k i D 12α ðx i ðtÞ 1 signðx i ðtÞÞjx i ðtÞj ފ
i51
ð10:14Þ
so by selecting the following control law:
12α ρ
u i ðtÞ 5 k i D ðx i ðtÞ 1 signðx i ðtÞÞjx i ðtÞj Þ 2 f i ðxðtÞÞ 2 s i ðtÞ ð10:15Þ
Then, (10.14) becomes:
n
_
X
VðtÞ 52 js i ðtÞj # 0 ð10:16Þ
i51
and this completes the proof.