Page 634 - Mathematical Techniques of Fractional Order Systems

Enhanced Fractional Order Chapter | 20 605

Assumption IV.1. The control gain gðX; tÞ is not zero and of known sign.
It is also strictly positive or strictly negative.

Assumption IV.2. The external disturbance is bounded: dðtÞ # D with D

an unknown positive constant.
Then the control law of the certainty equivalent controller is obtained as:
1 h i
T
u 5 2fðX; tÞ 1 y ðnqÞ 1 k e ð20:21Þ

d
gðX; tÞ
h i T
ðqÞ
where y 5 y d ; y ; ...; y ððn21ÞqÞ AR n
d d d
T
n
e 5 y 2 y 5 e; e ; .. .; e ððn21ÞqÞ AR ;
ðqÞ
d
d
e ðiqÞ 5 y ðiqÞ 2 y ðiqÞ is the tracking error vector.
d
Substituting Eq. (20.21) into Eq. (20.20), we have
e ðnqÞ 5 k n e ðn21Þq 1 ? 1 k 1 e 5 0 ð20:22Þ
which is the main objective of control, lim t-N eðtÞ 5 0:
The sliding surface is defined as:
sðX; tÞ 52 ðKeÞ
52 ðk 1 e 1 k 2 e ðqÞ 1 ... 1 k n21 e ððn22ÞqÞ ð20:23Þ
1 e ððn21ÞqÞ Þ
when eð0Þ 5 0, the tracking problem x 5 y d implies that the sliding surface
sðeÞ 5 0; ’t $ 0.
This later classic derivative can be decomposed into a fractional type,
q
_ sðX; tÞ 5 D ð12qÞ ðD ðsðX; tÞÞÞ 5 0
ð20:24Þ
q
then D ðsðX; tÞÞ 5 0
If the functions fðX; tÞ and gðX; tÞ are known and the system is free of
external disturbance, i.e., dðtÞ 5 0: The control signal in the following equa-
tion drives the dynamic to reach to the sliding surface:
n21 !
X
s ðqÞ 52 k i e ðiqÞ 1 e ðnqÞ
i51
n21 ð20:25Þ
X
52 k i e ðiqÞ ðnqÞ
d
1 fðX; tÞ 1 gðX; tÞu eq 2 y
i51
5 0
Therefore, the equivalent control law is given by:
1 X n21
u eq 5 k i e ðiqÞ 2 fðX; tÞ 1 y ðnqÞ ð20:26Þ
d
gðX; tÞ i51
Substituting Eq. (20.26) into Eq. (20.19), we have