Page 633 - Mathematical Techniques of Fractional Order Systems

604 Mathematical Techniques of Fractional Order Systems

If q 1 5 q 2 5 ? 5 q n 5 q the above system is called a commensurate order
system. Then equivalent form of the above system is described as:

x ðnqÞ 5 fðX; tÞ 1 gðX; tÞu 1 dðtÞ
ð20:17Þ
y 5 x 1
where fðX; tÞ and gðX; tÞ are unknown but bounded nonlinear functions
which express system dynamics and dðtÞis the external bounded disturbance.
The control objective is to force the system output y to follow a given
bounded reference signal y d , under the constraint that all signals involved
must be bounded.
The reference signal vector y and the tracking error vector e are defined as,
d
T
h i
ðqÞ
y 5 y d ; y ; ...; y ððn21ÞqÞ AR n
d d d
T n ð20:18Þ
ðqÞ
e 5 y 2 y 5 e; e ; ...; e ððn21ÞqÞ AR ;
d d
e ðiqÞ 5 y ðiqÞ 2 y ðiqÞ
d
T n
Let k 5 k 1 ; k 2 ; ...; k n Š AR be chosen such that the stable condition
½

argðeigðAÞÞ . qπ=2 is met, where 0 , q , 1 and eigðAÞ represents the

eigenvalues of the system state matrix given in Eq. (20.19).
By substituting Eq. (20.18) into Eq. (20.17) we obtain the closed-loop
control system in the state space domain as follows:
x ðnqÞ 5 Ax 1 BfðXÞ 1 gðXÞuŠ
½
ð20:19Þ
T
y 5 c x
where
2 3 2 3 2 3
0 1 0 0 ? 0 0 0 1
0 0 1 0 ? 0 0 0
6 0 7 6 7 6 7
6 7 6 7 6 7
A5 6 ^ ^ ^ ^ & ^ ^ 7 ;B5 ^ and c5 ^
6 7
6 7
6 7 6 7 6 7
0 0 0 0 ? 0 1 0 0
4 5 4 5 4 5
2k 1 2k 2 2k 3 2k 4 ? 2k ðn21Þ 2k n 1 0
By using the relation y ðqÞ 5 Ay d 1 By ðnqÞ ; the following equation is
d d
obtained:
h i

e ðqÞ 5 Ae 1 BfðXÞ 1 gðXÞu 2 y ðnqÞ
d
ð20:20Þ
T
e 5 c e
In what follows, a fuzzy adaptive control will be designed to stabilize the
system Eq. (20.16) or an equivalent system Eq. (20.19).
i. If the functions fðX; tÞ and gðX; tÞare known and the system is free of
external disturbance dðtÞ(i.e., dðtÞ 5 0),
The following assumptions are considered (Liu and Wang, 2009;
Boulkroune and M’saad, 2012),