Page 636 - Mathematical Techniques of Fractional Order Systems

Enhanced Fractional Order Chapter | 20 607

Note that the control law Eq. (20.28) is realizable only while fðX; tÞ and
gðX; tÞ are well known.
However, fðX; tÞ and gðX; tÞ are unknown and external disturbance
dðtÞ 6¼ 0, the ideal control effort Eq. (20.28) cannot be implemented. We

replace fðX; tÞ; gðX; tÞ by the fuzzy logic system fðX θ f Þ, gðX θ g Þ in a speci-

fied form as Eqs. (20.14, 20.15), i.e.,
T T
fðX θ f Þ 5 ξ ðXÞθ f ; gðX θ g Þ 5 ξ ðXÞθ g ; ð20:32Þ

T
Here the fuzzy basis function ξ ðXÞ depends on the fuzzy membership
functions and is supposed to be fixed, while θ f ; θ g and are adjusted by adap-
tive laws based on a Lyapunov stability criterion (Aguila-Camacho et al.,
2014; Duarte-Mermoud et al., 2015; Sastry and Bodson, 1989).

The optimal parameter estimations θ ; θ and are defined as:
f g

sup fðX θ f Þ 2 fðX; tÞ

f xAΩ x
θ 5 arg min θ f AΩ f
ð20:33Þ
sup
g xAΩ x
θ 5 arg min θ g AΩ g gðX θ g Þ 2 gðX; tÞ
where Ω f ; Ω g ; Ω p , and Ω x are constraint sets of suitable bounds on θ f ; θ g ; θ p ,

θ f # M f ,
and x respectively, and they are defined as Ω f 5 θ f

θ p # M p , and Ω x 5 x jj # M x gf
Ω g 5 θ g j x where
θ g # M g , Ω p 5 θ p
M f ; M g ; M p and M x are positive constants.
Assuming that the fuzzy parameters θ f ; θ g , and θ p never reach the
boundaries.
Let us define the minimum approximation error,
h i h i

ω 5 fðX; tÞ 2 fðX θ Þ 1 gðX; tÞ 2 gðX θ Þ u ð20:34Þ

f g

~
~
~

and define the errors: θ f 5 θ f 2 θ ; θ g 5 θ g 2 θ , and θ p 5 θ p 2 θ .
f g p
Then, the equation of the sliding surface Eq. (20.17) can be rewritten as:
h i h i

s ðqÞ 5 ω 1 fðX θ Þ 2 fðXθ f Þ 1 gðX θ Þ 2 gðXθ g Þ u

f g

2 pðs θ p Þ 1 pðs θ Þ 2 pðs θ Þ 1 dðtÞ

p p
T
T
T
~
~
~
5 ω 2 θ ξðsÞ 2 θ ξðXÞ 2 θ ξðXÞu ð20:35Þ
p f g

2 pðs θ Þ 1 dðtÞ
p

20.5 STABILITY ANALYSIS
Consider the commensurate fractional order SISO nonlinear dynamic system
Eq. (20.17) with control input Eq. (20.28), if the robust compensator u a and
the fuzzy-based adaptive laws are chosen as:
1
T
u a 52 B Pe ð20:36Þ
r