Page 89 - IJOCTA-15-2

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M. Yavuz, M. Ozt¨urk, B. Ya¸skıran / IJOCTA, Vol.15, No.2, pp.281-293 (2025)
and then the fractional order sliding mode con-
˙ s(t) = µ˙e(t) + ¨e(t). (6) trol using 3 different sliding surfaces defined by
the Caputo fractional order operator will be dis-
cussed and these will be used to control the 2-
Taking double derivative from Eq.(4) and sub-
DOF robot manipulator.
stituting into Eq.(6) produces:
Definition 1. The Caputo fractional derivative
¨
¨
˙ s(t) = µ˙e(t) + θ d − θ. (7) of a function f(t) is defined as follows: 67
Z t (n)
Substituting Eq.(2) into Eq.(7) generates: α 1 f (τ)
a D f(t) = dτ
t
Γ(n − α) a (t − τ) α−n+1
˙
¨
˙ s(t) = µ˙e(t) + θ d + Aθ + BG(θ) − u. (8) for (n − 1 < α < n),
R ∞ −t z−1
where Gamma function Γ(z) = e t dt,
By equaling ˙s(t) to zero, the equivalent con- 0
(Re(z) > 0).
trol can be obtained as:
Definition 2. The Caputo fractional integral is
¨
˙
u eq = µ˙e(t) + θ d + Aθ + BG(θ). (9) equivalent to the Riemann fractional integral and
is defined as follows: 67
The equivalent control solely is not able to 1 Z t
suppress the external disturbances. Therefore, a a D −α f(t) = (t−τ) α−1 f(τ)dτ for α > 0.
t
reaching control law will be implemented to solve Γ(α) a
that problem. The reaching control law is intro- 3.2.1. Approach 1
duced as: 66
u r = k r s(t), (10) As approach 1, the Caputo fractional order sliding
surface can be selected as follows: 63
α
s(t) = ˙e(t) + β a D e(t), (15)
where k r is the reaching control gain that is a t
positive constant.
Therefore, the SMC is defined as: where β is a positive constant.
Taking the derivative from Eq.(15), the fol-
u(t) = u eq (t) + u r (t). (11)
lowing equation is obtained:
The proposed control method’s stability can ˙ s(t) = ¨e(t) + β a D ˙e(t). (16)
α
t
be proved by using the Lyapunov theory:
1 T Taking double derivative from Eq.(4) and sub-
V (t) = s (t)s(t). (12) stituting into Eq.(16) produces:
2
α
¨
¨
The condition for stability satisfaction is as ˙ s(t) = θ d − θ + β a D ˙e(t). (17)
t
follows:
T
˙
V (t) = s (t) ˙s(t) < 0. (13)
Substituting Eq.(2) into Eq.(17) generates:
If the necessary equations above are substi-
tuted in Eq.(13) and simplified: ¨ ˙ α
˙ s(t) = θ d + Aθ + BG(θ) − u + β a D ˙e(t). (18)
t
˙
T
V (t) = s (t) (−k r s(t)) . (14)
By equaling ˙s(t) to zero, the equivalent con-
trol can be obtained as:
Eq.(14) satisfies the condition in Eq.(13).
Therefore, the proposed controller is stable.
α
˙
¨
u eq = θ d + Aθ + BG(θ) + β a D ˙e(t). (19)
t
3.2. Caputo fractional order sliding mode 66
The reaching control law is introduced as:
control
In this section, first, the definition of Caputo frac- u r = k r s(t). (20)
tional order derivative and integral will be given,
284