Page 90 - IJOCTA-15-2

Comparison of fractional order sliding mode controllers on robot manipulator
Therefore, the CFOSMC is defined as: Therefore, the CFOSMC is defined as:

u(t) = u eq (t) + u r (t). (30)
u(t) = u eq (t) + u r (t). (21)
The stability of the proposed control method Stability of approach 2 using Lyapunov the-
can be proved by using the Lyapunov theory.
ory; If similar procedures are performed to the
Substituting Eq.(18) into Eq.(13) produces: stability test of approach 1, it is seen that ap-
proach 2 is also stable.

˙
˙
¨
T
α
V (t) = s (t) θ d + Aθ + BG(θ) − u + β a D ˙e(t) .
t
(22)
3.2.3. Approach 3
If Eq.(21) and the necessary equations above As approach 3, the Caputo fractional order sliding
are substituted in Eq. (22) and simplified: surface can be selected as follows: 69
α
˙
T
V (t) = s (t) (−k r s(t)) . (23) s(t) = µe(t) + ˙e(t) + β a D e(t), (31)
t
Eq.(23) satisfies the condition in Eq.(13). where µ, β are positive constants.
Therefore, the proposed controller is stable. Taking the derivative from Eq.(31), the fol-
lowing equation is obtained:
3.2.2. Approach 2
α
˙ s(t) = µ˙e(t) + ¨e(t) + β a D ˙e(t). (32)
As approach 2, the Caputo fractional order sliding t
surface can be selected as follows: 68
Taking double derivative from Eq.(4) and sub-
α
s(t) = µe(t) + β a D e(t), (24) stituting into Eq.(32) produces:
t
¨
¨
α
where µ, β are positive constant. ˙ s(t) = µ˙e(t) + θ d − θ + β a D ˙e(t). (33)
t
Taking the derivative from Eq.(24), the following
equation is obtained:
Substituting Eq.(2) into Eq.(33) generates:
˙ s(t) = µ˙e(t) + β a D α−1 ¨ e(t). (25)
t
α
˙
¨
˙ s(t) = µ˙e(t) + θ d + Aθ + BG(θ) − u + β a D ˙e(t).
t
Taking double derivative from Eq.(4) and sub- (34)
stituting into Eq.(25) produces:
By equaling ˙s(t) to zero, the equivalent con-
˙ s(t) = µ˙e + β a D α−1 ¨ ¨ (26) trol can be obtained as:
[θ d − θ].
t
¨
˙
α
Substituting Eq.(2) into Eq.(26) generates: u eq = µ˙e(t) + θ d + Aθ + BG(θ) + β a D ˙e(t). (35)
t
˙ s(t) = µ˙e + β a D α−1 ¨ ˙
[θ d + Aθ + BG(θ) − u]. (27)
t
The reaching control law is introduced as: 66
If ˙s(t) is set equal to zero and the operator
D 1−α is applied to Equation (27), the equivalent u r = k r s(t). (36)
control can be obtained as follows: Therefore, the CFOSMC is defined as:

1
˙
¨
u eq = a D 1−α µ˙e(t) + θ d + Aθ + BG(θ). (28) u(t) = u eq (t) + u r (t). (37)
t
β
The reaching control law is introduced as: 66 Stability of approach 3 using Lyapunov the-
ory; If similar procedures are performed to the
u r = k r s(t). (29)
stability test of approach 1, it is seen that ap-
proach 3 is also stable.
285