Correction to: Stabilization of a class of fractionalorder chaotic systems using a nonsmooth control methodology
Correction to: Stabilization of a class of fractionalorder chaotic systems using a nonsmooth control methodology
Mohammad Pourmahmood Aghababa 0
0 M. P. Aghababa (
This note provides a corrigendum to the paper “Stabilization of a class of fractionalorder chaotic systems using a nonsmooth control methodology” [Nonlinear Dynamics, 89 (2017) 13571370]. It is pointed out that we have relied on a wrong formula in [2] to compute the upper bounds of the finite settling times of the proposed control methods in [1]. Fortunately, the minor errors appeared in [1] are readily corrected via the Mean Value Theorem for definite integrals. It is proved that the mentioned minor errors do not affect the main results and designs of the control methods in [1].
Sliding mode control; Fractionalorder system; Mean value theorem

In the recently published paper [
1
], the author obtained
the following inequality for the Lyapunov function of
the proposed control method (see Eq. (24) in [
1
]):
V1 (t ) − V1 (0)
βη
≤ − √0.5 (q) 0
(V1 (τ ))0.5 (t − τ )q−1 dτ
(1)
Then, the author relied on a formula published in [
2
]
(the formula appears in the last line of page 489 in [
2
])
to compute a bound for the finite settling time of the
proposed control approach (see Eq. (25) in [
1
]).
t
Letting V1 (t ) − V1 (0) = 0 V˙1 (τ ) dτ yields
t V˙1 (τ ) βη
0 (V1 (τ ))0.5 dτ ≤ − √0.5 (q) 0
t
(t − τ )q−1dτ
(2)
However, since (V1 (τ ))0.5 is part of the integrand
function, it is not possible to eliminate it by division. Here, it
is shown that although inequality (2) cannot be obtained
from (1), the main results obtained in [
1
] are still correct
and the finitetime stability is guaranteed.
In this note, we use the wellknown first Mean Value
Theorem (MVT) for definite integrals to achieve an
upper bound for the settling time of the system. First,
the MVT is restated below.
Theorem 1 [
3
]. Consider a continuous function f (t )
on the closed interval [a, b]. There is at least one
number c ∈ (a, b), such that the following equality holds:
a
b
f (τ ) dτ = f (c) (b − a)
(3)
Now, in order to obtain the finite settling time, replacing
t by T1 in (1) to obtain a definite integral and applying
MVT to the righthand side of Eq. (1) (Eq. (24) in [
1
]),
one obtains
T1
Remark 1 Based on the above discussions and
corrections, it is clear that the mentioned minor mistakes
do not influence the main results of the original work
[
1
] and the finitetime stability of the proposed control
methodologies in [
1
] is still ensured.
βηT1
= − √0.5 (q)
(V1 (c1))0.5 (T1 − c1)q−1 (4)
2 Conclusion
βηT1
√0.5 (q)
→ T1 (T1 − c1)q−1 ≤
where c1 ∈ (0, t ) is a finite number.
Setting V1 (T1) ≡ 0 for 0 < T1 ≤ t ≤ ∞, we have
(V1 (c1))0.5 (T1 − c1)q−1 ≤ V1 (0)
Owing to q − 1 < 0 and (T1 − c1) > 0, the
inequality T1 (T1 − c1)q−1 ≥ (T1 − c1) (T1 − c1)q−1 =
(T1 − c1)q is always satisfied. According to this fact
and using (5), one obtains
(T1 − c1)q ≤
√0.5 (q) V1 (0)
βη (V1 (c1))0.5
Therefore, the finite settling time for the sliding motion
of the method proposed in [
1
] is achieved as follows:
T1 ≤
q √0.5 (q) V1 (0)
βη (V1 (c1))0.5
Thus, the origin of the sliding mode dynamics (15) in
[
1
] is finitetime stable with the settling time (7) instead
of that given in (27) of [
1
].
Similarly, the settling times T2, T3, T4 and T5
expressed in Theorems 3, 4 and 5 in [
1
], respectively,
are modified as follows:
T2 ≤
T3 ≤
T4 ≤
T5 ≤
q
q
k (V2 (c2))0.5
q √0.5 (q) V4 (0)
βη (V4 (c4))0.5
√0.5 (q) V3 (0)
(K − ρ − θ ) (V3 (c3))0.5 + c3
√0.5 (q) V5 (0)
(L − ρ − θ − γ ) (V5 (c5))0.5 + c5
where c2, c3, c4, c5 ∈ (0, t ) are finite numbers.
(5)
(6)
(7)
(8)
(9)
(10)
(11)
In this note, some minor errors appeared in the
computation the settling times in [
1
] were pointed out. The
errors were due to the fact that some wrong results of
[
2
] were exploited. Accordingly, we readily modified
the finite settling times in [
1
] using the wellknown
Mean Value Theorem for definite integrals. We proved
that the introduced minor errors do not have serious
consequences on the main results of the original paper
[
1
].
Acknowledgements This work has received no funds from any
institutions. The author is most grateful to Associate Professor
Hasan P. Aghababa for his valuable discussions and useful
suggestions about the usage of MVT.
Compliance with Ethical Standards
Conflict of Interest The author declares that he has no conflict
of interest.
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2. Zhao , Y. , Wang , Y. , Liu , Z. : Finite time stability analysis for nonlinear fractional order differential systems . In: Control Conference (CCC) , 2013 32nd Chinese, Xi'an, China , 26  28 July 2013 . IEEE ( 2013 )
3. Rudin , W.: Principles of Mathematical Analysis, 3rd edn . McGrawHill , New York ( 1976 )