Correction to: Stabilization of a class of fractional-order chaotic systems using a non-smooth control methodology

Nonlinear Dynamics, Oct 2017

This note provides a corrigendum to the paper “Stabilization of a class of fractional-order chaotic systems using a non-smooth control methodology” [Nonlinear Dynamics, 89 (2017) 1357–1370]. 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].

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Correction to: Stabilization of a class of fractional-order chaotic systems using a non-smooth control methodology

Correction to: Stabilization of a class of fractional-order chaotic systems using a non-smooth control methodology Mohammad Pourmahmood Aghababa 0 0 M. P. Aghababa ( This note provides a corrigendum to the paper “Stabilization of a class of fractional-order chaotic systems using a non-smooth control methodology” [Nonlinear Dynamics, 89 (2017) 1357-1370]. 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; Fractional-order 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 finite-time stability is guaranteed. In this note, we use the well-known 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 right-hand 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 finite-time 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 finite-time 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 well-known 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. 1. Aghababa , M.P. : Stabilization of a class of fractional-order chaotic systems using a non-smooth control methodology . Nonlinear Dyn . 89 , 1357 - 1370 ( 2017 ) 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 . McGraw-Hill , New York ( 1976 )


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Mohammad Pourmahmood Aghababa. Correction to: Stabilization of a class of fractional-order chaotic systems using a non-smooth control methodology, Nonlinear Dynamics, 2017, 1-2, DOI: 10.1007/s11071-017-3838-x