- Email: [email protected]
Reply to “Comments on ‘Stabilization of a class of nonlinear systems with actuator saturation via active disturbance rejection control’ [Automatica 63 (2016) 302–310]”
Automatica (
)
–
Contents lists available at ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Correspondence Reply to ‘‘Comments on ‘Stabilization of a class of nonlinear systems with actuator saturation via active disturbance rejection control’ [Automatica 63 (2016) 302–310]’’✩ We are thankful to Guo, Wu and Zhou for pointing out four errors in the proof of Theorem 3 and Proposition 4 (see Guo, Wu, and Zhou, 2017). The purpose of this note is to present corrections for these errors in Ran, Wang, and Dong (2016). In the light of our correction of the first error, the corrections for the remaining three errors are much along lines suggested by Guo, Wu, and Zhou (2016). Correction to Error 1. Since f ∈ C 1 (Rn , R), for any x belonging to the domain containing the origin {x ∈ Rn ; ∥x∥ ≤ d}, where d is some positive constant, there exists a constant C1 > 0 such that |f (x)| ≤ C1 ∥x∥. Let ρ = min{d, |Cb| } and δ ∈ (0, ρ ). The sets Ω1 and 1 Ω2 are redefined as follows:
The inequality above guarantees that ∥η∥ → 0 as ε → 0 in the time interval [t1 , t2 ].
References Guo, B.Z., Wu, Z.H., & Zhou, H.C. (2016). Personal communication through the editorial staff of Automatica. Guo, B. Z., Wu, Z. H., & Zhou, H. C. (2017). Comments on ‘‘stabilization of a class of nonlinear systems with actuator saturation via active disturbance rejection control’’. Automatica (submitted for publication). Ran, M., Wang, Q., & Dong, C. (2016). Stabilization of a class of nonlinear systems with actuator saturation via active disturbance rejection control. Automatica, 63, 302–310.
{ } δ Ω1 = x ∈ Rn ; ∥x∥ ≤ ρ − δ , Ω2 = {x ∈ Rn ; ∥x∥ ≤ ρ − }.
2 With the new definitions above, Ω1 and Ω2 are bounded subsets of Rn . Remark 1. Note that if the expression of f (·) is known, the value of C1 can be accurately computed. In the case that f (·) is largely unknown, namely, the exact value of C1 cannot be obtained, its estimated value ˆ C1 satisfying ˆ C1 ≥ C1 can be used.
Maopeng Ran 1 School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, PR China E-mail address: [email protected].
Correction to Error 2.Let Ω0 = int(Ω1 ) = {x ∈ Rn ; ∥x∥ < ρ − δ}. The condition (x(0), η(0)) ∈ Ω1 × Ω0 in original Proposition 4 and Theorem 3 should be changed to x(0) ∈ Ω0 . Correction to Errors 3 and 4. Eq. (14) shows that the maximum over [t1 , t2 ] of the right-hand side of Eq. (14) goes to zero as ε → 0. Since x(0) is an interior point of Ω1 and (12) holds, there exists an ε -independent t0 ∈ (0, t1 ) such that x(t , ε) ∈ Ω1 , ∀t ∈ [0, t0 ]. Consequently, the inequality (14) can be further written as
Qing Wang School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, PR China E-mail address: [email protected].
√
Chaoyang Dong School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, PR China E-mail address: [email protected].
) λ13 N0 β1 λ12 ε exp − t0 + ∥η∥ ≤ √ 2λ12 ε λ11 λ13 λ11 ( ) n+1 2 λ12 ∑ |xi (0) − xˆ i (0)| λ13 ≤√ − exp t 0 λ11 ε 2(n+1−i) 2λ12 ε V1 (η(0))
(
4 March 2017 Available online xxxx
i=1
+
N0 β1 λ12 ε
λ11 λ13
, t ∈ [t1 , t2 ].
1
Fax: +86 010 82333287.
DOI of original article: http://dx.doi.org/10.1016/j.automatica.2015.10.010. ✩ This work was supported by the National Natural Science Foundation of China (Nos. 61273083 and 61374012). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Zongli Lin under the direction of Editor André L. Tits.
http://dx.doi.org/10.1016/j.automatica.2017.06.023 0005-1098/© 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Ran, M., et al., Reply to ‘‘Comments on ‘Stabilization of a class of nonlinear systems with actuator saturation via active disturbance rejection control’ [Automatica 63 (2016) 302–310]’’. Automatica (2017), http://dx.doi.org/10.1016/j.automatica.2017.06.023.
)
–
Contents lists available at ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Correspondence Reply to ‘‘Comments on ‘Stabilization of a class of nonlinear systems with actuator saturation via active disturbance rejection control’ [Automatica 63 (2016) 302–310]’’✩ We are thankful to Guo, Wu and Zhou for pointing out four errors in the proof of Theorem 3 and Proposition 4 (see Guo, Wu, and Zhou, 2017). The purpose of this note is to present corrections for these errors in Ran, Wang, and Dong (2016). In the light of our correction of the first error, the corrections for the remaining three errors are much along lines suggested by Guo, Wu, and Zhou (2016). Correction to Error 1. Since f ∈ C 1 (Rn , R), for any x belonging to the domain containing the origin {x ∈ Rn ; ∥x∥ ≤ d}, where d is some positive constant, there exists a constant C1 > 0 such that |f (x)| ≤ C1 ∥x∥. Let ρ = min{d, |Cb| } and δ ∈ (0, ρ ). The sets Ω1 and 1 Ω2 are redefined as follows:
The inequality above guarantees that ∥η∥ → 0 as ε → 0 in the time interval [t1 , t2 ].
References Guo, B.Z., Wu, Z.H., & Zhou, H.C. (2016). Personal communication through the editorial staff of Automatica. Guo, B. Z., Wu, Z. H., & Zhou, H. C. (2017). Comments on ‘‘stabilization of a class of nonlinear systems with actuator saturation via active disturbance rejection control’’. Automatica (submitted for publication). Ran, M., Wang, Q., & Dong, C. (2016). Stabilization of a class of nonlinear systems with actuator saturation via active disturbance rejection control. Automatica, 63, 302–310.
{ } δ Ω1 = x ∈ Rn ; ∥x∥ ≤ ρ − δ , Ω2 = {x ∈ Rn ; ∥x∥ ≤ ρ − }.
2 With the new definitions above, Ω1 and Ω2 are bounded subsets of Rn . Remark 1. Note that if the expression of f (·) is known, the value of C1 can be accurately computed. In the case that f (·) is largely unknown, namely, the exact value of C1 cannot be obtained, its estimated value ˆ C1 satisfying ˆ C1 ≥ C1 can be used.
Maopeng Ran 1 School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, PR China E-mail address: [email protected].
Correction to Error 2.Let Ω0 = int(Ω1 ) = {x ∈ Rn ; ∥x∥ < ρ − δ}. The condition (x(0), η(0)) ∈ Ω1 × Ω0 in original Proposition 4 and Theorem 3 should be changed to x(0) ∈ Ω0 . Correction to Errors 3 and 4. Eq. (14) shows that the maximum over [t1 , t2 ] of the right-hand side of Eq. (14) goes to zero as ε → 0. Since x(0) is an interior point of Ω1 and (12) holds, there exists an ε -independent t0 ∈ (0, t1 ) such that x(t , ε) ∈ Ω1 , ∀t ∈ [0, t0 ]. Consequently, the inequality (14) can be further written as
Qing Wang School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, PR China E-mail address: [email protected].
√
Chaoyang Dong School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, PR China E-mail address: [email protected].
) λ13 N0 β1 λ12 ε exp − t0 + ∥η∥ ≤ √ 2λ12 ε λ11 λ13 λ11 ( ) n+1 2 λ12 ∑ |xi (0) − xˆ i (0)| λ13 ≤√ − exp t 0 λ11 ε 2(n+1−i) 2λ12 ε V1 (η(0))
(
4 March 2017 Available online xxxx
i=1
+
N0 β1 λ12 ε
λ11 λ13
, t ∈ [t1 , t2 ].
1
Fax: +86 010 82333287.
DOI of original article: http://dx.doi.org/10.1016/j.automatica.2015.10.010. ✩ This work was supported by the National Natural Science Foundation of China (Nos. 61273083 and 61374012). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Zongli Lin under the direction of Editor André L. Tits.
http://dx.doi.org/10.1016/j.automatica.2017.06.023 0005-1098/© 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Ran, M., et al., Reply to ‘‘Comments on ‘Stabilization of a class of nonlinear systems with actuator saturation via active disturbance rejection control’ [Automatica 63 (2016) 302–310]’’. Automatica (2017), http://dx.doi.org/10.1016/j.automatica.2017.06.023.