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Comments on “Wirtinger-based integral inequality: Application to time-delay systems [Automatica 49 (2013) 2860–2866]”
Automatica 50 (2014) 300–301
Contents lists available at ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Correspondence
Comments on ‘‘Wirtinger-based integral inequality: Application to time-delay systems [Automatica 49 (2013) 2860–2866]’’✩ Min Zheng a,1 , Kang Li b , Minrui Fei a a
School of Mechatronic Engineering and Automation, Shanghai University, Shanghai 200072, China
b
School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, Belfast BT9 5AH, UK
article
info
Article history: Received 11 August 2013 Received in revised form 29 August 2013 Accepted 4 September 2013 Available online 8 November 2013
abstract In a recent paper (Automatica 49 (2013) 2860–2866), the Wirtinger-based inequality has been introduced to derive tractable stability conditions for time-delay or sampled-data systems. We point out that there exist two errors in Theorem 8 for the stability analysis of sampled-data systems, and the correct theorem is presented. © 2013 Elsevier Ltd. All rights reserved.
Keywords: Stability analysis Wirtinger based inequality Sampled data systems
1. Errors and corrections Recently, the Wirtinger-based integral inequality was introduced for the stability analysis of time-delay or sampled-data systems, which can effectively reduce the conservatism, compared with the traditional Jensen inequality when dealing with the cross b term a w( ˙ u)T Rw( ˙ u)du. In Seuret and Gouaisbaut (2013), a novel Wirtinger-based integral inequality was presented in Corollary 5 and applied to the stability analysis of sampled-data systems. As one of the main results, Theorem 8 has however two errors which are detrimental to the correctness of the theorem. Remark. The symbols and notations used in this comment are consistent with Seuret and Gouaisbaut (2013).
According to the proof of Theorem 8, the following inequality should hold
¯ 4 < 0. Π (τ , TK ) = Π1 + (TK − τ )Π2 + (TK − 2τ )Π3 + τ Π
(1)
Since Π (τ , TK ) is affine, and consequently convex, with respect to τ ∈ [0, TK ], it is sufficient to ensure the two vertices satisfy (1), that is
Π (0, TK ) = Π1 + (TK − 0)Π2 + (TK − 0)Π3 + 0 = Θ1 < 0
(2)
¯4 Π (TK , TK ) = Π1 + (TK − TK )Π2 + (TK − 2TK )Π3 + TK Π −1 T = Π1 − TK Π3 + TK Π4 + TK (Y1 R Y1 + 3Y2 R−1 Y2T ) < 0. (3)
✩ The authors of the original paper were contacted, but did not have any
Obviously, (2) is equal to the first LMI in (23) in Seuret and Gouaisbaut (2013). By using the Schur Complement Lemma, (3) should be equal to the second LMI in (23) in Seuret and Gouaisbaut (2013). Note the two middle terms in the last line of (3), which are −TK Π3 + TK Π4 , so there is a sign error in Θ2 (T ) < 0 in Theorem 8 where the first element in the matrix should be Π1 − T (Π3 − Π4 ), instead of Π1 − T (Π3 + Π4 ).
comment. This work is supported by the National Nature Science Foundation of China under grant Nos. 61074032, 61271347, Shanghai Key Laboratory of Power Station Automation Technology, and Shanghai Science Technology Commission under grant Nos. 10JC1405000, 11ZR1413100. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Graziano Chesi under the direction of Editor André Tits. E-mail addresses: [email protected] (M. Zheng), [email protected] (K. Li), [email protected] (M. Fei). 1 Tel.: +86 15026546706; fax: +44+(0) 28 90 975449.
B. The second error is the faulty definition of matrix variable Π4 in (24) in Theorem 8 of Seuret and Gouaisbaut (2013). According to (25) of Seuret and Gouaisbaut (2013), it can be deduced that the correct definition of matrix Π4 should be Π4 = Y3 M4 + M4T Y3T = He{Y3 M4 }, instead of Π4 = Y3 M4 as stated in Theorem 8. In summary, the correct version of Theorem 8 is given as follows.
A. The first error appears in the LMI condition Θ2 (T ) < 0 of (23) in Theorem 8 in Seuret and Gouaisbaut (2013).
DOI of original article: http://dx.doi.org/10.1016/j.automatica.2013.05.030.
0005-1098/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.automatica.2013.09.047
M. Zheng et al. / Automatica 50 (2014) 300–301
Theorem 8. Let 0 < Tmin ≤ Tmax be two positive scalars. Assume that there exist n × n matrices P > 0, R > 0, S = S T , Q = Q T and X = X T and 3n × n matrices Y1 , Y2 , Y3 , that satisfy
Θ1 (T ) = Π1 + T (Π2 + Π3 ) < 0 Π1 − T (Π3 − Π4 ) TY1 ∗ −TR Θ2 ( T ) = ∗ ∗
3TY2 0 −3TR
For T ∈ {Tmin , Tmax } and where
Π1 = Π10 − He{(Y1 + Y3 )W1 + 3Y2 W2 }, Π10 = He{M1T PM0 − W1T QM2 } − W1T SW1 ,
< 0.
301
Π2 = M0T RM0 + He{M0T SW1 + M0T QM2 }, Π3 = M2T XM2 , Π4 = He{Y3 M4 } with M0 = [Ac As 0], M1 = [I 0 0], M2 = [0 I 0], M3 = [0 0 I ], M4 = [0 As Ac ], W1 = [I − I 0], and W2 = [I I − 2]. Then the system (22) is asymptotically stable for all sequence {tk }k≥0 satisfying (21). References Seuret, A., & Gouaisbaut, F. (2013). Wirtinger-based integral inequality: application to time-delay systems. Automatica, 49, 2860–2866.
Contents lists available at ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Correspondence
Comments on ‘‘Wirtinger-based integral inequality: Application to time-delay systems [Automatica 49 (2013) 2860–2866]’’✩ Min Zheng a,1 , Kang Li b , Minrui Fei a a
School of Mechatronic Engineering and Automation, Shanghai University, Shanghai 200072, China
b
School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, Belfast BT9 5AH, UK
article
info
Article history: Received 11 August 2013 Received in revised form 29 August 2013 Accepted 4 September 2013 Available online 8 November 2013
abstract In a recent paper (Automatica 49 (2013) 2860–2866), the Wirtinger-based inequality has been introduced to derive tractable stability conditions for time-delay or sampled-data systems. We point out that there exist two errors in Theorem 8 for the stability analysis of sampled-data systems, and the correct theorem is presented. © 2013 Elsevier Ltd. All rights reserved.
Keywords: Stability analysis Wirtinger based inequality Sampled data systems
1. Errors and corrections Recently, the Wirtinger-based integral inequality was introduced for the stability analysis of time-delay or sampled-data systems, which can effectively reduce the conservatism, compared with the traditional Jensen inequality when dealing with the cross b term a w( ˙ u)T Rw( ˙ u)du. In Seuret and Gouaisbaut (2013), a novel Wirtinger-based integral inequality was presented in Corollary 5 and applied to the stability analysis of sampled-data systems. As one of the main results, Theorem 8 has however two errors which are detrimental to the correctness of the theorem. Remark. The symbols and notations used in this comment are consistent with Seuret and Gouaisbaut (2013).
According to the proof of Theorem 8, the following inequality should hold
¯ 4 < 0. Π (τ , TK ) = Π1 + (TK − τ )Π2 + (TK − 2τ )Π3 + τ Π
(1)
Since Π (τ , TK ) is affine, and consequently convex, with respect to τ ∈ [0, TK ], it is sufficient to ensure the two vertices satisfy (1), that is
Π (0, TK ) = Π1 + (TK − 0)Π2 + (TK − 0)Π3 + 0 = Θ1 < 0
(2)
¯4 Π (TK , TK ) = Π1 + (TK − TK )Π2 + (TK − 2TK )Π3 + TK Π −1 T = Π1 − TK Π3 + TK Π4 + TK (Y1 R Y1 + 3Y2 R−1 Y2T ) < 0. (3)
✩ The authors of the original paper were contacted, but did not have any
Obviously, (2) is equal to the first LMI in (23) in Seuret and Gouaisbaut (2013). By using the Schur Complement Lemma, (3) should be equal to the second LMI in (23) in Seuret and Gouaisbaut (2013). Note the two middle terms in the last line of (3), which are −TK Π3 + TK Π4 , so there is a sign error in Θ2 (T ) < 0 in Theorem 8 where the first element in the matrix should be Π1 − T (Π3 − Π4 ), instead of Π1 − T (Π3 + Π4 ).
comment. This work is supported by the National Nature Science Foundation of China under grant Nos. 61074032, 61271347, Shanghai Key Laboratory of Power Station Automation Technology, and Shanghai Science Technology Commission under grant Nos. 10JC1405000, 11ZR1413100. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Graziano Chesi under the direction of Editor André Tits. E-mail addresses: [email protected] (M. Zheng), [email protected] (K. Li), [email protected] (M. Fei). 1 Tel.: +86 15026546706; fax: +44+(0) 28 90 975449.
B. The second error is the faulty definition of matrix variable Π4 in (24) in Theorem 8 of Seuret and Gouaisbaut (2013). According to (25) of Seuret and Gouaisbaut (2013), it can be deduced that the correct definition of matrix Π4 should be Π4 = Y3 M4 + M4T Y3T = He{Y3 M4 }, instead of Π4 = Y3 M4 as stated in Theorem 8. In summary, the correct version of Theorem 8 is given as follows.
A. The first error appears in the LMI condition Θ2 (T ) < 0 of (23) in Theorem 8 in Seuret and Gouaisbaut (2013).
DOI of original article: http://dx.doi.org/10.1016/j.automatica.2013.05.030.
0005-1098/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.automatica.2013.09.047
M. Zheng et al. / Automatica 50 (2014) 300–301
Theorem 8. Let 0 < Tmin ≤ Tmax be two positive scalars. Assume that there exist n × n matrices P > 0, R > 0, S = S T , Q = Q T and X = X T and 3n × n matrices Y1 , Y2 , Y3 , that satisfy
Θ1 (T ) = Π1 + T (Π2 + Π3 ) < 0 Π1 − T (Π3 − Π4 ) TY1 ∗ −TR Θ2 ( T ) = ∗ ∗
3TY2 0 −3TR
For T ∈ {Tmin , Tmax } and where
Π1 = Π10 − He{(Y1 + Y3 )W1 + 3Y2 W2 }, Π10 = He{M1T PM0 − W1T QM2 } − W1T SW1 ,
< 0.
301
Π2 = M0T RM0 + He{M0T SW1 + M0T QM2 }, Π3 = M2T XM2 , Π4 = He{Y3 M4 } with M0 = [Ac As 0], M1 = [I 0 0], M2 = [0 I 0], M3 = [0 0 I ], M4 = [0 As Ac ], W1 = [I − I 0], and W2 = [I I − 2]. Then the system (22) is asymptotically stable for all sequence {tk }k≥0 satisfying (21). References Seuret, A., & Gouaisbaut, F. (2013). Wirtinger-based integral inequality: application to time-delay systems. Automatica, 49, 2860–2866.