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Comments on “PBH tests for nonlinear systems”
Automatica xxx (xxxx) xxx
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Comments on ‘‘PBH tests for nonlinear systems’’✩ ∗
Mohammad Amin Sarafrazi a , , Zbigniew Bartosiewicz b , Ülle Kotta c a
Somayeh st., Azadi Blvd., 71439-46936 Shiraz, Iran Bialystok University of Technology, Faculty of Computer Science, Wiejska 45A, 15-351 Białystok, Poland c Department of Software Science, Tallinn University of Technology, Tallinn 12618, Estonia b
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info
a b s t r a c t A counterexample is presented to show that Corollary 3.6 of Theorem 3.4 in Kawano and Ohtsuka (2017) is incorrect. The proof of Theorem 3.4 could be easily fixed, but use of condition (11) is inevitable, even in the single-input case. © 2019 Published by Elsevier Ltd.
Article history: Received 14 August 2018 Received in revised form 21 May 2019 Accepted 6 August 2019 Available online xxxx Keywords: Nonlinear systems Accessibility Nonlinear eigenvalues PBH tests
In Theorem 3.4 of Kawano and Ohtsuka (2017) a new strong accessibility criterion is proposed in terms of non-existence of left nonlinear eigenvector v , satisfying equations (10) and (11). Then, in Corollary 3.6, based on the proof of Theorem 3.4, it is claimed that for single input systems the condition (11) can be dropped from Theorem 3.4. The following counterexample shows that Corollary 3.6 of Kawano and Ohtsuka (2017) is incorrect. Counterexample: Consider the nonlinear system x2 0 0
[ ] Σ : x˙ = f (x) + ug(x) =
⎡ ⎤ 0
+ u ⎣x1 ⎦ x21
over a domain U := {x ∈ C3 | x1 ̸ = 0}, and also consider the vector v = [x2 − x1 1]T . One can easily check that v is a left eigenvector for the system Σ , with the eigenvalue α = 0. Moreover, v T g = 0. Therefore, according to Corollary 3.6, the system Σ cannot be strongly accessible. However, after computing the first few brackets from the standard strong accessibility distribution of the system Σ , we obtain g = [0 x1 x21 ]T , [f , g ] = [−x1 x2 2x2 x1 ]T , [g , [f , g ]] = [0 2x1 4x21 ]T , which shows that the strong accessibility distribution is full rank, and therefore the system Σ is strongly accessible. DOI of original article: https://doi.org/10.1016/j.automatica.2017.02.027. ✩ The work of Z. Bartosiewicz has been supported by the grant S/WI/1/2016. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor A. Pedro Aguiar under the direction of Editor André L. Tits. ∗ Corresponding author. E-mail addresses: [email protected] (M.A. Sarafrazi), [email protected] (Z. Bartosiewicz), [email protected] (Ü. Kotta).
This inconsistency is caused by a non-standard definition of strong accessibility distribution C , and then the use of this nonstandard accessibility distribution in the proof of Theorem 3.4. In fact, from the first paragraph of Section 3.1, and also Proposition 3.1, one infers that the standard notion of accessibility, as in Isidori (1995) and Nijmeijer and Van der Schaft (1990) is studied. Moreover, Proposition 3.1 in Kawano and Ohtsuka (2017) holds for the standard definition of strong accessibility distribution. But, the distribution C , defined as the ‘‘accessibility distribution’’ in the second paragraph of Section 3.1, does not match the standard strong accessibility distribution, as in Isidori (1995) and Nijmeijer and Van der Schaft (1990). For example, unlike the standard accessibility distribution, C does not need to be involutive, and does not contain vector fields such as adg adf g. The proof of Theorem 3.4 is based on the non-standard definition of accessibility distribution C . So, for Theorem 3.4 to be applicable as a test for standard accessibility notion, the proof of this theorem needs to be revised. The proof of (b) ⇒ (a) relies on accessibility decomposition and accessibility rank condition of Proposition 3.1, which does not hold for C . However, validity of the proof is easily recovered once the nonstandard accessibility distribution C is replaced by the standard one. In the proof of (a) ⇒ (b), the authors show that the existence of a left eigenvector v satisfying (10) and (11), is equivalent to v being an annihilator of C . But since C is a subset of the standard strong accessibility distribution, and is not necessarily equal to it, to recover the validity of this part of the theorem, one needs to show that such a left eigenvector v also annihilates the other vector fields of the standard accessibility distribution that are missing from C . Fortunately, by replacing C in the proof of (a)
https://doi.org/10.1016/j.automatica.2019.108617 0005-1098/© 2019 Published by Elsevier Ltd.
Please cite this article as: M.A. Sarafrazi, Z. Bartosiewicz and Ü. Kotta, Comments on ‘‘PBH tests for nonlinear systems’’. Automatica (2019) 108617, https://doi.org/10.1016/j.automatica.2019.108617.
2
M.A. Sarafrazi, Z. Bartosiewicz and Ü. Kotta / Automatica xxx (xxxx) xxx
⇒ (b) by the standard accessibility distribution, and using the same approach as used in the proof of (a) ⇒ (b) in Kawano and
References
Ohtsuka (2017), the claim can be proved easily, but in the revised proof, the use of condition (11) is inevitable, even for the singleinput case. Therefore, Theorem 3.4 is correct, but Corollary 3.6 is not.
Isidori, A. (1995). Nonlinear control systems. New York: Springer-Verlag. Kawano, Y., & Ohtsuka, T. (2017). PBH tests for nonlinear systems. Automatica, 80, 135–142. Nijmeijer, H., & Van der Schaft, A. J. (1990). Nonlinear dynamical control systems. New York: Springer-Verlag.
Please cite this article as: M.A. Sarafrazi, Z. Bartosiewicz and Ü. Kotta, Comments on ‘‘PBH tests for nonlinear systems’’. Automatica (2019) 108617, https://doi.org/10.1016/j.automatica.2019.108617.
Contents lists available at ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Correspondence
Comments on ‘‘PBH tests for nonlinear systems’’✩ ∗
Mohammad Amin Sarafrazi a , , Zbigniew Bartosiewicz b , Ülle Kotta c a
Somayeh st., Azadi Blvd., 71439-46936 Shiraz, Iran Bialystok University of Technology, Faculty of Computer Science, Wiejska 45A, 15-351 Białystok, Poland c Department of Software Science, Tallinn University of Technology, Tallinn 12618, Estonia b
article
info
a b s t r a c t A counterexample is presented to show that Corollary 3.6 of Theorem 3.4 in Kawano and Ohtsuka (2017) is incorrect. The proof of Theorem 3.4 could be easily fixed, but use of condition (11) is inevitable, even in the single-input case. © 2019 Published by Elsevier Ltd.
Article history: Received 14 August 2018 Received in revised form 21 May 2019 Accepted 6 August 2019 Available online xxxx Keywords: Nonlinear systems Accessibility Nonlinear eigenvalues PBH tests
In Theorem 3.4 of Kawano and Ohtsuka (2017) a new strong accessibility criterion is proposed in terms of non-existence of left nonlinear eigenvector v , satisfying equations (10) and (11). Then, in Corollary 3.6, based on the proof of Theorem 3.4, it is claimed that for single input systems the condition (11) can be dropped from Theorem 3.4. The following counterexample shows that Corollary 3.6 of Kawano and Ohtsuka (2017) is incorrect. Counterexample: Consider the nonlinear system x2 0 0
[ ] Σ : x˙ = f (x) + ug(x) =
⎡ ⎤ 0
+ u ⎣x1 ⎦ x21
over a domain U := {x ∈ C3 | x1 ̸ = 0}, and also consider the vector v = [x2 − x1 1]T . One can easily check that v is a left eigenvector for the system Σ , with the eigenvalue α = 0. Moreover, v T g = 0. Therefore, according to Corollary 3.6, the system Σ cannot be strongly accessible. However, after computing the first few brackets from the standard strong accessibility distribution of the system Σ , we obtain g = [0 x1 x21 ]T , [f , g ] = [−x1 x2 2x2 x1 ]T , [g , [f , g ]] = [0 2x1 4x21 ]T , which shows that the strong accessibility distribution is full rank, and therefore the system Σ is strongly accessible. DOI of original article: https://doi.org/10.1016/j.automatica.2017.02.027. ✩ The work of Z. Bartosiewicz has been supported by the grant S/WI/1/2016. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor A. Pedro Aguiar under the direction of Editor André L. Tits. ∗ Corresponding author. E-mail addresses: [email protected] (M.A. Sarafrazi), [email protected] (Z. Bartosiewicz), [email protected] (Ü. Kotta).
This inconsistency is caused by a non-standard definition of strong accessibility distribution C , and then the use of this nonstandard accessibility distribution in the proof of Theorem 3.4. In fact, from the first paragraph of Section 3.1, and also Proposition 3.1, one infers that the standard notion of accessibility, as in Isidori (1995) and Nijmeijer and Van der Schaft (1990) is studied. Moreover, Proposition 3.1 in Kawano and Ohtsuka (2017) holds for the standard definition of strong accessibility distribution. But, the distribution C , defined as the ‘‘accessibility distribution’’ in the second paragraph of Section 3.1, does not match the standard strong accessibility distribution, as in Isidori (1995) and Nijmeijer and Van der Schaft (1990). For example, unlike the standard accessibility distribution, C does not need to be involutive, and does not contain vector fields such as adg adf g. The proof of Theorem 3.4 is based on the non-standard definition of accessibility distribution C . So, for Theorem 3.4 to be applicable as a test for standard accessibility notion, the proof of this theorem needs to be revised. The proof of (b) ⇒ (a) relies on accessibility decomposition and accessibility rank condition of Proposition 3.1, which does not hold for C . However, validity of the proof is easily recovered once the nonstandard accessibility distribution C is replaced by the standard one. In the proof of (a) ⇒ (b), the authors show that the existence of a left eigenvector v satisfying (10) and (11), is equivalent to v being an annihilator of C . But since C is a subset of the standard strong accessibility distribution, and is not necessarily equal to it, to recover the validity of this part of the theorem, one needs to show that such a left eigenvector v also annihilates the other vector fields of the standard accessibility distribution that are missing from C . Fortunately, by replacing C in the proof of (a)
https://doi.org/10.1016/j.automatica.2019.108617 0005-1098/© 2019 Published by Elsevier Ltd.
Please cite this article as: M.A. Sarafrazi, Z. Bartosiewicz and Ü. Kotta, Comments on ‘‘PBH tests for nonlinear systems’’. Automatica (2019) 108617, https://doi.org/10.1016/j.automatica.2019.108617.
2
M.A. Sarafrazi, Z. Bartosiewicz and Ü. Kotta / Automatica xxx (xxxx) xxx
⇒ (b) by the standard accessibility distribution, and using the same approach as used in the proof of (a) ⇒ (b) in Kawano and
References
Ohtsuka (2017), the claim can be proved easily, but in the revised proof, the use of condition (11) is inevitable, even for the singleinput case. Therefore, Theorem 3.4 is correct, but Corollary 3.6 is not.
Isidori, A. (1995). Nonlinear control systems. New York: Springer-Verlag. Kawano, Y., & Ohtsuka, T. (2017). PBH tests for nonlinear systems. Automatica, 80, 135–142. Nijmeijer, H., & Van der Schaft, A. J. (1990). Nonlinear dynamical control systems. New York: Springer-Verlag.
Please cite this article as: M.A. Sarafrazi, Z. Bartosiewicz and Ü. Kotta, Comments on ‘‘PBH tests for nonlinear systems’’. Automatica (2019) 108617, https://doi.org/10.1016/j.automatica.2019.108617.