Author’s reply to “A counterexample to Generalized eigenvalue-based stability tests for 2-D linear systems: Necessary and sufficient conditions”

✍ Scribed by Peilin Fu; Jie Chen; Silviu-Iulian Niculescu

• Publisher: Elsevier Science
• Year: 2010
• Tongue: English
• Weight: 635 KB
• Volume: 46
• Category: Article
• ISSN: 0005-1098
• DOI: 10.1016/j.automatica.2009.10.003

✦ Synopsis

Author's reply to ''A counterexample to Generalized eigenvalue-based stability tests for 2-D linear systems: Necessary and sufficient conditions'' $

We would like to thank the authors of Rapisarda and Takaba (2009), who correctly pointed out an error in our earlier paper (Fu, Chen, & Niculescu, 2006) and provided a valid counterexample. It is clear from both papers that due to the authors' oversight, the error occurred with the side condition det(K (1)) > 0 in Theorem 1 (Fu et al., 2006), which as given in Bose (1982), should be stated as K (1) > 0. The error led further to the incorrect conditions det(∆(1)) > 0 and det(P -Q ) > 0 in the same theorem, resulted directly from the equivalence of these three conditions established in Fu et al. (2006).

To rectify the error, it suffices to replace the aforementioned three side conditions with Bose's original statement K (1) > 0.

Specifically, the incorrect statements Theorem 1(iii), (iv) and (v) can be remedied as follows:

(iii) det(K (z 1 )) = 0 for all z 1 ∈ ∂D, and K (1) > 0;

(iv) det(∆(z 1 )) = 0 for all z 1 ∈ ∂D, and K (1) > 0;

(v) σ (P, Q ) ∩ ∂D = ∅, and K (1) > 0.

Note that we only reinstalled the condition K (1) > 0 in Theorem 1(iii), (iv) and (v).

References