Abstract
We address in this chapter the invariance property for a specific case where all the critical imaginary roots are simple. Although such an invariance property has been proved in the literature, some new perspectives are obtained. We first study the asymptotic behavior of the frequency-sweeping curves by means of the dual Puiseux series, a new concept proposed in this chapter. Then, a useful equivalence relation between the Puiseux series and the dual Puiseux series is found. Based on it, we give a new proof on the invariance property. The aforementioned new perspectives are crucial to confirm the invariance property in the general case.
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Notes
- 1.
Notice that “\(\mathrm{Re} ({C_g}) = 0\)” is the degeneracy condition for the specific case where \(n=1\). The general degeneracy condition will be presented in Appendix B.
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Li, XG., Niculescu, SI., Çela, A. (2015). Invariance Property for Critical Imaginary Roots with Index \(n=1\) . In: Analytic Curve Frequency-Sweeping Stability Tests for Systems with Commensurate Delays. SpringerBriefs in Electrical and Computer Engineering(). Springer, Cham. https://doi.org/10.1007/978-3-319-15717-7_7
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DOI: https://doi.org/10.1007/978-3-319-15717-7_7
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Publisher Name: Springer, Cham
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Online ISBN: 978-3-319-15717-7
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