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Invariance properties in the root sensitivity of time-delay systems with double imaginary roots. (English) Zbl 1191.93049
Summary: If i$$\omega \in {\text i}\mathbb R$$ is an eigenvalue of a time-delay system for the delay $$\tau _{0}$$ then i$$\omega$$ is also an eigenvalue for the delays $$\tau _k:= \tau _{0}+k2\pi /\omega$$, for any $$k\in \mathbb Z$$. We investigate the sensitivity, periodicity and invariance properties of the root i$$\omega$$ for the case that i$$\omega$$ is a double eigenvalue for some $$\tau _k$$. It turns out that under natural conditions (namely, the condition that the root exhibits the completely regular splitting property if the delay is perturbed), the presence of a double imaginary root i$$\omega$$ for some delay $$\tau _{0}$$ implies that i$$\omega$$ is a simple root for the other delays $$\tau _k$$, $$k\neq 0$$. Moreover, we show how to characterize the root locus around i$$\omega$$. The entire local root locus picture can be completely determined from the square root splitting of the double root. We separate the general picture into two cases depending on the sign of a single scalar constant; the imaginary part of the first coefficient in the square root expansion of the double eigenvalue.

##### MSC:
 93B60 Eigenvalue problems 93C73 Perturbations in control/observation systems 93C15 Control/observation systems governed by ordinary differential equations
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##### References:
 [1] Chen, J.; Gu, G.; Nett, C.N., A new method for computing delay margins for stability of linear delay systems, Systems and control letters, 26, 2, 107-117, (1995) · Zbl 0877.93117 [2] Cooke, K.L.; Grossman, Z., Discrete delay, distributed delay and stability switches, Journal of mathematical analysis and applications, 86, 592-627, (1982) · Zbl 0492.34064 [3] Cooke, K.L.; van den Driessche, P., On zeroes of some transcendental equations, Funkcialaj ekvacioj. serio internacia, 29, 77-90, (1986) · Zbl 0603.34069 [4] Fu, P., Chen, J., & Niculescu, S. -I. (2007). High-order analysis of critical stability properties of linear time-delay systems. In Proceedings of the 2007 American control conference. New York City, USA: July 11-13 (pp. 4921-4926). [5] Fu, P.; Niculescu, S.-I.; Chen, J., Stability of linear neutral time-delay systems: exact conditions via matrix pencil solutions, IEEE transactions on automatic control, 51, 6, 1063-1069, (2006) · Zbl 1366.34091 [6] Jarlebring, E.; Hochstenbach, M.E., Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations, Linear algebra and its applications, 431, 3, 369-380, (2009) · Zbl 1170.65063 [7] Louisell, J., A matrix method for determining the imaginary axis eigenvalues of a delay system, IEEE transactions on automatic control, 46, 12, 2008-2012, (2001) · Zbl 1007.34078 [8] Michiels, W.; Niculescu, S.-I., Stability and stabilization of time-delay systems: an eigenvalue-based approach, () · Zbl 1052.93029 [9] Niculescu, S.-I., Delay effects on stability. A robust control approach, (2001), Springer-Verlag London [10] Rekasius, Z. (1980). A stability test for systems with delays. In Proc. of joint Autom. Contr. Conf San Francisco, pages TP9-A. · Zbl 0429.93017 [11] Sipahi, R.; Delice, I.I., Extraction of 3D stability switching hypersurfaces of a time delay system with multiple fixed delays, Automatica, 45, 6, 1449-1454, (2009) · Zbl 1166.93317 [12] Sipahi, R.; Olgac, N., Degenerate cases in using the direct method, Journal of dynamic systems, measurement, and control, 125, 2, 194-201, (2003) [13] Thowsen, A., An analytic stability test for class of time-delay systems, IEEE transactions on automatic control, 26, 3, 735-736, (1981) · Zbl 0481.93049 [14] Walton, K.; Marshall, J., Direct method for TDS stability analysis, IEE proc., part D, 134, 101-107, (1987) · Zbl 0636.93066
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