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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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