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## Section: New Results

### Poles placement for reduced order Time-delay systems

Participants : Souad Amrane [University Mouloud Mammeri] , Islam Boussaada, Fazia Bedouhen [University Mouloud Mammeri] , Silviu-Iulian Niculescu, Matej Kure [Czech technical university in Prague] , Wim Michiels [KU Leuven] , Thomas Vyhlidal [Czech technical university in Prague] .

It is well known in dynamical system theory that real spectral values correspond to non oscillating solutions. In the paper [11]we made a connexion between the degree of a given quasipolynomial and the admissible number of non oscillating modes for the corresponding Time-delay system. More precisely, we have shown that the assignment of at most $n$ real spectral values is possible for generic quasipolynomial function of degree $n$. Namely, explicit formulas on the quasipolynomial's coefficients guaranteeing the coexistence of $n$ negative spectral values are obtained. Furthermore, a new quasipolynomial factorization technique, analogous to the one we developed for multiple spectral values for the proof of the dominancy of $n$ distinct negative spectral values is obtained.

In the paper [23] a robust alternative of the delayed resonator is proposed by spectral approach where a double root assignment at the excitation frequency is proposed. Such an excitation frequency is projected to widening the stop-band in the active absorber frequency response. It is shown that the performance sensitivity to the mismatch between the design and true excitation frequency is considerably decreased. Additionally, the overall scheme is supplemented by a control loop which improves the stability margin.