This work revisits recent results on maximal multiplicity induced-dominancy for spectral values in reduced-order time-delay Systems and extends it to the general class of second-order retarded differential equa- tions. A parametric multiplicity-induced-dominancy property is characterized, allowing to a delayed stabilizing design with reduced complexity. As a matter of fact, the approach is merely a delayed-output-feedback where the candidates’ delays and gains result from the manifold defining the maximal multiplicity of a real spectral value, then, the dominancy is shown using the argument principle. Sensitivity of the control design with respect to the pa- rameters uncertainties/variation is discussed. Various reduced order examples illustrate the applicative perspectives of the approach.
In Panteley and Loria (2017) a framework for the study of synchronization and collective behaviour of networked heterogeneous systems was introduced. It was underlined that in such scenario an emergent collective behaviour arises, one that is inherent to the network and that is independent of the interconnection strength. Therefore, the natural way to make complete study of synchronization is by investigating, on one hand, the stability of the emergent dynamical system and, on the other, by assessing the difference between the motion of each individual system and that of the emergent one. Thus, if all systems' motions approach that of the emergent dynamics, we say that they reach dynamic consensus. In this paper we study dynamic consensus of a fairly general class of nonlinear heterogeneous oscillators, called Stuart-Landau. We establish that the emergent dynamics consists in that of an "averaged" oscillator with a global attractor that consists in a limit-cycle and, moreover, we determine its frequency of oscillation. Then, we show that the heterogeneous oscillators achieve practical dynamic consensus that is, their synchronization errors measured relative to the collective motion, are ultimately bounded.
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