P3δ

P3δ (Partial pole placement via delay action) is a Python implementation of recent methods for the stability analysis and stabilization of linear time-delay systems exploiting the delay action. Its control design strategy is based on properties of the spectral distribution of the time-delay system.

The idea to design such an implementation dates back to the first publications on the topic where the characterization of multiple spectral values of a time-delay system in terms of confluent Vandermonde and Birkhoff matrices, and their structural properties, enabled a constructive alternative to the Pólya and Szegő count for the maximal admissible multiplicity [1, 2].

The multiplicity of a spectral value itself is not important as such, but its connection with the dominancy of this spectral value is a meaningful tool for control synthesis. Indeed, two main properties called respectively the multiplicity-induced-dominancy (MID) and the coexisting-real-roots-induced-dominancy (CRRID) were introduced, respectively in [3, 4] and [5, 6], and applied, for instance, in the control of active vibrating flexible structures.