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.
- I. Boussaada, G. Mazanti, S.-I. Niculescu, A. Hammoumou, T. Millet, J. Raj, J. Huynh. New Features of P3δ Software. Insights and Demos. 17th IFAC Workshop on Time Delay Systems (TDS 2022). IFAC-PapersOnLine, 55(36):246–251, 2022. [doi] [arXiv] [HAL]
- I. Boussaada, G. Mazanti, S.-I. Niculescu, A. Leclerc, J. Raj, M. Perraudin. New Features of P3δ software: Partial Pole Placement via Delay Action. 16th IFAC Workshop on Time Delay Systems (IFAC TDS 2021). IFAC-PapersOnLine, 54(18):215–221, 2021. [doi] [arXiv] [HAL]
- I. Boussaada, G. Mazanti, S.-I. Niculescu, J. Huynh, F. Sim, M. Thomas. Partial pole placement via delay action: A Python software for delayed feedback stabilizing design. 2020 24th International Conference on System Theory, Control and Computing (ICSTCC), 196–201, 2020. [doi] [arXiv] [HAL]