Premium
Pseudospectral method for assessing stability robustness for linear time‐periodic delayed dynamical systems
Author(s) -
Borgioli Francesco,
Hajdu David,
Insperger Tamas,
Stepan Gabor,
Michiels Wim
Publication year - 2020
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.6368
Subject(s) - floquet theory , robustness (evolution) , linear system , control theory (sociology) , mathematics , computation , lyapunov function , solver , eigenvalues and eigenvectors , mathematical analysis , mathematical optimization , computer science , nonlinear system , algorithm , physics , biochemistry , chemistry , control (management) , artificial intelligence , gene , quantum mechanics
Summary The article presents a pseudospectral approach to assess the stability robustness of linear time‐periodic delay systems, where periodic functions potentially present discontinuities and the delays may also periodically vary in time. The considered systems are subject to linear real‐valued time‐periodic uncertainties affecting the coefficient matrices, and the presented method is able to fully exploit structure and potential interdependencies among the uncertainties. The assessment of robustness relies on the computation of the pseudospectral radius of the monodromy operator, namely, the largest Floquet multiplier that the system can attain within a given range of perturbations. Instrumental to the adopted novel approach, a solver for the computation of Floquet multipliers is introduced, which results into the solution of a generalized eigenvalue problem which is linear w.r.t. (samples of) the original system matrices. We provide numerical simulations for popular applications modeled by time‐periodic delay systems, such as the inverted pendulum subject to an act‐and‐wait controller, a single‐degree‐of‐freedom milling model and a turning operation with spindle speed variation.