Pseudospectral method for assessing stability robustness for linear time‐periodic delayed dynamical systems

• Published in: International journal for numerical methods in engineering 2020-08, Vol.121 (16), p.3505-3528
• Main Authors: Borgioli, Francesco, Hajdu, David, Insperger, Tamas, Stepan, Gabor, Michiels, Wim
• Format: Article
• Language: English
• Subjects: Computation; Computer simulation; differential equations; Dynamic stability; dynamical systems; Eigenvalues; Mathematical analysis; Mathematical models; Matrix methods; Operators (mathematics); optimization; Pendulums; Periodic functions; Robustness (mathematics); spectral; Spectral methods; stability; Stability analysis; Uncertainty; vibrations
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.6368

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.