The paper 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-andwait controller, a single-degree-of-freedom milling model and a turning operation with spindle speed variation.
In this paper we propose an algorithm to compute the distance to instability of a linear system of delay differential equations (DDEs) containing uncertainties in the delay terms as well as in matrices coefficients. For what regards the system matrices, any structure on the perturbation can be considered in order to allow only specific parameters to change; moreover, real-valued matrix perturbations are taken into account. The algorithm relies on the computation of the pseudospectral abscissa of the system and performs a bisection-Newton's method to find the minimum size of the perturbation that generates instability. A few illustrative examples, including a model for a rotating cutting machine, finally show the correctness and the efficiency of the method.
Ménière’s disease (MD) is a pathology of the inner ear, the symptoms of which include tinnitus, vertigo attacks, fluctuating hearing loss, and nausea. Neither cause nor cure are currently known, though animal experiments suggest that disruption of the inner ear circulation, including venous hypertension and endolymphatic hydrops, to be hallmarks of the disease. Recent evidence for humans suggests a potential link to strictures in the extracranial venous outflow routes. The purpose of the present work is to demonstrate that the inner-ear circulation in humans is disrupted by extracranial venous outflow stricture and to discuss the implications of this finding for MD. The hypothesis linking extracranial venous outflow strictures to the altered dynamics of central nervous system (CNS) fluid compartments is investigated theoretically via a global, closed-loop, multiscale mathematical model for the entire human circulation, interacting with the brain parenchyma and cerebrospinal fluid (CSF). The fluid dynamics model for the full human body includes submodels for the heart, pulmonary circulation, arterial system, microvasculature, venous system and the CSF, with a specially refined description of the inner ear vasculature. We demonstrate that extracranial venous outflow strictures disrupt inner ear circulation, and more generally, alter the dynamics of fluid compartments in the whole CNS. Specifically, as compared to a healthy control, the computational results from our model show that subjects with extracranial outflow venous strictures exhibit: altered inner ear circulation, redirection of flow to collaterals, increased intracranial venous pressure and increased intracranial pressure. Our findings are consistent with recent clinical evidence in humans that links extracranial outflow venous strictures to MD, aid the mechanistic understanding of the underlying features of the disease and lend support to recently proposed biophysically motivated therapies aimed at reducing the overall pressure in the inner ear circulation. More work is required to understand the finer details of the condition, such as the associated dynamics of fluids in the perilymphatic and endolymphatic spaces, so as to incorporate such knowledge into the mathematical models in order to reflect the real physiology more closely.
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