Spectrum-based stability analysis and stabilization of a class of time-periodic time delay systems

TL;DR

This paper introduces an eigenvalue-based method for analyzing and stabilizing linear time-periodic systems with delays, combining global and local techniques for computing Floquet multipliers and deriving derivatives for optimization.

Contribution

It presents a novel dual eigenvalue approach for stability analysis of periodic delay systems, including new characterizations of eigenvectors and derivatives for optimization.

Findings

Effective computation of Floquet multipliers demonstrated

New eigenvector characterizations introduced

Method applicable to stability optimization tasks

Abstract

We develop an eigenvalue-based approach for the stability assessment and stabilization of linear systems with multiple delays and periodic coefficient matrices. Delays and period are assumed commensurate numbers, such that the Floquet multipliers can be characterized as eigenvalues of the monodromy operator and by the solutions of a finite-dimensional non-linear eigenvalue problem, where the evaluation of the characteristic matrix involves solving an initial value problem. We demonstrate that such a dual interpretation can be exploited in a two-stage approach for computing dominant Floquet multipliers, where global approximation is combined with local corrections. Correspondingly, we also propose two novel characterizations of left eigenvectors. Finally, from the nonlinear eigenvalue problem formulation, we derive computationally tractable expressions for derivatives of Floquet multipliers with respect to parameters, which are beneficial in the context of stability optimization. Several numerical examples show the efficacy and applicability of the presented results.