Critical frequencies of controlled second-order linear systems with delayed feedback

This note focuses on stability issues of linear time-invariant second-order systems with delayed control input. We provide a computational method to determine the set of purely imaginary characteristic roots which reveals the critical values of frequency and time-delay where the stability switching of the controlled system may takes place. This method requires computing only the solutions of an eigenvalue quadratic problem with constant matrices. Based on this method, we further provide a methodology to compute the largest first time-delay interval such that the system under consideration is stable for all delay values that lie in the computed interval. Numerical examples are provided in order to illustrate the applicability of the results.