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.

本笔记重点讨论具有延迟控制输入的线性时不变二阶系统的稳定性问题。我们提供了一种计算方法来确定一组纯虚特征根，它揭示了可能发生受控系统稳定性切换的频率和时延的临界值。此方法只需要计算具有常数矩阵的特征值二次问题的解。基于这种方法，我们进一步提供了一种计算最大第一时间延迟间隔的方法，以便所考虑的系统对于位于计算间隔内的所有延迟值都是稳定的。提供了数值示例以说明结果的适用性。