The present study discusses an optimal design method of a proportional integral derivative (PID) control system for a continuous-time second-order plus dead-time system (SOPDT) that includes an under-damping system. The proposed PID control system is designed to minimize the performance index defined as the tracking performance for the set-point or the regulation performance for the disturbance, where the assigned stability margin is also achieved in order to attain robust stability for the modeling error. Because of the relationship between tracking performance and robust stability, the PID parameters are decided such that the tracking performance is optimized subject to the user-specified stability margin. In the proposed design method, in order to obtain the most general optimal PID possible parameters for controlled plants, we discuss a design method based on a normalized plant model. As a result, the PID parameters are seamlessly optimized between over-damping, critical-damping, and under-damping systems. The effectiveness of the proposed method is demonstrated through numerical examples.

本研究讨论了包含欠阻尼系统的连续时间二阶加停滞时间系统（SOPDT）的比例积分微分（PID）控制系统的最佳设计方法。提出的PID控制系统旨在将性能指标最小化，该性能指标定义为设定点的跟踪性能或扰动的调节性能，在该性能指标中还可以实现分配的稳定性裕度，从而获得对建模误差的鲁棒稳定性。由于跟踪性能与鲁棒稳定性之间的关系，因此确定PID参数，以便根据用户指定的稳定性裕量优化跟踪性能。在建议的设计方法中，为了获得受控工厂的最通用的最佳PID可能参数，我们讨论一种基于标准化植物模型的设计方法。结果，PID参数在过阻尼，临界阻尼和欠阻尼系统之间进行了无缝优化。通过数值算例证明了该方法的有效性。