Vibration PID control design by receptance-based partial eigenstructure assignment

In recent decades, the partial eigenstructure assignment problem has attracted considerable attention due to its practicality and significant benefits in vibration control. For linear vibration systems, enhancing system response can be achieved by strategically assigning zeros and poles. To meet the dynamic requirements of vibration systems, this paper presents a receptance-based approach for partial eigenstructure assignment using a novel PID controller. A comprehensive description and validation of the proposed method are provided. The PID controller is applied to the free vibration equation of a multi-degree-of-freedom linear system. Through the use of the Laplace transform and the Sherman-Morrison formula, the flexibility matrix of the closed-loop system is obtained. Under certain assumptions and constraints, the theoretical gain vector for PID control is determined using the Moore-Penrose generalized inverse. Furthermore, a practical strategy for optimizing sensor placement is proposed. The introduction of integral control inevitably results in the generation of new zero-pole pairs. A comprehensive explanation and analysis of these newly introduced zeros and poles, as well as the modified system, are provided. Two numerical examples validate the effectiveness of the proposed methodology.