Active control of linear vibrating systems for antiresonance assignment with regional pole placement

This paper proposes a novel method for antiresonance assignment and regional pole placement in linear time-invariant vibrating systems, by means of state feedback control. The method also handles asymmetric systems and unstable ones too. Additionally, it works with both point and cross-receptances and handles the simultaneous assignment of more antiresonances in the same receptance. The method relies on two stages. In the first stage, the desired pairs of closed-loop zeros of a prescribed receptance are exactly assigned. In the second stage, all the closed-loop system poles are placed within the desired region of the complex plane. This feature allows the controller to impose the system stability and to feature the desired dynamic properties through a regional pole placement. Since the gain correction computed in the second stage is obtained as a solution of the homogeneous system related to the zero-assignment problem, it does not cause any spillover on the assigned zeros. The first step exploits the receptance method for gain computing, while the second step uses the first-order model formulation to exploit all the benefits of the Linear Matrix Inequality theory, by formulating a bilinear matrix problem solved as a semidefinite optimization aimed at reducing the control effort. The chief original contribution of the proposed method is that it embeds an a-priori imposition of both the closed-loop stability and the pole clustering in the desired regions, by overcoming the limitations of most of the methods appeared in the literature. The method effectiveness is demonstrated through five meaningful test cases.