A discussion on the collocated active control of beams with piezoelectric patches

The structure of the control matrix of a direct strain rate feedback, based on collocated piezoelectric patches operated in surface mode, is analytically obtained in the elementary case of a simply supported beam. This kind of control, often improperly called "velocity feedback" because of a formal analogy with the equation of a classical velocity controlled system, has been exploited by several authors with contradictory results. The study of the measurement and actuation matrices, together with the parametric eigenvalue analysis of the control matrix, allowed the identification of an instability mechanism that could justify experimental evidences. The most critical parameter seems to be the error in collocation, that can reduce the stability as much as wider is the frequency range of interest.