Minimum Norm Partial Quadratic Eigenvalue Assignment with Time Delay in Vibrating Structures Using the Receptance and the System Matrices

The partial quadratic eigenvalue assignment problem (PQEAP) is to compute a pair of feedback matrices such that a small number of unwanted eigenvalues in a structure are reassigned to suitable locations while keeping the remaining large number of eigenvalues and the associated eigenvectors unchanged. The problem arises in active vibration control of structures. For real-life applications, it is not enough just to compute the feedback matrices. They should be computed in such a way that both closed-loop eigenvalue sensitivity and feedback norms are as small as possible. Also, for practical effectiveness, the time-delay between the measurement of the state and implementation of the feedback controller should be considered while solving the PQEAP. These problems are usually solved using only system matrices and do not necessarily take advantage of the receptances which are available by measurements. In this paper, we propose hybrid methods, combining the system matrices and measured receptances, for solutions of the multi-input PQEAP and the minimum-norm PQEAP, both for systems with and without time-delay. These hybrid methods are more efficient than the standard methods which only use the system matrices and no the receptances. These hybrid methods also offer several other computational advantages over the standard methods. Our results generalize the recent work by Ram, Mottershead, and Tehrani [Linear Algebra Appl., 434 (2011), pp. 1689–1696]. The results of numerical experiments demonstrate the effectiveness of the proposed methods.