Robust partial quadratic eigenvalue assignment for the damped vibroacoustic system

In this paper, we consider the robust partial quadratic eigenvalue assignment problem for the damped vibroacoustic system by active feedback control. Based on the special structures of the mass and stiffness matrices of the damped vibroacoustic system, we provide a new orthogonality relation and give solution to partial quadratic eigenvalue assignment problem which only needs a few unwanted eigenvalues of the open-loop pencil and associated eigenvectors. With the QR decomposition of the control matrix, we propose a new method to characterize the solution to the quadratic eigenvalue assignment problem, which does not need to solve the Sylvester equation. Finally, we propose two different gradient-based methods for the robust and minimum norm partial quadratic eigenvalue assignment problems, in which the closed-loop eigenvalue sensitivity and the feedback norms can be minimized simultaneously. Numerical experiments demonstrate the effectiveness of our methods.