Optimization methods for partial quadratic eigenvalue assignment in vibrations

Vibrating structures, e.g., buildings, bridges, highways, and others, sometime experience dangerous vibrations when acted upon by external forces. A smart way to control such vibrations is to apply active vibration control. The most important aspect of an active vibration control strategy is to effectively compute the feedback control force to absorb these vibrations. For practical applications, the feedback control force must be computed in a numerically robust way. It is, therefore, desired that these computed feedback matrices have small norms and the closed-loop condition number is as small as possible. These considerations give rise to some beautiful but extremely difficult (usually nonconvex) nonlinear optimization problems. In this paper, we survey some of the recent developments on numerical solutions of the optimization problems arising in partial eigenvalue assignment for second-order control systems.