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No Spillover Eigenvalues Assignment of Second-Order Systems with Dense Force Actuators Matrices Using Brauer’s Theorem

This paper presents an approach for eigenvalue assignment in second-order linear systems with no spillover property. Second-order differential equations arise from dynamical modeling of vibrating structures by finite element or lumped parameter first principles approach in several practical problems. Certain structures can face practical issues when subjected to external perturbation forces, as resonance or flutter type vibrations. The control of excessive vibrations can be attempted by techniques of active vibration control using linear feedback. To change only a few eigenvalues and eigenvectors that cause excessive vibrations, the requirement of no spillover property is a somewhat attractive issue. Furthermore, only the part of the eigenstructure whose eigenvalues must be reassigned is necessary to be known for an efficient parametrization of the feedback matrices. Brauer’s theorem, a milestone result of linear algebra, as well as some recent related results, is applied here to achieve partial eigenvalue assignment using dense force actuator (influence) matrices. The proposal can be applied to general systems with no restriction on the mass, damping, and stiffness with symmetry or definiteness. The procedures to implement the proposal are synthesized in a step-by-step form, and some numerical examples are given to illustrate its application.