No spillover partial eigenvalue assignment in second-order linear systems using the Brauer’s theorem and dense influence matrices

¹ Universidade Federal da Bahia, Departamento de Engenharia Elétrica, Rua Aristides Novis, 2, Federação, Salvador-BA, Brazil
² Instituto Federal de Educação, Ciência e Tecnologia da Bahia, Rua Emídios dos Santos, S/N, Barbalho, Salvador-BA, Brazil

^* e-mail: Jessicalimamota@gmail.com
^** e-mail: moliveiradealmeida1986@gmail.com
^*** e-mail: tlsantos@ufba.br
^**** e-mail: araujo@ieee.org

Abstract

The control of vibrations for systems with second-order linear dynamics is a cornerstone for several engineering application, as structural vibration control in civil and aerospace structures. In such systems, sometimes only a few eigenvalues must be mitigated in situations as resonances, flutter, and another dangerous vibratory phenomenon. The use of active vibration control is an attractive alternative due to its flexibility and the availability of general design approaches. In the case of partial eigenvalue assignment, no spillover can be achieved, that is, only a small spectrum or eigenstructure partition is reallocated, with the remaining being unperturbed. In this paper, the no spillover property of Brauer’s spectrum perturbation theorem is applied to compute a dense influence matrix structure of actuation, reassigning then the dangerous target eigenvalues and kept the remaining of the spectrum unchanged. A Sylvester equation must be solved using only the part of the spectrum to be reassigned. The results are tested in a benchmark taken from the specialized literature, and thus confirming its effectiveness.