Stability estimation of high dimensional vibrating systems under state delay feedback control

The paper presents a method of assessing the stability of high dimensional vibrating systems under state feedback control with a short time delay. It is first proved that if the time delay is sufficiently short, an n -degree-of-freedom system with feedback delay maintains 2n eigenvalues near those of the corresponding system without feedback delay. A perturbation approach is then proposed to determine the first order variation of an arbitrary eigenvalue and corresponding eigenvector of the system with feedback delay by solving a set of linear algebraic equations only. The computation in this approach can be simplified to a matrix multiplication provided that the product of the time delay and the modulus of the eigenvalue is much less than 1. Furthermore, the approach is directly related to the Newton-Raphson iteration in the continuation of eigenvalues for long time delay. The efficacy of the approach is demonstrated via a number of case studies on two feedback delay systems of two degrees of freedom and ten degrees of freedom respectively.