State feedback control with time delay

We consider the problem of assigning eigenvalues of a linear vibratory system by state feedback control in the presence of time delay. It is shown that for a system with n degrees of freedom we may assign 2n eigenvalues. Assigning 2n eigenvalues in a time-delayed system does not necessarily regulate the dynamics of the system or even guarantee its stability. We therefore separate the eigenvalues into two groups, primary and secondary eigenvalues. The primary eigenvalues are the 2n finite eigenvalues of the system without time delay. The secondary eigenvalues are the other eigenvalues emerging from infinity due to the delay. A method of a posteriori analysis to identify the primary eigenvalues and to ensure that they have been properly assigned is proposed in the paper. The method is demonstrated by various examples.