This paper proposes a method to analyze, beyond stability, the performances of linear time-delay systems. Using robust analysis techniques, a sufficient condition that analyzes the location of eigenvalues in the complex plane is presented. More precisely, a set of quadratic inequality constraints are designed to define an admissible region for the infinitely many eigenvalues of a time-delay system and the quadratic separation theorem is applied to assess that the eigenvalues are effectively belonging to that stability region. This method is then used for the control of an active mass damper. A standard state feedback control is replaced with a static output feedback plus a static delayed output feedback. This strategy avoids the full measurement of the state and shows that delays in the dynamic may be helpful for stabilization. The closed-loop system is then expressed as a time-delay system and the performance criterion is exploited to analyze the stability and the damping properties. Simulations and experimental tests support the approach.