Partial eigenvalue assignment and its stability in a time delayed system

• An algorithm for Quadratic Partial Eigenvalue Assignment problem (QPEVAP) for the time delay second order control system is proposed.
• This algorithm generalizes the earlier work on the single input case.
• This method directly works in a given second order system without a-priori transformation to a standard state space system.
• A detailed posterior stability analysis of the closed loop system is carried out with respect to partial eigenvalue assignment.
• Critical time delays as well as stability pockets/switches of the closed loop system are identified.

Active vibration control strategy is an effective way to control dangerous vibrations in a structure, caused by resonance and to manipulate the dynamics of vibrational response. Implementation of this strategy requires real-time computations of two feedback control matrices such that a small amount of eigenvalues of the associated quadratic matrix pencil are replaced by suitably chosen ones while the remaining large number of eigenvalues and eigenvectors remain unchanged ensuring the no spill-over. This mathematical problem is referred to as the Quadratic Partial Eigenvalue Assignment problem. The greatest challenge there is to solve the problems using the knowledge of only a small number of eigenvalues and eigenvectors that are computable using state-of-the-art techniques. This paper generalizes the earlier work on partial assignment to constant time-delay systems. Furthermore, a posterior stability analysis is carried out to identify the ranges of the time-delay that maintains the closed-loop assignment while keeping the stability of the infinite number of eigenvalues for the time-delayed systems. The practical features of the proposed methods are that it is implemented in the second-order setting itself using only those small number of eigenvalues and the eigenvectors that are to be assigned and the no spill-over is established by means of mathematical results. The results of our numerical experiments support the validity of our proposed methods.