A discrete optimal control method for a flexible cantilever beam with time delay

Time delay inevitably exists in active control systems. It may cause unsynchronized control forces that can not only degrade the performance of the control systems, but also induce instability of the dynamic systems. In this paper, an active vibration controller (with time delay) for a flexible cantilever beam is studied and a method for treating the time delay is proposed. The dynamic equation of the controlled mode with time delay is first presented using the independent modal space control, and is then discretized and transformed into a standard discrete form with no explicit time delay by augmenting the state variables. The continuous performance index is also transformed into a discrete form. Then, a discrete optimal control algorithm is designed based on the augmented state system.' Since time delay effect is incorporated in the mathematical model of the dynamic system throughout the control design and no approximations and assumptions are made in the control algorithm derivation, system stability is guaranteed. Furthermore, the extraction of modal coordinate from actual physical measurements and the conversion to actual control force from modal one are presented. The feasibility and efficiency of the proposed control algorithm are demonstrated by numerical simulation studies, which indicate that the vibration of the beam may be suppressed significantly using the proposed control algorithm. Instability may occur if the time delay is neglected in control design.