This paper studies the vibration control problem for systems with delayed state under persistent disturbances. In order to simply the vibration control problem for system with delayed states, a free-delayed-state method is proposed to reform the original problem into an equivalent optimal regulate problem with delay-free ones. Then, an approximate optimal vibration controller is obtained from a Riccati equation and a Sylvester equation. The results of numerical simulations illustrate that the proposed vibration controller can eliminate the unstable influences caused by the persistent disturbances and delayed state, and improve control performance effectively.
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