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The linear quadratic Gaussian (LQG) control in a discrete-time form for the systems having an input delay is applied to the active vibration suppression of a flexible cantilever beam to solve the instability problem of its boundary control due to a delayed boundary feedback. The dynamic equation of an Euler-Bernoulli beam with a delayed control input is first described, and then is discretized and transformed into a standard state-space equation, which does not contain the time delay apparently. An LQG controller is designed for the standardized system. In the proposed method, the dimensions of the state vector after the standardization keep the same as those of the original time delay systems, although the other existing discrete-time control approaches for systems with an input delay may increase the dimensions of the control system. Simulation results demonstrate that the proposed approach is a feasible and effective way to solve the instability problem of the boundary control due to the time delay existing in the boundary feedback.