In this brief the design and analysis of an optimal robust minimax linear quadratic Gaussian (LQG) control of vibration of a flexible beam is studied. The analysis is performed by transforming the minimax LQG control design problem to its equivalent mixed sensitivity design problem. The first six modes of the beam in the frequency range of 0-700 Hz are selected for control purpose. Among these modes, three modes in the frequency range of 100-400 Hz are used for control, while the other three modes are left as the uncertainty of modeling. Both the model and the uncertainty are measured based on experimental data. The nominal model is identified from frequency response data and the uncertainty is presented by frequency weighted multiplicative modeling method. For the augmented plant consisting of the nominal model and its accompanied uncertainty, a minimax LQG controller is designed. Analysis and tradeoff between robust stability and robust performance is shown by selecting two different choices of uncertainty modeling. Simulation results used to show how the uncertainty weights can be tuned so that the proposed robust controller increase the damping of the system in its resonance frequencies and maintain the robust stability of the feedback system at the same time.