Effect of weighting a stroke of an active mass damper in the linear quadratic regulator problem

The effect of weighting on the linear quadratic regulator problem is analytically described for a single-degree-of-freedom undamped structure controlled by an Active Mass Damper (AMD). The feedback control law is introduced in a closed form after both the state equation and the performance index are non-dimensionalized. The closed expression proves that the feedback gain associated with a stroke of the AMD depends only on its weighting parameter and the mass and frequency ratios of the AMD to the structure, and the gain becomes zero when the stroke is neglected in weighting. Weighting the stroke apparently increases the AMD damping with increasing natural frequency. This pole allocation leads to ineffective control, which compels the auxiliary mass to be fixed to the structure. The essential feedback gain associated with the structural velocity changes its sign according to the weighting state. The effect of tuning the AMD to the structural natural frequency can be seen when the stroke is weighted a little. From a viewpoint of stability, the optimal weighting parameter for the stroke is proposed when an active tuned mass damper is installed.