Optimal control of linear buildings under seismic excitations

In this paper, an optimal feedback control law will be designed for linear building structures under seismic excitation. Using the proposed method, the vibration control of a single-degree-of-freedom structure with an active tuned mass damper (ATMD) will be studies. Analytical expressions of the linear quadratic regulator (LQR) feedback gains for the ATMD are derived by solving the Ricatti equation straightforwardly. The mean-square responses of the structure as well as the auxiliary mass against Gaussian white noise excitations are evaluated by solving the Lyapunov equation analytically based on the stochastic optimal control theory. Using these analytical solutions, the optimal damping parameter for the auxiliary mass is also derived. Finally, the optimal displacement feedback control law is presented.