Problem of controlling the vibrations of the structures and thereby minimizing their damages due to earthquakes is becoming increasingly important. This has led to the use of active and passive energy dissipation devices and base isolation techniques in the structures. This book presents Linear Quadratic Gaussian approach to design active control system to perform vibration control of tall buildings undergoing seismic excitations. The motion of the structure is formulated as State Space Model. The mathematical interpretation of the quadratic optimal theory to design the control algorithm in time-domain is presented. An eight-storeyed building is considered to illustrate the effectiveness of passive and active control systems. Optimum values of Active Tuned Mass Damper (ATMD) parameters are decided based on various response characteristics and stability of the systems verified by Bode's and Nyquist criteia.