Optimal Structural Control via Training on Ensemble of Earthquakes

In civil engineering, the design of active vibration control systems for structures subjected to earthquake excitation is usually done using linear-quadratic optimal control theory. However, when this theory is applied to a system with an external forcing function, the function must be either neglected, known a priori, or treated as white noise. If it is treated as white noise, the control is optimized for steady-state response. For seismic analyses of structures, these three assumptions—that the earthquake input is known in advance, is neglected, or is white noise—are questionable. This represents a serious deficiency in using standard methods of linear optimal control for reducing structural vibrations under seismic loading. This paper presents a new method of addressing the issue of including the earthquake-type excitation explicitly in the development of control systems, by designing feedback and feedforward controllers whose gains are optimized by training on an ensemble of earthquakes. Two different control strategies are presented: in the first, the controller is composed of a state feedback term only (closed loop); in the second, a control term proportional to the external excitation is fed forward (open loop) in addition to the closed loop term. The development of the controller follows the general formalism developed by Kabamba and Longman (1981, 1983) for the design of optimal controllers of arbitrary prescribed order with quadratic cost functionals. In this formalism, the gradients of the cost functional are obtained in explicit form and involve Liapunov equations. The results of this study indicate that inclusion of the forcing function explicitly in the development of the controller provides better results than the standard Riccati solution and drastically reduces the peak structural response.