Symbolic and Numeric Solutions of Modified Bang-Bang Control Strategies for Performance-Based Assessment of Base-Isolated Structures

This work explores symbolic and numeric solutions to the Lyapunov matrix equation as it applies to performance-based assessment of base-isolated structures supplemented by modified bang-bang control. Traditional studies of this type rely on numeric simulations alone. This study is the first to use symbolic analysis as a means of identifying key "cause and effect" relationships existing between parameters of the active control problem and the underlying differential equations of motion. We show that symbolic representations are very lengthy, even for structures having a small number of degrees of freedom. However, under certain simplifying assumptions, symbolic solutions to the Lyapunov matrix equation assume a greatly simplified form (thereby avoiding the need for computational solutions). Regarding the behavior of the bang-bang control strategy, further analysis shows: (1) for a 1-DOF system, the actuator force acts very nearly in phase, but in opposite direction to the velocity (90 degrees out of phase and in opposite direction to the displacement), and (2) for a wide range of 2-DOF nonlinear base-isolated models, bang-bang control is insensitive to nonlinear deformations in the isolator devices. Through nonlinear time-history analysis, we see that one- and two-DOF models are good indicators of behavior in higher DOF models. An analytical framework for system assessment through energy- and power-balance analysis is formulated. Computational experiments on base-isolated systems are conducted to identify and quantitatively evaluate situations when constant stiffness bang-bang control can significantly enhance overall performance, compared to base isolation alone, and assess the ability of present-day actuator technologies to deliver actuator power requirements estimated through simulation