Symbolic Analysis of Sub-Optimal Bang-Bang Control Mechanisms in Base Isolated Structures

In the performance-based design of earthquake-resistant structures, researchers have recently proposed protection systems where base isolation devices are supplemented by active control mechanisms. Established approaches to understanding this problem domain rely on numerical and experimental analyses, which have the disadvantage of obscuring potential insight into cause-and-effect relationships existing between parameters of sub-optimal control and their affect on linear and nonlinear system response. As a first step toward mitigating this limitation, this paper explores the role of symbolic analysis in understanding how sub-optimal bang-bang control mechanisms depend on design objectives and their impact on performance of base isolated structures. New results are obtained through three avenues of investigation: (1) Single- and two-degree-of-freedom systems, (2) Restricted classes of multi-degree-of-freedom systems, and (3) Sensitivity of parameters in modified bang-bang control to localized nonlinear deformations in the base isolation devices. The principle outcome is matrices of symbolic expressions for bang-bang control expressed in terms of the structural system parameters and state. We identify modeling constraints and limits (e.g., perfect isolation) where lengthy symbolic expressions simplify to the point where relationships between the inner workings of the bang-bang control strategy and specific design objectives become evident.