Optimal bounded-state control with application to building structure

Abstract

To prevent structural damage, it is safer to bound the vibration amplitude rather than to minimize the vibration energy as in an LQR formulation. This is due to the fact that even if the sum of the squares of the vibration amplitude over a period is minimal, at some instants vibration is still possible over the tolerable limit of the structure. Therefore, in this paper, bounded-state control is addressed by using linear state/output feedback in a numerical optimization. The worst-case maximum magnitude of the system response is first calculated with the assumption that the external excitation and the control force are infinity-norm-bounded and that the initial condition is uncertain. Based on this calculated worst-case maximum magnitude, a non-linear optimization procedure is then developed to determine an optimal bounded-state controller. This non-linear optimization procedure can also be extended to the design of a dynamic controller. It is also shown that the matching condition is both necessary and sufficient for controlling the state arbitrarily close to zero in the presence of an unknown but bounded disturbance. The developed bounded-state controller is applied to a building structure for damage protection. Two different types of control implementation considered are an active tendon control and an active mass driver.