Optimal input shaping filters for non-zero initial states

This paper presents an approach to design optimal vibration reduction input shapers for systems with non-zero initial conditions. The problem is first formulated as an optimal control problem and the optimal solution is shown to be bang-bang. Once the structure of the optimal shaper is known, a parametric problem formulation is presented for the computation of the switching times. For digital implementation, discrete time approximate solutions are derived by solving a quasi convex linear program. Simulation results are shown for closed-form implementation of these filters on flexible structures. The digital solutions are experimentally verified on a portable bridge crane.