Two-Dimensional Input Shaping for One-Dimensional Continua

This paper extends input shaping control to one-dimensional continua. Unlike discrete systems where the input is shaped only in the temporal domain, temporal and spatial input shaping can produce zero residual vibration in setpoint position control of distributed strings and beams. For collocated and noncollocated boundary control of strings and domain control of strings and pinned beams, the response to step inputs is solved in closed form using delays. For a clamped beam model, a closed form infinite modal series is used. The boundary controlled string can be setpoint regulated using two-pulse zero vibration (ZV) and three-pulse zero vibration and derivative (ZVD) shapers but ZVD is not more robust to parameter variations than ZV, a unique characteristic of second-order partial differential equations systems. Noncollocated ZV and ZVD boundary control enables rigid body translation of a string with zero residual vibration. Domain controlled strings and pinned beams with spatial input distributions that satisfy certain orthogonality conditions (e.g., midspan point load or uniformly distributed load) can be setpoint regulated with shaped inputs. For the cantilevered beam, modal shaping of the input distribution and ZV or ZVD temporal shaping drives the tip to the desired position with zero residual vibration.