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This paper derives shaped time-optimal servomechanism (STOS) closed-loop control laws using a phase-plane approach to address flexible structures having multiple flexible modes. The STOS control laws for systems having two and three flexible modes are detailed. The flexible modes are addressed by using multimode input shapers to shape the time-optimal control for the rigid body portion of the system. The altered rigid body phase-plane trajectories resulting from this shaping are solved and used to determine an STOS closed-loop controller. Both "convolution" and "simultaneous" method multimode input shapers are considered. We also improve the robustness to modeling errors of the STOS control. The more robust zero vibration and derivative (ZVD) shaper is used. The ZVD shaper consists of more impulses and has a longer time duration than the simpler zero vibration (ZV) shaper which leads to the more robust STOS (RSTOS) control derived from the ZVD shaper having more switches and longer time response than the STOS control derived from the ZV shaper. Simulation results show that using the ZVD shaper in deriving the RSTOS control improves the robustness to modeling errors in frequency and damping of the flexible mode.
Date of Publication: Oct. 2005