Discrete time-optimal command shaping

Finite impulse-response “input shapers” may be used to prefilter inputs to eliminate or minimize residual vibration when maneuvering flexible structures. Vibration is reduced at the expense of adding a delay equal to the length of the shaper. This paper first presents results that aid in the design of “single-input shapers” in the discrete domain. Next, MIMO shaping is considered. Given certain a priori information about the inputs, shorter shapers (which introduce less delay) can be derived. A time-optimal scheme is developed here, and existence of solutions is proved for both SISO and MIMO cases. The method is direct, general, easy to implement, and allows for the addition of derivative constraints to improve robustness. Finally, traditional optimal control and input shaping are compared, and it is shown that under certain conditions optimal discrete-time input shaping and discrete time-optimal control are equivalent.