Discrete time-optimal command shapers and controls for multi-input multi-output systems

Flexible modes in many systems make fast maneuvers difficult. In order not to excite such modes, a finite impulse-response (FIR) "input shaper" can be used to prefilter inputs, reducing vibration at the expense of adding a delay equal to the length of the shaper. Given a system model in discrete time, the discrete time-optimal input shaper is the shortest prefilter that leads to zero vibration. The two cases where inputs are either unknown or known a priori are qualitatively different because the designer can take advantage of this knowledge to derive shorter shapers. The paper first reviews the case where inputs are unknown ("single-input shaping"). It then presents a method by which the search for optimal input shapers in both the known and unknown cases can be phrased as a quasiconvex optimization problem. The method is general, direct, and allows for the addition of derivative constraints to improve robustness. Finally, traditional optimal control and input shaping are compared for the case where inputs are known a priori, and it is shown that under certain conditions optimal discrete time input shaping and discrete time-optimal control are equivalent.