Closed-loop input shaping for flexible structures using time-delay control

Input shaping techniques reduce the residual vibration in flexible structures by convolving the command input with a sequence of impulses. The exact cancellation of the residual structural vibration via input shaping is dependent on the amplitudes and instances of impulse application. A majority of the current input shaping schemes are inherently open-loop where impulse application at inaccurate instances can lead to system performance degradation. In this paper, we develop a closed-loop control design framework for input shaped systems. This framework is based on the realization that the dynamics of input shaped systems give rise to time delays in the input. Thus, we exploit the feedback control theory of time delay systems for the closed-loop control of input shaped flexible structures. A Riccati equation-based and a linear matrix inequality-based frameworks are developed for the stabilization of systems with uncertain, multiple input delays. Next, the aforementioned framework is applied to an input shaped flexible structure system. This framework guarantees closed-loop system stability and performance when the impulse train is applied at inaccurate instances. A simulation study demonstrates that the closed-loop system with the proposed time delay controller outperforms the open-loop, input shaped system and the standard linear quadratic regulator when the impulse is applied at an inexact instance.