Time-varying LQ terminal control for compliant morphing structure using acceleration-based objective function

Abstract

Flexible structures integrated with smart materials, like piezoelectric actuators, can constitute an active compliant morphing system with the advantages of lightweight, hinge-less, high precision, and adaptability. Such a system is desired to realize rapid, smooth point-to-point shape control without transient and residual vibrations. A time-varying linear quadratic terminal (TVLQT) scheme can be designed to satisfy such finite time control requirements with time-varying feedforward signal and feedback gains. However, the regular formulation of the objective function, with the quadratic expression of state and input, cannot ensure the control performance of the terminal point and especially residual responses after it. The sharp change of the input at the terminal time will induce a high level of residual structural vibration of the compliant structures. To solve such an issue, the key contribution of this study is presenting a new formulation of the objective function utilizing the acceleration of the system. This new objective function can be transformed in terms of state and input, but with the coupling term xTNu, and fortunately can still be solved via differential Riccati equations (DREs). Two compliant structures actuated by piezoelectric materials, a piezocompsite-actuated plate, and a smart space truss, are used to demonstrate the dynamic shape control problem and verify the proposed control scheme. The numerical simulations show that the sharp change phenomenon of the input has been canceled out, and the associated input at the terminal time is much closer to the analytical steady-state value so that the residual vibrations can be avoided. Finally, an experimental setup for shape control of the MFC-actuated plate is established and the effectiveness of the proposed control scheme is experimentally verified. The results imply that improved, smooth dynamic shape control performance of the compliant morphing structure can be realized both during morphing processes and after the terminal point.