Piezoelectric materials are known to exhibit nonlinear effects if they are operated outside their linear small-signal regime, which significantly degrades the performance of structural control concepts. In this contribution, these nonlinear effects are experimentally investigated in the context of a feedforward control application, where a control command is designed to steer the tip of a cantilever beam by means of a piezoelectric patch actuator from initial to a prescribed desired final stationary deflection. As expected, the encountered nonlinear effects degrade the control performance with increasing applied electrical field. Hence, a modified feedforward control design procedure is proposed. The overall nonlinear system model is recast in a series connection of an input nonlinearity and the linear dynamics of the mechanical structure, which the feedforward control to be designed in two steps: First, a feedforward control for the linear model part is derived based on an approach exploting the notion of flatness in combination with modal analysis of the linear dynamics. It uses the finite-element method to derive the linear dynamics of the piezoelectric structure. Secondly, an inverse filter is designed to compensate for the nonlinear piezoelectric hysteresis and creep effects. By insertion of this inverse filter at the system input, i.e. by filtering the feedforward control, very good tracking control performance is recovered in both small and large-signal operation of the piezoelectric actuator. This filter itself is derived by inversion of a model of the nonlinearities in the discrete-time domain. The chosen model for the hysteresis is based on polynomial approximations of the hysteresis loops, appropriate scaling of these loops to the actual point of operation by keeping track of the input signal reversals and on implementation of the physically motivated Madelung rules that the piezoelectric hysteresis obeys. The creep is found to behave according to a Kelvin-Voigt viscoelastic model. Various experiments for the piezoelectrically actuated beam show that the modified feedforward control design yields very good tracking performance also outside of the small-signal regime by application of the compensation filter. The excellent feedforward tracking control performance predicted by simulations of the full (linear) finite-element model is verified. The designed feedforward control realizes very fast rest-to-rest transition in less than half of the period of the first structural mode. As an interesting application of such a feedforward control, a two-degree-of-freedom control concept combining the presented feedforward control and additional feedback control is investigated. By use of the feedforward control, the feedback can be relatively simple because it is only responsible for disturbance rejection and adding robustness.
Damping Force Tracking Control of MR Damper System Using a New Direct Adaptive Fuzzy Controller
