High frequency bandwidth polarization and strain control using a fractional derivative inverse model

The present paper proposes an inverse dynamical model based on fractional derivatives in order to simulate the electrical field versus the polarization field E(P) as well as the electrical field versus the mechanical strain E(S) of ferroelectric hysteresis. By considering a fractional derivative term, the frequency bandwidth of the inverse model is greatly increased. As a consequence, the model became suited for the usual inverse model applications, such as adaptive inverse control of piezoelectric actuators, high-speed positioning or high precision positioning. The proposed high-accuracy inverse model rendered it possible to avoid standard feedback mechanisms that usually exhibit restrain frequency bandwidths due mainly to high frequency noise. Starting from a quasi-static inverse model, and based on a simple mechanism related to the dry-friction concept, a polarization fractional derivative term was added in order to take into account the dynamical effects. The order of the fractional derivative was, using a direct model P(E) and experimental data on a large frequency bandwidth (10−3 Hz < f < 102 Hz), found to be equal to 0.5. A quadratic relation was used to link the ferroelectric strain and the polarization field. Moreover, a dynamical strain control based on an inverse dynamic polarization field contribution was proposed. Experimental procedures were developed to verify the accuracy of the polarization as well as of the strain control. Good results were obtained and exposed for sinus and triangular polarization-imposed waveforms. Various frequencies and amplitudes were tested in both cases