High frequency bandwidth polarization and strain control using a fractional derivativeinverse model

Abstract

The present paper proposes an inverse dynamical model based on fractionalderivatives in order to simulate the electrical field versus the polarization fieldE(P) as well as the electrical field versus the mechanical strainE(S) of ferroelectric hysteresis. By considering a fractional derivative term, the frequency bandwidthof the inverse model is greatly increased. As a consequence, the model became suited forthe usual inverse model applications, such as adaptive inverse control of piezoelectricactuators, high-speed positioning or high precision positioning. The proposed high-accuracyinverse model rendered it possible to avoid standard feedback mechanisms that usuallyexhibit restrain frequency bandwidths due mainly to high frequency noise. Starting from aquasi-static inverse model, and based on a simple mechanism related to the dry-frictionconcept, a polarization fractional derivative term was added in order to take into accountthe dynamical effects. The order of the fractional derivative was, using a direct modelP(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 ferroelectricstrain and the polarization field. Moreover, a dynamical strain control based onan inverse dynamic polarization field contribution was proposed. Experimentalprocedures were developed to verify the accuracy of the polarization as well as of thestrain control. Good results were obtained and exposed for sinus and triangularpolarization-imposed waveforms. Various frequencies and amplitudes were tested in bothcases.