This paper addresses the development of energy-based models and model-based control designs necessary
to achieve present and projected applications involving atomic force microscopy. The models are based
on a combination of energy analysis at the mesoscopic level with stochastic homogenization techniques
to construct low-order macroscopic models. Approximate model inverses are then employed as filters to
linearize transducer responses for linear robust control design.
This paper focuses on the development of parameter estimation techniques
for models quantifying hysteresis and constitutive nonlinearities in
ferroelectric materials. These models are formulated as integral equations
with known kernels and unknown densities to be identified through least squares
fit to data. Due to the compactness of the integral operators, the resulting
discretized models inherit ill-posedness which often must be accommodated
through regularization. The accuracy of regularized finite-dimensional
models is illustrated through comparison with experimental data.