Model Development and Control Design for Atomic Force Microscopy

Abstract

The development of energy-based models and model-based control designs necessary to achieve present and projected applications involving atomic force microscopy is investigated. Applications include real-time product diagnostics or monitoring of biological processes, nanoelectromechanical systems (NEMS) and employment of atomic force microscope (AFM) technology for spintronics. A crucial component in the AFM design is the piezoceramic (PZT)-based stage used to position the sample. Whereas PZT actuators provide the broadband and extremely high set point capabilities required by the AFM stages, they also exhibit frequency-dependent hysteresis and constitutive nonlinearities.

To characterize the field-polarization relation in PZT, low-order macroscopic models are constructed based on a combination of energy analysis at the mesoscopic level along with stochastic homogenization techniques. To account for nonuniformity and inhomogeneities in the material, local coercive field values are assumed to be distributed. Due to interactions among the dipoles, the effective field is also assumed to be distributed. Previous work has employed specific functions to describe these distributions. However, the fact that these choices are not based on energy considerations, motivates the use of general densities.

The dynamics of the actuator must be incorporated as well. A rod model is suitable for a stacked actuator whose cross-section is small compared to the length. The equation of motion for the rod can be derived using force balancing with boundary conditions determined from the fact that the rod is fixed at one end and pushes against the stage at the other.

At low frequencies, the hysteresis and constitutive nonlinearities inherent in PZT can be accommodated through PID or robust control designs. However, at the higher frequencies required by the previously outlined applications, increasing noise-to-data ratios and diminishing high-pass characteristics of control filters preclude a sole reliance on feedback laws to eliminate hysteresis. This motivates the development of control designs that incorporate and approximately compensate for hysteresis through model inverses employed as filters to linearize transducer responses for linear robust control design and PID control design. The inverse models are also tested in an open loop control experiment on a PZT stacked actuator.