Repetitive Control for Hysteretic Systems: Theory and Application in Piezo-Based Nanopositioners
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This dissertation studies the design and analysis of repetitive controllers for hysteretic systems. An example hysteretic system is a piezoelectric actuator (piezoactuator), the workhorse of actuators used for positioning and manipulating objects and tools at the micro and nano scale. For example, in scanning probe microscopes (SPMs) a piezoactuator is used to raster (back and forth in a repetitive fashion) a probe tool with sub-nanometer precision relative to a sample surface for imaging, manipulating, and fabricating organic or inorganic nano-scale features. Likewise, piezoactuatorsare used to position optics in space telescopes and tools in micro-machining systems. Despite their importance, piezoactuators exhibit hysteresis effect, a nonlinear behavior between the applied input voltage and the resulting output displacement of the piezoactuator. If left uncompensated for, hysteresis (and dynamic effects) cause positioning error that significantly limits the performance of piezo-based positioning systems. Repetitive control, a feedback-based approach which exploits the process of repetition, is commonly applied to track periodic reference trajectories and/or to reject periodic disturbances. The major challenges in the design of RC are closed-loop stability, robustness, and minimizing the steady-state tracking error. For hysteretic systems such as piezo-based nanopositioners, the nonlinearity can drastically limit the performance of RC designed around a linear dynamics model. In this work, the effect of hysteresis on the closed-loop stability of RC is analyzed and the allowable size of the hysteresis nonlinearity for a stable RC is quantified. In the stability analysis, the bounded-input bounded-output (BIBO) stability of the repetitive controlled hysteretic system in the L2-norm sense is shown. Combining this result withthe Small-Gain Theorem, an acceptable size of the hysteresis nonlinearity is determined that guarantees closed-loop stability. Therefore, one main contribution of this study is to provide a theoretical framework for analyzing the performance of RC for hysteretic systems. When the hysteresis effect exceeds the maximum bound, a new inverse-hysteresis feedforward controller based on the Prandtl-Ishlinskii hysteresis model is proposed. The control approach is applied to a custom-designed piezoactuatordriven nanopositioning stage, and experimental tracking and nanofabrication results are presented to validate the RC and inverse model design. The tracking results at 1 kHz show that by adding hysteresis compensation the stability margin and rate of convergence of RC are improved by 14%. Likewise, the maximum tracking error is reduced from 13.7% (using industry-standard integral control) to 3.9% (using RC and hysteresis compensation), a 71% reduction. The RC approach is also applied to nanofabrication, where it is shown that by using RC with hysteresis compensationthe error during fabrication is substantially reduced.