Nonlinear Control Design for a Piezoelectric-Driven Nanopositioning Stage

Abstract

Abstract : A model-based nonlinear optimal control design is developed and applied to a piezoelectric-driven nanopositioning stage to demonstrate high accuracy displacement control when ferroelectric nonlinearities, hysteresis and relaxation behavior are present in the piezoelectric actuator. The nonlinear control design is compared to Proportional-Integral-Derivative (PID) control to illustrate performance enhancements when ferroelectric material behavior is directly incorporated into the control design. A multi-scale ferroelectric constitutive law is used to model the nonlinear, hysteretic and relaxation behavior of the piezoelectric actuator. The constitutive model is implemented in a structural model of a nanopositioning stage that used piezoelectric stack actuators. Both moderate frequency (100 Hz) and quasi-static (0.2 Hz) operating regimes are considered to address issues associated with high accuracy atomic force microscopy applications. The nonlinear control design includes and open-loop nonlinear control input determined from finite-dimensional optimal control theory and linear perturbation feedback determined from Proportional Integral (PI) control. The Hybrid control design reduces errors by over two orders of magnitude when nonlinearities, hysteresis and relaxation behavior is present. Additionally, the magnitude of control input is considerably smaller when the hybrid control design is implemented.