Inversion-based Hysteresis Compensation Using Adaptive Conditional Servocompensator for Nanopositioning Systems

Nanopositioning stages are widely used in high-precision positioning applications. However, they suffer from an intrinsic hysteretic behavior, which deteriorates their tracking performance. This study proposes an adaptive conditional servocompensator (ACS) to compensate the effect of the hysteresis when tracking periodic references. The nanopositioning system is modeled as a linear system cascaded with hysteresis at the input side. The hysteresis is modeled with a Modified Prandtl-Ishlinskii (MPI) operator. With an approximate inverse MPI operator placed before the system hysteresis operator, the resulting system takes a semi-affine form. The design of the adaptive conditional servocompensator consists of two stages: firstly, we design a continuously-implemented sliding mode control (SMC) law. The hysteresis inversion error is treated as a matched disturbance and an analytical bound on the inversion error is used to minimize the conservativeness of the SMC design. The second part of the controller is the adaptive conditional servocompensator. Under mild assumptions, we establish the well-posedness and periodic stability of the closed-loop system. In particular, the solution of the closed-loop error system will converge exponentially to a unique periodic solution in the neighborhood of zero. The efficacy of the proposed controller is verified experimentally on a commercial nanopositioning device under different types of periodic reference inputs, via comparison with multiple inversion-based and inversion-free approaches.

Approximate inverse limits and (m,n)-dimensions

This paper introduces shape boundary regions in descriptive proximity forms of CW (Closure-finite Weak) spaces as a source of amiable fixed subsets as well as almost amiable fixed subsets of descriptive proximally continuous (dpc) maps. A dpc map is an extension of an Efremovič-Smirnov proximally continuous (pc) map introduced during the early-1950s by V.A. Efremovič and Yu.M. Smirnov. Amiable fixed sets and the Betti numbers of their free Abelian group representations are derived from dpc's relative to the description of the boundary region of the sets. Almost amiable fixed sets are derived from dpc's by relaxing the matching description requirement for the descriptive closeness of the sets. This relaxed form of amiable fixed sets works well for applications in which closeness of fixed sets is approximate rather than exact. A number of examples of amiable fixed sets are given in terms of wide ribbons. A bi-product of this work is a variation of the Jordan curve theorem and a fixed cell complex theorem, which is an extension of the Brouwer fixed point theorem.