Online Koopman Operator Learning to Identify Cross-Coupling Effect of Piezoelectric Tube Scanners in Atomic Force Microscopes

Abstract

As one of the most significant applications of nanopositioning technology, the tremendous development of atomic force microscopes (AFMs) has been witnessed these years. Essentially, the scanning motions of AFMs are generally driven by piezoelectric tube scanners (PTSs), whose cross-coupling effect hinders their high speed and high-precision positioning. Therefore, it becomes an urgent yet challenging mission to establish a niche model for the nonlinear cross-coupling dynamics of the PTS. As an emergent pure data-driven learning approach, the Koopman operator sheds some light on the PTS modeling methodology, which is thus adopted in this article to approximate the nonlinear cross-coupling dynamics of PTSs in an infinite-dimensional space. Moreover, an online high-order extended dynamic mode decomposition algorithm is proposed for the finite-dimensional approximation of the Koopman operator online. The merit of the present model lies in updating the identified cross coupling upon the arrival of new data in an incremental way. Finally, experiments are conducted to approximate the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X–Y</i> axes cross coupling of PTSs of an NTMDT Prima AFM, which verifies the effectiveness and superiority of the proposed modeling algorithm. This article is expected to pave the way from the Koopman operator learning theory to real applications in dynamics modeling of abundant nanoscale measurement systems.