Linearization of Recurrent-Neural-Network- Based Models for Predictive Control of Nano-Positioning Systems Using Data-Driven Koopman Operators

Shengwen Xie,Juan Ren

doi:10.1109/access.2020.3013935

Abstract

Recent studies have shown that the nonlinear dynamics of nano-positioning systems (e.g., piezo-electric actuators (PEAs)) can be accurately captured by recurrent neural networks (RNNs). One direct application of this technique is PEA system control for precision positioning: linearize the nonlinear RNN model and then apply model predictive control (MPC). However, due to the linearization approach commonly used (e.g., Taylor series), the control bandwidth and the control performance are quite limited as the obtained linear system is only guaranteed to be accurate within small neighborhood of the linearization point. To address this issue, we propose a Koopman operator-based approach for linearization and then use the obtained linear parameter-varying model for predictive control. This linearization scheme can significantly decrease the overall approximation error within the MPC prediction horizon, and thus, lead to improved tracking performance. The proposed approach was validated through two applications-trajectory tracking of PEA, and deformation control of polymers during atomic force microscope nano-indentation.

Highlights

Due to the frequency- and amplitude-dependent nonlinearity of piezo-electric actuators (PEAs), it is challenging to design broadband, high-precision, and computationally efficient real-time controllers [1], [2]
A discrete nonlinear system represented by a recurrent neural networks (RNNs) is linearized using data-driven Koopman operator approach, and a predictive controller is designed based on the linearized model [21]
SIMULATION RESULTS In the simulation, the RNN model [10] which captures the nonlinear dynamics of a PEA was used

Summary

INTRODUCTION

Due to the frequency- and amplitude-dependent nonlinearity of piezo-electric actuators (PEAs), it is challenging to design broadband, high-precision, and computationally efficient real-time controllers [1], [2]. A discrete nonlinear system represented by a RNN is linearized using data-driven Koopman operator approach, and a predictive controller is designed based on the linearized model [21]. The main contributions of this paper are: 1) we explained why the Koopman approach still works when the linear model has low order (i.e., the dimension of the lifted space is low), 2) a solution has been provided to deal with the surge in approximation error when the prediction horizon becomes large, 3) the proposed approach has been implemented for deformation control of materials which is very challenging for other controllers, 4) detailed comparison studies between Taylor approximation and Koopman approximation in terms of principles and computation efficiency have been conducted, which provided helpful information for controller design. Structure of the RNN model and how Eq (6) can model the nonlinearities of the PEA system are detailed in [10]

LINEARIZATION WITH TAYLOR SERIES

LINEARIZATION BASED ON KOOPMAN OPERATORS

SIMULATION RESULTS

EXPERIMENTAL RESULTS

COMPUTATION EFFICIENCY

CONCLUSION AND FUTURE WORK