Multivariable Negative-Imaginary Controller Design for Damping and Cross Coupling Reduction of Nanopositioners: A Reference Model Matching Approach

Abstract

This paper presents the design and experimental implementation of a novel multi-input multi-output (MIMO) control structure using two negative-imaginary damping controllers to damp the first resonant mode and attenuate cross coupling effects between the axes of a piezoelectric tube scanner. The dynamics of the scanner in the lateral and longitudinal axes are identified from measured data and the design of the controller using a MIMO framework is done based on a reference model matching approach. The controller proposed in the paper is able to achieve a bandwidth near to the first resonance frequency of the scanner and the proposed controller is robust against changes in resonance frequencies that are due to load changes on the scanner. Experimental results are presented to show the effectiveness of the proposed controller.Index Terms-Atomic force microscope, damping control, integral resonant control, negative-imaginary controller, piezoelectric tube scanner.

A Fast Algebraic Estimator for System Parameter Estimation and Online Controller Tuning—A Nanopositioning Application

“…It is clear that the estimation of ω n converges very quickly to its actual value, and that the convergence criterion is a monotonically decreasing function which tends to zero (a feature not achieved with the condition proposed in [17]). Note that the limit tolerance δ is reached in a very short time, and thus the condition (14) is verified. Consequently, the estimate of the natural frequency is produced in less than one cycle of the high-frequency oscillations experienced by the system.…”

“…Additionally, conditions Z σ (n) > 0 and Z ζ (n) > 0 are imposed to ensure σ > 0 and ζ > 0. Considering T s to be the sampling time, the sample under which condition 14is verified is denoted asn, and the timen•T s is the operation time interval corresponding to the interval of time from the beginning of the estimation process (it is assumed that n = 0 when the estimation starts) until (14) is satisfied. The estimates provided by the algorithm are:…”

“…3) Compute (12), (13) using successive estimates of ω n σ, and ζ. 4) Check the re-initiation criterion (14). If this criterion is verified, the last value of each EWMA will be considered a valid estimation of the system parameters (15).…”

“…The proposed algebraic estimator involves two algorithms: the estimation procedure and the convergence criterion. In Subsection IV-A, simulations are carried out to illustrate the accuracy, small convergence time of the estimator and the effectiveness of the convergence criterion given by (14). Consequently, the experimental setup is utilized to validate the efficacy of the proposed technique using a practical application in Subsection IV-B.…”

“…Till date, this parameter uncertainty was addressed by ascertaining that the designed control scheme was robust to parameter variations within a certain operational limit. Consequently, several control techniques that possess adequate robustness have been proposed, [10], [11], [12], [13], [14], [15]. These techniques, though robust, undergo considerable degradation in positioning performance when the system moves away from the nominal model parameters on which the controller design was initially based.…”