Exponential convergence bounds in least squares estimation: Identification of viscoelastic properties in atomic force microscopy

Abstract

Abstract-Using atomic force microscopy (AFM) for studying soft, biological material has become increasingly popular in recent years. New approaches allow the use of recursive least squares estimation to identify the viscoelastic properties of a sample in AFM. As long as the regressor vector is persistently exciting (PE), exponential convergence of the parameters to be identified can be guaranteed. However, even exponential convergence can be slow. In this article, upper bounds on the parameter convergence is found, completely determined by the PE properties and least squares update law parameters. Furthermore, for a parameter vector which is piecewise constant at regular intervals, the time interval necessary for the error to converge to any specified upper limit is determined. For a soft sample in AFM, the viscoelastic properties can be spatially inhomogeneous. These properties can be spatially resolved by periodically tapping at discrete points along the sample. The results of this article then allows us to determine the time interval necessary at each tap in order to guarantee convergence to any specified fraction of the step-change in the parameters. Simulation results are presented, demonstrating the applicability of the approach.

Model-Based Identification of Nanomechanical Properties in Atomic Force Microscopy: Theory and Experiments

“…The question then remains, how long does the parameter estimator need to run during a given indentation in order to guarantee convergence to some specified error? This topic was investigated in detail in [29] for the general case of the recursive least squares method. Some results are restated here and extended for the case of covariance reset between each interval.…”

“…Furthermore, in this article we present an upper bound on the exponential convergence of the parameter error, for the recursive least squares estimator in general. Preliminary results on this were presented in [29]. In this article, the results are extended by determining the estimation time interval for which the parameter error is guaranteed to have been reduced to some fraction of the initial error, after performing a covariance reset.…”