Exponential Convergence Bounds in Least Squares Estimation: Identification of Viscoelastic Properties in Atomic Force Microscopy

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.