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Adaptive control of the nonlinear dynamic behavior of the cantilever-sample system of an atomic force microscope

The paper presents a model reference adaptive control (MRAC) of first and second order to control the nonlinear dynamics of an atomic force microscope (AFM) cantilever, which is operated in contact mode. The AFM is a powerful tool to measure the topography of a sample at the scale of a few nanometers, where a small sharp tip supported in a micro cantilever scans the surface. In the contact mode the sample's topography is obtained by using the closed-loop control that holds the tip sample force constant. The nonlinear dynamics of the tip-sample system is very complex with different kinds of nonlinear forces that act between the tip and the sample. Here the dominated force depends on the distance tip-sample. In the present work we use a modified Hertz model to describe the nonlinear force when the distance tip-sample is less than 1 nm. First the complex nonlinear tip-sample system is controlled with a nonlinear MRAC of 1st order and after with a nonlinear MRAC of 2nd order. The results of both control strategies were compared in order to see which one gives a better control perfomance. Here a stability proof for both MRAC methods is present. A variety of simulation results are presented to demonstrate the efficacy of the proposed methods. The procedure is general and can be applied to any nonlinear system.