Integrated design of the feedback controller and topography estimator for atomic force microscopy S. Kuiper a,n , P.M.J. Van den Hof b,c , G. Schitter d a TNO Technical Sciences, Stieltjesweg 1, 2628 CK Delft, The Netherlands b Delft Center for Systems and Control Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands c Control Systems Group, Department of Electrical Engineering, University of Technology Eindhoven, Den Dolech 2, 5612 AZ Eindhoven, The Netherlands d Automation and Control Institute, Vienna University of Technology, Gusshausstrasse 27–29, 1040 Vienna, Austria article info Article history: Received 4 October 2012 Accepted 19 March 2013 Available online 15 May 2013 Keywords: Atomic force microscopy Model-based control Robust control Topography estimation Optimal filter design abstract In atomic force microscopy (AFM) the force between the measurement tip and the sample is controlled in a feedback loop to prevent damage of the tip and sample during imaging, and to convert the measurement of the tip–sample force into an estimate of the sample topography. Dynamical uncertain- ties of the system limit the achievable control bandwidth and the accuracy of the topography estimation. This paper presents an integrated approach to design a feedback controller and topography estimator, taking into account the dynamical uncertainties of the system. The proposed methodology is experimentally demonstrated on a commercial AFM system, showing a direct trade-off between the control bandwidth and the accuracy of the topography estimation. & 2013 Elsevier Ltd. All rights reserved. 1. Introduction Since its invention in 1986 (Binnig, Quate, & Gerber, 1986) the atomic force microscope (AFM) has become a very important tool in micro-biology, material science, and nanotechnology to mea- sure various sample properties at the nanometer scale. In Fig. 1 the working principle of the AFM is depicted. The sample is probed by a very sharp tip that has an end-radius on the order of a few nanometers, which is mounted on the free end of a micro- cantilever. When the tip is brought in close proximity to the sample, the interaction forces between the tip and the sample can be detected by measuring the deflection of the cantilever beam via an optical sensing system (Alexander et al., 1989). During imaging the sample is scanned relative to the measurement tip in a lateral raster scanning pattern in order to image the area of interest. During scanning the interaction force between the tip and the sample is controlled by a feedback loop, manipulating the distance between the tip and the sample based on the measured cantilever deflection. This feedback loop prevents damage of the tip and the sample due to large interaction forces, and also allows us to convert the force measurement into an estimate of the sample topography. In order to provide the lateral scanning motion and to allow the control of the tip–sample interaction force, a high precision positioning stage is used that can position the sample relative to the measurement tip in all the three spatial directions. While scanning, the measured cantilever deflection and the compensating actions of the feedback loop are recorded by the system's data acquisition in order to obtain a map of the sample topography. The blockdiagrams (a)–(c) of Fig. 2 (solid lines) depict the feedback loop in AFM controlling the tip–sample force, with the actuator dynamics G and the feedback controller K. The dynamics of the cantilever, the tip–sample interaction and the optical cantilever deflection detection are governed by B. While scanning, the sample topography enters the feedback loop as an unknown input signal, denoted h(t) in Fig. 2. Signal n(t) in Fig. 2 denotes the noise and disturbances acting on the system, which may be stemming from various sources, such as electronic noise from the sensor electronics and power amplifier, vibrations stemming from the environment, temperature drifts within the instrument, and Brownian noise observable from the small cantilever (Sarid, 1994). Although stemming from various sources, these noise contributions can be viewed as a signal entering the feedback loop at the same location as the sample topography signal h(t). During imaging, the feedback controller is aimed to minimize the variations of the tip–sample force with respect to the constant setpoint level r, in some sense minimizing the control error eðt Þ¼ r-dðtÞ. Due to this feedback action, the actuator follows the varying sample topography over the scanned area; hðtÞ≈-xðtÞ as long as eðtÞ≈0. Different methods can be used to estimate the sample topo- graphy in AFM, as outlined by the different blockdiagrams of Fig. 2. Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/conengprac Control Engineering Practice 0967-0661/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conengprac.2013.03.006 n Corresponding author. Tel.: +31 (0)888664806. E-mail addresses: stefan.kuiper@tno.nl (S. Kuiper), p.m.j.vandenhof@tue.nl (P.M.J. Van den Hof), schitter@acin.tuwien.ac.at (G. Schitter). Control Engineering Practice 21 (2013) 1110–1120