Michael R. P. Ragazzon

Lyapunov Estimator for High-Speed Demodulation in Dynamic Mode Atomic Force Microscopy

In dynamic mode atomic force microscopy (AFM), the imaging bandwidth is governed by the slowest component in the open-loop chain consisting of the vertical actuator, cantilever, and demodulator. While the common demodulation method is to use a lock-in amplifier (LIA), its performance is ultimately bounded by the bandwidth of the postmixing low-pass filters. This brief proposes an amplitude and phase estimation method based on a strictly positive real Lyapunov design approach. The estimator is designed to be of low complexity while allowing for high bandwidth. In addition, suitable gains for high performance are suggested such that no tuning is necessary. The Lyapunov estimator is experimentally implemented for amplitude demodulation and shown to surpass the LIA in terms of tracking bandwidth and noise performance. High-speed AFM images are presented to corroborate the results.

On Amplitude Estimation for High-Speed Atomic Force Microscopy

Amplitude estimation or demodulation plays a vital part in the control loop of dynamic mode high-speed atomic force microscopy (AFM). The closed-loop bandwidth will be limited by the convergence speed of the estimator. Recent developments have introduced new ways of demodulating the measured deflection signal. This article reviews and compares present methods for AFM amplitude demodulation and introduces a new Lyapunov based estimator. The performance of the techniques are discussed in terms of bandwidth, measurement noise, convergence time, unwanted harmonics, and complexity.

Topography and force imaging in atomic force microscopy by state and parameter estimation

A novel imaging method for atomic force microscopy based on estimation of state and parameters is presented. The cantilever dynamics is modeled as a linear system augmented by the tip-sample interaction force. The states of this augmented system are observed. The tip-sample force function is based on the Lennard-Jones potential with a nonlinearly parameterized unknown topography parameter. By estimating this parameter together with the tip-sample force using a nonlinear observer approach, the topography of the sample can be found. The observer and parameter estimator is shown to be exponentially stable. Simulation results are presented and compared to a more conventional extended Kalman filter.

A review of demodulation techniques for amplitude-modulation atomic force microscopy

In this review paper, traditional and novel demodulation methods applicable to amplitude-modulation atomic force microscopy are implemented on a widely used digital processing system. As a crucial bandwidth-limiting component in the z-axis feedback loop of an atomic force microscope, the purpose of the demodulator is to obtain estimates of amplitude and phase of the cantilever deflection signal in the presence of sensor noise or additional distinct frequency components. Specifically for modern multifrequency techniques, where higher harmonic and/or higher eigenmode contributions are present in the oscillation signal, the fidelity of the estimates obtained from some demodulation techniques is not guaranteed. To enable a rigorous comparison, the performance metrics tracking bandwidth, implementation complexity and sensitivity to other frequency components are experimentally evaluated for each method. Finally, the significance of an adequate demodulator bandwidth is highlighted during high-speed tapping-mode atomic force microscopy experiments in constant-height mode.

Imaging topography and viscoelastic properties by constant depth atomic force microscopy

Identification of mechanical properties of cells is known to be an effective tool for medical diagnosis, and holds potential for future developments in treatment of various diseases. In this paper a novel method for identification of viscoelastic properties of a soft sample using atomic force microscopy in dynamic mode is presented. The estimation scheme is based on parameter identification of a lumped spring-damper system model. The estimator guarantees exponentially fast parameter convergence. The indentation depth of the tip into the sample must be constant for viscoelastic properties to be consistent during a scan. A depth controller is designed to keep the indentation constant by utilizing the online estimates of the sample spring constant and topography. Simulations show the effectiveness of the presented method.

Frequency domain analysis of robust demodulators for high-speed atomic force microscopy

A fundamental but often overlooked component in the z-axis feedback loop of the atomic force microscope (AFM) operated in dynamic mode is the demodulator. Its purpose is to obtain a preferably fast and low-noise estimate of amplitude and phase of the cantilever deflection signal in the presence of sensor noise and additional distinct frequency components. In this paper, we implement both traditional and recently developed robust methods on a LabVIEW digital processing system for high-bandwidth demodulation. The techniques are rigorously compared experimentally in terms of measurement bandwidth, implementation complexity and robustness to noise. We conclude with showing high-speed tapping-mode AFM images in constant height, highlighting the significance of an adequate demodulator bandwidth.

Lyapunov estimation for high-speed demodulation in multifrequency atomic force microscopy

An important issue in the emerging field of multifrequency atomic force microscopy (MF-AFM) is the accurate and fast demodulation of the cantilever-tip deflection signal. As this signal consists of multiple frequency components and noise processes, a lock-in amplifier is typically employed for its narrowband response. However, this demodulator suffers inherent bandwidth limitations as high-frequency mixing products must be filtered out and several must be operated in parallel. Many MF-AFM methods require amplitude and phase demodulation at multiple frequencies of interest, enabling both z-axis feedback and phase contrast imaging to be achieved. This article proposes a model-based multifrequency Lyapunov filter implemented on a field-programmable gate array (FPGA) for high-speed MF-AFM demodulation. System descriptions and simulations are verified by experimental results demonstrating high tracking bandwidths, strong off-mode rejection and minor sensitivity to cross-coupling effects. Additionally, a five-frequency system operating at 3.5 MHz is implemented for higher harmonic amplitude and phase imaging up to 1 MHz.

Higher-harmonic AFM imaging with a high-bandwidth multifrequency Lyapunov filter

A major difficulty in multifrequency atomic force microscopy (MF-AFM) is the accurate estimation of amplitude and phase at multiple frequencies for both z-axis feedback and material contrast imaging. A lock-in amplifier is typically chosen for its narrowband response and ease of implementation. However, its bandwidth is limited due to post mixing low-pass filters and multiple are required in parallel for MF-AFM. This paper proposes a multifrequency demodulator in the form of a model-based Lyapunov filter implemented on a Field Programmable Gate Array (FPGA). System modelling and simulations are verified by experimental results demonstrating high tracking bandwidth and off-mode rejection at modelled frequencies. Additionally, AFM scans with a five-frequency-based system are presented wherein higher harmonic imaging is performed up to 1 MHz.

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

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.

H∞ Reduced Order Control for Nanopositioning: Numerical Implementability

Abstract In this paper we discuss how model reduction affects the stability and computational complexity of controllers for nanopositioning systems. A robust H ∞ multiple-input multiple-output controller is designed and implemented for the lateral stage of an atomic force microscope. A model-based controller can often be of high order and may be difficult to run in real-time on hardware with limited computational power. The resulting controller can be considered to be stiff, which is characterized by a large spread of eigenvalues. Continuous-time systems running in real-time are often solved using explicit Runge-Kutta (ERK) methods, which easily becomes unstable for stiff systems. We show how small the time-step for a given controller needs to be for a selection of ERK methods. We also consider how model reduction affects the computational complexity of the controller, and show how the reduction can alter the placement of the eigenvalues and thus the required step-size for implementability. We demonstrate that the original 18th-order H ∞ controller can be reduced to a 10th-order controller without any significant reduction in performance or stability, which results in a 46.7% reduction in execution time, partly because the order reduction enables the use of a simpler solver type.