J. Tommy Gravdahl

Damping and Tracking Control Schemes for Nanopositioning

Fast and accurate tracking of reference trajectories is highly desirable in many nanopositioning applications, including scanning probe microscopy. Performance in common positioning stage designs is limited by the presence of lightly damped resonances and actuator nonlinearities such as hysteresis and creep. To improve the tracking performance in such systems, several damping and tracking control schemes have been presented in the literature. In this paper, six different control schemes are presented and applied to a nanopositioning system for experimental comparison. They include schemes applying damping control in the form of positive position feedback, integral resonant control, integral force feedback, and passive shunt-damping. Also, general pole placement in the form of model reference control, as well as a control scheme requiring only a combination of a low-pass filter and an integrator, is presented. The control schemes are fixed-structure, low-order control laws, for which few results exist in the literature with regard to optimal tuning. A practical tuning procedure for obtaining good tracking performance for five of the control schemes is, therefore, presented. Experimental results show that the schemes provide similar performance, and the main differences are due to the specific implementation of each scheme.

Discrete-time repetitive control with model-less FIR filter inversion for high performance nanopositioning

Repetitive control (RC) is used to track and reject periodic exogenous signals. RC achieves tracking by incorporating a model of a periodic signal in the feedback path, which provides infinite loop-gain at the harmonic frequencies of the periodic signal. To improve robustness, the periodic signal model is bandwidth limited, and to improve the performance, an inverse plant response filter is used. This filter can either be an infinite impulse response (IIR) filter or a finite impulse response (FIR) filter. The accuracy of the filter typically determines the allowable bandwidth of the periodic signal model, and it is therefore desirable to obtain the most accurate inverse possible. In this paper a model-less method for synthesizing an FIR filter for the inverse response is presented, and it is compared to the common approach of using an inverse model-based IIR filter. An experimental comparison of the two approaches is presented, and it is demonstrated that the two methods produce identical results, but where the model-less FIR filter approach has the added benefit of avoiding the modeling effort needed to obtain the IIR filter.

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 Simplified Method for Discrete-Time Repetitive Control Using Model-Less Finite Impulse Response Filter Inversion

Repetitive control (RC) achieves tracking and rejection of periodic exogenous signals by incorporating a model of a periodic signal in the feedback path. To improve the performance, an inverse plant response filter (IPRF) is used. To improve robustness, the periodic signal model is bandwidth-limited. This limitation is largely dependent on the accuracy of the IPRF. A new method is presented for synthesizing the IPRF for discrete-time RC. The method produces filters in a simpler and more consistent manner than existing bestpractice methods available in the literature, as the only variable involved is the selection of a windowing function. It is also more efficient in terms of memory and computational complexity than existing methods. Experimental results for a nanopositioning stage show that the proposed method yields the same or better tracking performance compared to existing methods. [DOI: 10.1115/1.4033274]

Design of a nonlinear damping control scheme for nanopositioning

The application of a nonlinear control law for vibration damping on a typical nanopositioning system is investigated. The nonlinear control law is an augmentation of the linear integral force feedback scheme, where the constant gain used in integral force feedback, is replaced by a passive nonlinear operator. The nonlinear control law improves the performance of integral force feedback as it provides more rapid suppression of large disturbances, while maintaining low noise sensitivity. L2-stability for the control law is established. Experimental results are presented, showing improved performance when applying the nonlinear augmentation of the integral force feedback scheme, compared to the original linear integral force feedback scheme.

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.

Analog Robust Repetitive Control for Nanopositioning Using Bucket Brigade Devices

Abstract In many applications of nanopositioning, such as scanning probe microscopy, tracking fast periodic reference trajectories with high accuracy is highly desirable. Repetitive control is a simple and effective control scheme to obtain good tracking of such reference trajectories. In order to implement repetitive control, a method for introducing time-delay is necessary. This can easily be implemented using a memory buffer with digital signal processing equipment. To achieve fast, high accuracy, and low noise performance, fast microcontrollers or field-programmable gate array hardware with fast high-resolution analog-to-digital and digital-to-analog converters are needed. As an inexpensive alternative to digital signal processing, the use of an analog bucket brigade device to implement the time-delay is investigated in this paper. Bucket brigade devices use switching to carry the input voltage over an array of capacitors, achieving a specified time-delay. Low-noise bucket brigade devices can achieve a signal-to-noise ratio around 80 dB, comparable to the actual performance when using 16-bit analog-to-digital converters. In this paper, the proposed control scheme utilizes a modified integral control law in conjunction with the repetitive control law. The overall control scheme ensures robustness towards plant uncertainty. Experimental results demonstrate the effectiveness of the overall control scheme and the analog implementation.

Optimal Controller Gain Tuning for Robust Stability of Spacecraft Formation

The spacecraft formation control problem sets high demands to the performance, especially with respect to positional accuracy. The problem is further complicated due to scarce fuel resources and limited actuation effects, in addition to the many sources of disturbances. This paper addresses the problem of finding the optimal gains of spacecraft formation controllers. By optimal, we mean the gains that minimizes a cost functional which penalizes both the control efforts and the state deviation, while still guaranteeing stability of the closed-loop systems in the presence of disturbances.

Nonlinear tracking control scheme for a nanopositioner

The ability to track periodic reference trajectory signals fast and with good accuracy, is highly required in many different nanopositioning applications. Since different factors can affect the performance of such devices, like lightly damped resonances and actuator nonlinearties including hysteresis and creep, a number of control schemes have been presented in order to overcome these difficulties, in the recent literature. In the present paper a nonlinear feedback controller is proposed that includes both force and tracking control of a nanopositioner. The nonlinear controller is an augmentation of a linear integral force controller where the constant gain used in the integral force feedback, is replaced by a passive nonlinear operator. The nonlinear control law provides improved performance with regards to disturbance rejection and vibration damping over the linear control law. In addition, a feedback component is added. The stability of the overall closed loop system is analysed using the multivariable Popov criterion.