Robust Control for High Speed Atomic Force Microscopy

Abstract

Atomic force microscopes (AFMs) are used in many nanopositioning applications in order to measure the topography of various specimens at an atomic level through surface imaging. The imaging of samples in AFMs is carried out by using a three degree-of-freedom positioning unit called a piezoelectric tube scanner (PTS). The majority of the commercially available AFMs use PTS for x, y and z positioning because of its simplicity, large achievable scan range (>100 um) and smaller capacitance. In spite of having such good qualities, there are some limitations of the PTS which adversely affect the scanning speed and limit the overall performance of the AFM. The PTS of the AFM suffers from the problem of vibration, cross coupling effects between the axes of the scanner and nonlinear effects such as creep and hysteresis. This thesis presents several ways to compensate for the above mentioned problems of the PTS to improve the speed and accuracy of the PTS for high speed atomic force microscopy using robust control. The first contribution of this dissertation is the design of damping controllers to compensate for the effect of vibrations induced by the PTS. The first design uses a damping controller namely the resonant controller to damp the first resonant mode of the scanner. The design of the controller is presented both in single-input single-output (SISO) and multiple-input multiple-output (MIMO) forms. Experimental results show that the resonant controller significantly damps the first resonant mode of the scanner. One of the limitations of the use of the resonant controller is its high pass nature. The high pass nature of the resonant controller may add high frequency sensor noise and destabilise the system if there are unmodelled high frequency dynamics. In order to make the system robust to high frequency dynamics we next propose the design of a passive damping controller to damp the resonant modes of the scanner. The motivation to design a passive damping controller for the PTS is its bandpass nature. The bandpass nature of the passive damping controller not only reduces the addition of high frequency sensor noise but also results in large gain and phase margins. In order to design the passive damping controller for the PTS we have proposed a new analytical framework. The analytical framework examines the finite gain stability for a positive feedback interconnection between two stable linear systems where one system has mixed negative-imaginary (NI), passivity, and smallgain properties and the other system has mixed NI, negative-passivity (NP) and small-gain properties. The closed-loop system with this passive damping controller is robust against changes in the plant dynamics. Although the closed-loop system using the passive damping controller is robust, the performance of this controller is not same for all possible changes in plant dynamics. In order to achieve robust performance against changes in the plant dynamics, we propose another damping controller namely a minimax linear quadratic Gaussian (LQG) controller to compensate for induced vibrations in the PTS. This type of controller not only provides robust performance against changes in the plant dynamics but also results in large gain and phase margins. Due to its bandpass nature the minimax LQG controller also reduces the addition of high frequency sensor noise. A second contribution of this thesis is the design of an integral minimax LQG controller to improve the tracking performance of the PTS. The tracking accuracy of the PTS is hampered due to the low resonance frequency of the PTS. Here, we have proposed the design of a minimax LQG controller with integral action to track the reference triangular signal used for raster scanning in most commercial AFMs. The design of the controller includes uncertainty which arises due to the spill over dynamics of the system at high frequencies. Experimental results are compared with an integral controller to demonstrate the effectiveness of the proposed controller. The experimental results show that the integral minimax LQG controller achieves four times better performance as compared to the integral controller. The third and final contribution of this thesis is the design of a multi-variable controller for damping, tracking and cross coupling reduction of PTSs. At first we propose a design of multi-variable NI controllers for damping and cross coupling reduction of the PTS using a resonant and an integral resonant controller. Secondly, we propose a design of a double resonant controller with integral action to damp the resonant mode of the scanner, reduce the cross coupling effects of the scanner and improve the tracking performance of the PTS. The design of controllers in this case is done using a reference model matching approach. In all cases a performance comparison is made by implementing the controllers on the PTS. Experimental results presented in the thesis show that the proposed controllers provide significant improvement in the performance of the AFM.