The Atomic Force Microscope (AFM) is a powerful tool for studying structure and dynamics at the nanometer scale. Despite its wide application in many applications, the slow imaging rate of AFM remains a severe limitation. Non-raster methods seek to overcome this limitation by appealing to alternative scan patterns, either designed to be easier for the actuators to follow or to reduce the amount of sampling needed. One particular example in this latter category is the local circular scan (LCS). LCS reduces the imaging time by scanning less sample area rather than scanning faster. It drives the tip of the AFM along a circular trajectory, using feedback to center that circle on a sample edge, and moving the circle along the feature, thus concentrating the samples to the region of interest. While this approach can have a significant impact on improving the imaging rate of any AFM, its impact is further enhanced when it is combined with high speed scanners. Due to its unique scanning pattern, a high-speed, Dual-Stage Actuator (DSA) system is a natural fit. DSAs consist of the serial combination of a (relatively) low-speed, long-range piezoelectric actuator (LRA) and a high-speed, short-range piezoelectric actuator (SRA). The SRA can be dedicated to implementing the local circular motion and the LRA to tracking the underlying sample. However, the control of a DSA scanner is challenging for at least three reasons: it is a multi-input, single-output system, it is a highly resonant system due to the underlying piezoelectric actuators, and it is a high-speed system. In this thesis, we address these challenges. First, we establish the controllability and observability of a general N-stage system whose outputs are summed to produce a single signal. This property allows us to develop individual controllers for the LRA and SRA of a DSA system so that we can focus our design on the specific requirements of each component and its desired action. While we apply both a Model Predictive Control (MPC) and simple state feedback approach to the LRA, our primary focus is on the SRA element as its high speed character makes it the more challenging component. Here we turn to receding horizon Linear Quadratic Tracking (LQT) control and develop methods to implement this approach at high speed using a Field Programmable Gate Array (FPGA). We develop three variants of LQT that differ in the required sample rates, memory resources, and computing power. Implementing and testing all three in both simulation and on a DSA scanning stage in our lab, we compare their performance and address the practical implementation considerations under the limitations imposed by the hardware. Finally, we combine the control of the LRA and SRA in two axes to demonstrate the LCS scanning approach. Overall, this thesis achieves a practical implementation of a model-based receding LQT design on a dual-stage, high speed, highly resonant actuator system. Through both simulation and experimental results, we demonstrate that this approach is robust to modeling error and disturbances and suitable for high-speed implementation of the LCS approach to non-raster AFM. / 2023-08-29T00:00:00Z