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

The motivation of this research is to propose innovative nonlinear and optimal control design algorithms, which can be used in real life. The algorithms need to be computationally efficient, should deal with control constraints and should operate under state feedback. To show the efficacy of algorithms, automatic therapy for different cancer problems is chosen to be the field of application. In this thesis, first an advanced control design technique called ’optimal dynamic in-version’ has been successfully experimented with control constraints. The proposed approach has subsequently been shown to be quite effective in proposing automatic drug delivery schemes with simultaneous application of chemo and immunotherapy drugs for complete elimination of cancer cells in melanoma (a skin cancer) as well as glioma (a brain cancer). As per the current practice, the amount of drug dosages are generally given based on some apriori statistical study with a very small sample size, which in reality may either also lead to drug toxicity (due to excessive drug) or may become ineffective (due to insufficient drug) for a particular patient. Subject to the fidelity of the mathematical model (which has been taken from published literature), it has been shown in this thesis that nonlinear control theory can be used for computation of drug dosages, which can then be used in a feedback strategy, thereby customizing the drug for the patient’s condition, to cure the disease successfully. Next, attention has been shifted to impulsive control of systems. Such impulsive con-trol systems appear in many other applications such as control of swings, control of spacecrafts and rockets using reaction control system, radiotherapy in cancer treatment and so on. Two impulsive control design philosophies are proposed in this thesis. In one approach, recently proposed model predictive static programming (MPSP) has been extended for impulsive control systems and has been named as impulsive-MPSP (I-MPSP). In other approach, another recent development, namely the Pseudospectral method has been utilized to consider both the magnitude of the control impulses as well as the time instants at which they are applied as the decision variables. It can be noted, that to the best of the knowledge of the author, the time instants of control application, being considered as decision variables is being proposed for the first time in the nonlinear and optimal control framework. Both I-MPSP and Pseudospectral methods are computationally quite efficient and hence can be used for feedback control (I-MPSP happens to be computationally more efficient than the Pseudospectral method). Applicability of the proposed extensions have been shown by solving various benchmark problems such as (i) a scalar linear problem, (ii) Van der Pol’s oscillator problem and (iii) an inverted pendulum problem. Finally the applicability of the proposed I-MPSP strategy has been shown by solving challenging problems such as radiotherapy treatment of head and neck and adenocarcimona cancers. Radio-therapy model is considered with oxygen effect, in which radiosensitivity parameters are considered in different forms. Head and neck cancer is considered with constant radiosensitivity parameters and adenocarcinoma is considered with constant, linear, quadratic and saturation model of radiosensitivity parameters. Note that toxicity constraints on normal tissue, which are nonlinear control constraints, are also successfully incorporated in this control design.