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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

[pt] Considera-se, neste trabalho, um problema de controle ótimo no qual o critério ( relativo à atenuação de sinais de perturbação) e a restrição (relativa à margem de estabilidade) são, respectivamente, normas e H2 e H(infinito) ponderadas de funções de transferência em malha fechada. Um procedimento iterativo para o obtenção de soluções aproximadas, no qual somente problemas H2 irrestritos são resolvidos, é apresentado. Vários exemplos de aplicação deste procedimento são discutidos. / [en] In this work an optimal control problem is considered in which the cost function (pertaining to disturbance attenuation) and the constraint (due to unstructured, stability margin requirements) are, respectively, weighted H2 e H(infinity) norms of closed-loop transfer functions. An iterative scheme is described for generating aproximate soluctions in which only unconstrained H2 problems are solved. Some numerical examples are then discussed.

Nonlinear constrained optimal trajectory control is an important and fundamental area of research that continues to advance in numerous fields. Many attempts have been made to present new methods that can solve for optimal trajectories more efficiently or to improve the overall performance of existing techniques. This research presents a recently developed bio-inspired method called the Virtual Motion Camouflage (VMC) method that offers a means of quickly finding, within a defined but varying search space, the optimal trajectory that is equal or close to the optimal solution. The research starts with the polynomial-based VMC method, which works within a search space that is defined by a selected and fixed polynomial type virtual prey motion. Next will be presented a means of improving the solution’s optimality by using a sequential based form of VMC, where the search space is adjusted by adjusting the polynomial prey trajectory after a solution is obtained. After the search space is adjusted, an optimization is performed in the new search space to find a solution closer to the global space optimal solution, and further adjustments are made as desired. Finally, a B-spline augmented VMC method is presented, in which a B-spline curve represents the prey motion and will allow the search space to be optimized together with the solution trajectory. It is shown that (1) the polynomial based VMC method will significantly reduce the overall problem dimension, which in practice will significantly reduce the computational cost associated with solving nonlinear constrained optimal trajectory problems; (2) the sequential VMC method will improve the solution optimality by sequentially refining certain parameters, such as the prey motion; and (3) the B-spline augmented VMC method will improve the solution iv optimality without sacrificing the CPU time much as compared with the polynomial based approach. Several simulation scenarios, including the Breakwell problem, the phantom track problem, the minimum-time mobile robot obstacle avoidance problem, and the Snell’s river problem are simulated to demonstrate the capabilities of the various forms of the VMC algorithm. The capabilities of the B-spline augmented VMC method are also shown in a hardware demonstration using a mobile robot obstacle avoidance testbed.

With the recent trend for systems to be more and more autonomous, there is a growing need for cooperative trajectory planning. Applications that can be considered as cooperative systems such as surveying, formation flight, and traffic control need a method that can rapidly produce trajectories while considering all of the constraints on the system. Currently most of the existing methods to handle cooperative control are based around either simple dynamics and/or on the assumption that all vehicles have homogeneous properties. In reality, typical autonomous systems will have heterogeneous, nonlinear dynamics while also being subject to extreme constraints on certain state and control variables. In this thesis, a new approach to the cooperative control problem is presented based on the bio-inspired motion strategy known as local pursuit. In this framework, decision making about the group trajectory and formation are handled at a cooperative level while individual trajectory planning is considered in a local sense. An example is presented for a case of an autonomous farming system (e.g. scouting) utilizing nonlinear vehicles to cooperatively accomplish various farming task with minimal energy consumption or minimum time. The decision making and trajectory generation is handled very quickly while being able to consider changing environments laden with obstacles

Current trends in spacecraft are leading to smaller, more inexpensive options whenever possible. This shift has been primarily pursued for the opportunity to open a new frontier for technologies with a small financial obligation. Limited power, processing, pointing, and communication capabilities are all common issues which must be considered when miniaturizing systems and implementing low-cost components. This thesis addresses some of these concerns by applying two methods, in attitude estimation and control. Additionally, these methods are not restricted to only small, inexpensive satellites, but offer a benefit to large-scale spacecraft as well. First, star cameras are examined for the tendency to generate streaked star images during maneuvers. This issue also comes into play when pointing capabilities and camera hardware quality are low, as is often the case in small, budget-constrained spacecraft. When pointing capabilities are low, small residual velocities can cause movement of the stars in the focal plane during an exposure, causing them to streak across the image. Additionally, if the camera quality is low, longer exposures may be required to gather sufficient light from a star, further contributing to streaking. Rather than improving the pointing or hardware directly, an algorithm is presented to retrieve and utilize the endpoints of streaked stars to provide feedback where traditional methods do not. This allows precise attitude and angular rate estimates to be derived from an image which, with traditional methods, would return large attitude and rate error. Simulation results are presented which demonstrate endpoint error of approximately half a pixel and rate estimates within 2% of the true angular velocity. Three methods are also considered to remove overlapping star streaks and resident space objects from images to improve performance of both attitude and rate estimates. Results from a large-scale Monte Carlo simulation are presented in order to characterize the performance of the method. iii Additionally, a rapid optimal attitude guidance method is experimentally validated in a groundbased, pico-scale satellite test bed. Fast slewing performance is demonstrated for an incremental step maneuver with low average power consumption. Though the focus of this thesis is primarily on increasing the capabilities of small, inexpensive spacecraft, the methods discussed have the potential to increase the capabilities of current and future large-scale missions as well.

This thesis is concerned with electronic stability of an all-wheel drive electric vehicle with independent motors mounted in each wheel. The additional controllability and speed permitted using independent motors can be exploited to improve the handling and stability of electric vehicles. In this thesis, these improvements arise from employing a direct yaw moment control (DYC) system that seeks to adapt the understeer gradient of the vehicle and achieve neutral steer by employing a supervisory controller and simultaneously tracking an ideal yaw rate and ideal sideslip angle. DYC enhances vehicle stability by generating a corrective yaw moment realized by a torque vectoring controller which generates an optimal torque distribution among the four wheels. The torque allocation at each instant is computed by finding a solution to an optimization problem using gradient descent, a well-known algorithm that seeks the minimum cost employing the gradient of the cost function. A cost function seeking to minimize excessive wheel slip is proposed as the basis of the optimization problem, while the constraints come from the physical limitations of the motors and friction limits between the tires and road. The DYC system requires information about the tire forces in real-time, so this study presents a framework for estimating the tire force in all three coordinate directions. The sideslip angle is also a crucial quantity that must be measured or estimated but is outside the scope of this study. A comparative analysis of three different formulations of sliding mode control used for computation of the corrective yaw moment and an evaluation of how successfully they achieve neutral steer is presented. IPG Automotive’s CarMaker software, a high-fidelity vehicle simulator, was used as the plant model. A custom electric powertrain model was developed to enable any CarMaker vehicle to be reconfigured for independent control of the motors. This custom powertrain, called TVC_OpenXWD uses the torque/speed map of a Protean Pd18 implemented with lookup tables for each of the four motors. The TVC_OpenXWD powertrain model and controller were designed in MATLAB and Simulink and exported as C code to run them as plug-ins in CarMaker. Simulations of some common maneuvers, including the J-turn, sinusoidal steer, skid pad, and mu-split, indicate that employing DYC can achieve neutral steer. Additionally, it simultaneously tracks the ideal yaw rate and sideslip angle, while maximizing the traction on each tire[CB1] . The control system performance is evaluated based on its ability to achieve neutral steer by means of tracking the reference yaw rate, stabilizing the vehicle by means of reducing the sideslip angle, and to reduce chattering. A comparative analysis of sliding mode control employing a conventional switching function (CSMC), modified switching function (MSMC), and PID control (HSMC) demonstrates that the MSMC outperforms the other two methods in addition to the open loop system.

Every year the automotive industry strives to increase fuel efficiency in vehicles. When most vehicles are designed, fuel efficiency cannot always come first. The Shell Eco-marathon changes that by challenging students everywhere to develop the most fuel-efficient vehicle possible. There are many different factors that affect fuel efficiency, and different teams focus on different vehicle parameters. Currently, there is no straightforward design tool that can be used to help in Shell Eco-marathon vehicle design. For this reason, it is difficult to optimize every vehicle parameter for maximum fuel efficiency. In this study, a simulation is developed by using basic vehicle models and experimental data to accurately represent any prototype-class vehicle in the Shell Eco-marathon. This simulation is verified using different experimental data from an on-vehicle data acquisition system. An easy-to-use design tool is developed, and this tool is used to optimize driving strategy and final drive ratio to maximize fuel efficiency.

This thesis treats aspects of two fundamental problems in applied financial mathematics: calibration of a given stochastic process to observed marketprices on financial instruments (which is the topic of the first paper) and strategies for hedging options in financial markets that are possibly incomplete (which is the topic of the second paper). Calibration in finance means choosing the parameters in a stochastic process so as to make the prices on financial instruments generated by the process replicate observed market prices. We deal with the so called local volatility model which is one of the most widely used models in option pricing across all asset classes. The calibration of a local volatility surface to option marketprices is an ill-posed inverse problem as a result of the relatively small number of observable market prices and the unsmooth nature of these prices in strike and maturity. We adopt the practice advanced by some authors to formulate this inverse problem as a least squares optimization under the constraint that option prices follow Dupire’s partial differential equation. We develop two algorithms for performing the optimization: one based on techniques from optimal control theory and another in which a numerical quasi-Newton algorithmis directly applied to the objective function. Regularization of the problem enters easily in both problem formulations. The methods are tested on three months of daily option market quotes on two major equity indices.The resulting local volatility surfaces from both methods yield excellent replications of the observed market prices. Hedging is the practice of offsetting the risk in a financial instrument by taking positions in one or several other tradable assets. Quadratic hedging is a well developed theory for hedging contingent claims in incomplete markets by minimizing the replication error in a suitable L2-norm. This theory, though, is not widely used among market practitioners and relatively few scientific papers evaluate how well quadratic hedging works on real marketdata. We construct a framework for comparing hedging strategies, and use it to empirically test the performance of quadratic hedging of European call options on the Euro Stoxx 50 index modeled with an affine stochastic volatility model with and without jumps. As comparison, we use hedging in the standard Black-Scholes model. We show that quadratic hedging strategies significantly outperform hedging in the Black-Scholes model for out of the money options and options near the money of short maturity when only spot is used in the hedge. When in addition another option is used for hedging, quadratic hedging outperforms Black-Scholes hedging also for medium dated options near the money. / Den här avhandlingen behandlar aspekter av två fundamentala problem i tillämpad finansiell matematik: kalibrering av en given stokastisk process till observerade marknadspriser på finansiella instrument (vilket är ämnet för den första artikeln) och strategier för hedging av optioner i finansiella marknader som är inkompletta (vilket är ämnet för den andra artikeln). Kalibrering i finans innebär att välja parametrarna i en stokastisk process så att de priser på finansiella instrument som processen genererar replikerar observerade marknadspriser. Vi behandlar den så kallade lokala volatilitets modellen som är en av de mest utbrett använda modellerna inom options prissättning för alla tillgångsklasser. Kalibrering av en lokal volatilitetsyta till marknadspriser på optioner är ett illa ställt inverst problem som en följd av att antalet observerbara marknadspriser är relativt litet och att priserna inte är släta i lösenpris och löptid. Liksom i vissa tidigare publikationer formulerar vi detta inversa problem som en minsta kvadratoptimering under bivillkoret att optionspriser följer Dupires partiella differentialekvation. Vi utvecklar två algoritmer för att utföra optimeringen: en baserad på tekniker från optimal kontrollteori och en annan där en numerisk kvasi-Newton metod direkt appliceras på målfunktionen. Regularisering av problemet kan enkelt införlivas i båda problemformuleringarna. Metoderna testas på tre månaders data med marknadspriser på optioner på två stora aktieindex. De resulterade lokala volatilitetsytorna från båda metoderna ger priser som överensstämmer mycket väl med observerade marknadspriser. Hedging inom finans innebär att uppväga risken i ett finansiellt instrument genom att ta positioner i en eller flera andra handlade tillgångar. Kvadratisk hedging är en väl utvecklad teori för hedging av betingade kontrakt i inkompletta marknader genom att minimera replikeringsfelet i en passande L2-norm. Denna teori används emellertid inte i någon högre utsträckning av marknadsaktörer och relativt få vetenskapliga artiklar utvärderar hur väl kvadratisk hedging fungerar på verklig marknadsdata. Vi utvecklar ett ramverk för att jämföra hedgingstrategier och använder det för att empiriskt pröva hur väl kvadratisk hedging fungerar för europeiska köpoptioner på aktieindexet Euro Stoxx 50 när det modelleras med en affin stokastisk volatilitetsmodell med och utan hopp. Som jämförelse använder vi hedging i Black-Scholes modell.Vi visar att kvadratiska hedgingstrategier är signifikant bättre än hedging i Black-Scholes modell för optioner utanför pengarna och optioner nära pengarna med kort löptid när endast spot används i hedgen. När en annan option används i hedgen utöver spot är kvadratiska hedgingstrategier bättre än hedging i Black-Scholes modell även för optioner nära pengarna medmedellång löptid. / <p>QC 20141121</p>

<p>Current indirect optimal control problem (IOCP) solvers, like beluga or PINs, use symbolic math to derive the necessary conditions to solve the IOCP. This limits the capability of IOCP solvers by only admitting symbolically representable functions. The purpose of this thesis is to present a framework that extends those solvers to derive the necessary conditions of an IOCP with fully numeric methods. With fully numeric methods, additional types of functions, including conditional logic functions and look-up tables can now be easily used in the IOCP solver.</p> <p><br></p> <p>This aim was achieved by implementing algorithmic differentiation (AD) as a method to derive the IOCP necessary conditions into a new solver called Giuseppe. The Brachistochrone problem was derived analytically and compared Giuseppe to validate the automatic derivation of necessary conditions. Two additional problems are compared and extended using this new solver. The first problem, the maximum cross-range problem, demonstrates a trajectory can be optimized indirectly while utilizing a conditional density function that switches as a function of height according to the 1976 U.S. atmosphere model. The second problem, the minimum time to climb problem, demonstrates a trajectory can be optimized indirectly while utilizing 6 interpolated look up tables for lift, drag, thrust, and atmospheric conditions. The AD method yields the exact same precision as the symbolic methods, rather than introducing numeric error as finite difference derivatives would with the benefit of admitting conditional switching functions and look-up tables. </p>