Department of Mathematics and Statistics Graduate Theses
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Item Model Predictive Control: Shortcomings and Resolutions(2024-10-07) Fuernsinn, Annika; Mathematics and Statistics; Gharesifard, BahmanThis thesis is concerned with model predictive control (MPC); a method that approximates solutions of an infinite-horizon optimal control problem by considering a sequence of finite-horizon optimal control problems implemented in a receding horizon fashion. In the standard scheme, one control input is implemented in each iteration, and a new initial state is set for the next finite-horizon. One critical issue within the MPC setup is guaranteeing stability, and much of the literature on the topic is devoted to this, mainly by using classical tools from Lyapunov theory. Such results often rely on suitable terminal ingredients, whose design significantly impacts the performance of MPC. As we explore in this thesis, the issue mentioned is more critical than only performance, and leads to major shortcomings. The main objective of this work is to mathematically describe the source of some of these issues, and provide some resolutions. In particular, we present a novel MPC scheme with relaxed stability criteria, based on generalized control Lyapunov functions. Most notably, this scheme allows for implementing a flexible number of control inputs in each iteration, in a computationally attractive manner, while guaranteeing recursive feasibility and stability. The advantages of our flexible-step implementation are demonstrated on nonholonomic systems, switched systems and lastly, in the setting of adaptive control. Specifically, we provide a systematic method for constructing generalized control Lyapunov functions for the novel MPC scheme in the case of linear systems. When the true linear system is unknown, generating terminal conditions is not possible. In fact, even when the estimation of the unknown system matrices is done through solving a least-squares problem, existing results that guarantee stabilizing controls rely on unnatural choices of terminal costs. We present an extension of our MPC scheme to the unknown setting and show convergence to the origin of the closed-loop system.Item Using a Neural Network Autoencoder Framework for Time Series Interpolation(2024-09-25) Callaghan, Evan James; Mathematics and Statistics; Takahara, Glen; Burr, WesleyThis thesis focuses on constructing a robust and effective time series data interpolation method; one that makes no assumptions about the underlying structure of the series and outperforms current state-of-the-art techniques. For this task, we turn to advanced neural network models. In this work, we detail the development, implementation, and testing of a neural network-based autoencoder model designed for time series interpolation. This method aims to extract key elements from the sequential input data and recover the missing data points by leveraging patterns found within complete sections of the time series. In the development stage, we discuss the proposed algorithm and summarize each step. In the implementation stage, we outline the Python and R code development, wrapping up with a proof-of-concept example. Lastly, in the testing stage, we perform a series of interpolation simulations to evaluate performance. Simulation results show several instances where our technique provides improved performance compared to other methods, most notably, the Hybrid Wiener Interpolator (HWI). Following the evaluation of our newly developed method, we apply adaptations to the algorithm, inspired by the HWI, to create a hybrid approach. These adaptations involve an extra data preprocessing step in which we detect trend and periodic components from the input series. Simulations of the hybrid approach show significant improvements over the original implementation for highly structured time series, and remain consistent for input time series with few structural components.Item Optimized Binary Constellation Design for Distributed Detection over Gaussian MAC Sensor Networks(2024-09-11) Sardellitti, Luca; Mathematics and Statistics; Alajaji, Fady; Takahara, GlenWe consider a distributed detection system transmitting a binary source over a Gaussian multiple access channel (MAC). We model the network via binary sensors whose outputs are generated by binary symmetric channels of different noise levels. We aim to find the optimal constellation design for each sensor under individual power constraints. We begin by analyzing a version of the problem where there are two sensors sending a uniform source over the channel. In an introductory investigation, we assume each sensor uses all available power, but can change the relative angle between the constellations. Although explicit analysis of error probability in this setup is infeasible, an upper bound on the error probability is optimized, and is numerically shown to coincide very well with the true optimal rotation angle. Interestingly, this upper bound ignores all information in the imaginary axis, and could be equivalently achieved by using less power if both sensors were sharing a one-dimensional MAC. This led to the problem formulation of optimal power allocation for one-dimensional constellation designs. We next consider two binary sensors sending a non-uniform source over a one dimensional Gaussian MAC. We prove an optimal constellation design under individual sensor power constraints which minimizes the error probability of detecting the source. Three distinct cases arise for this optimization based on the parameters in the problem setup. In the most notable case (Case III), the optimal signaling design is to not necessarily use all of the power allocated to the more noisy sensor (i.e., whose output has less correlation to the source). We compare the error performance of the optimal one dimensional constellation to orthogonal signaling. The results show that the optimal one dimensional constellation achieves lower error probability than using orthogonal channels. Finally, we extend the problem to N sensors by making a simplifying assumption that the detection (fusion) center will not use maximum-a-posteriori (MAP) detection, but instead will use a simplified rule where the real line can only be split into left and right decision regions by a single decision boundary. Under this assumption, we characterize the optimization of any individual sensor's power allocation when the rest are fixed. This optimization is also divided into three distinct cases which directly mirror those from the two sensor problem. In the equivalent situation to Case III, the sensor should not necessarily use all of its power. However, unlike the two sensor case, the optimal power allocation does not generally have a closed form expression, and instead must be found numerically. We use the individual sensor optimization results to form an iterative algorithm for the joint optimization of all N sensors. We use numerical examples to compare the algorithm to other signaling designs, observing that the algorithm achieves consistently lower error probability than any design that also uses the simplified detection. Further, the algorithm is only slightly outperformed in some situations, particularly at high signal-to-noise ratio (SNR), by signaling techniques such as orthogonal signaling or using MAP detection, which typically require more power, bandwidth and overall implementation complexity.Item The Orthogonal Weingarten Calculus and Free Probability(2024-09-05) Gawlak, Dylan; Mathematics and Statistics; Mingo, JamesWe provide a self-contained derivation of the Weingarten calculus for compact matrix groups. We also provide a self-contained introduction to the combinatorics of the symmetric group needed for computations in free probability. We introduce partitioned pairings, an analogue of partitioned permutations. Using partitioned pairings, we will derive expressions for the leading and subleading terms of the Weingarten function for the orthogonal group. Using the combinatorial theory that we have developed, as well as what is called second-order real freeness, we will determine the first and second-order limiting distributions of the random matrix ensemble Xm = O1+Ot1+· · +·Om+Otm where {Oi} are independent Haar-distributed orthogonal matrices.Item Curves of genus 2 and quadratic forms(2024-09-05) Kir, Harun; Mathematics and Statistics; Kani, ErnstOur approach focuses on a special associated integral quadratic form qC, which is intrinsically attached to a curve C of genus 2. This form, known as the refined Humbert invariant, was introduced by Kani (1994). Our first main result is the classification of those imprimitive ternary quadratic forms which can occur as one of the forms qC, for some curve C of genus 2. We use this classification to classify the qC’s for curves with a prescribed group of automorphisms, and/or with some additional structure such as having an elliptic subcover of prescribed degree. This study, along with other results, has concrete applications on understanding the nature of intersections of Humbert surfaces. For instance, we prove that the intersection of two Humbert surfaces is not empty, and we determine the intersection of infinitely many prescribed Humbert surfaces in terms of ternary qC’s. To obtain these results, and many others, it is useful to study generalized Humbert sets. A generalized Humbert set H(q), associated with a given quadratic form q, was introduced by Kani in order to understand the nature of curves of genus 2. Regarding the structure of these sets, we find all ternary quadratic forms q for which H(q) is irreducible by relying on SAGE. Additionally, in our joint work with Kani, we give formulas for the cardinality of H(q) for any ternary form q. As an application of this formula, we establish formulas for the number of isomorphism classes of genus 2 curves with certain specified properties, such as curves with a prescribed automorphism group and having an elliptic subcover of a prescribed degree. We examine Shimura curves following the approaches of Lin and Yang, who studied these curves by associating positive binary quadratic forms with them. We clarify the connection between these forms and the refined Humbert invariant. By doing so, we obtain interesting results, such as improving a criterion for the existence of CM points on Shimura curves given by Lin and Yang.