226 research outputs found

    Undergraduate Catalog of Studies, 2023-2024

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    Exploiting Structural Properties in the Analysis of High-dimensional Dynamical Systems

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    The physical and cyber domains with which we interact are filled with high-dimensional dynamical systems. In machine learning, for instance, the evolution of overparametrized neural networks can be seen as a dynamical system. In networked systems, numerous agents or nodes dynamically interact with each other. A deep understanding of these systems can enable us to predict their behavior, identify potential pitfalls, and devise effective solutions for optimal outcomes. In this dissertation, we will discuss two classes of high-dimensional dynamical systems with specific structural properties that aid in understanding their dynamic behavior. In the first scenario, we consider the training dynamics of multi-layer neural networks. The high dimensionality comes from overparametrization: a typical network has a large depth and hidden layer width. We are interested in the following question regarding convergence: Do network weights converge to an equilibrium point corresponding to a global minimum of our training loss, and how fast is the convergence rate? The key to those questions is the symmetry of the weights, a critical property induced by the multi-layer architecture. Such symmetry leads to a set of time-invariant quantities, called weight imbalance, that restrict the training trajectory to a low-dimensional manifold defined by the weight initialization. A tailored convergence analysis is developed over this low-dimensional manifold, showing improved rate bounds for several multi-layer network models studied in the literature, leading to novel characterizations of the effect of weight imbalance on the convergence rate. In the second scenario, we consider large-scale networked systems with multiple weakly-connected groups. Such a multi-cluster structure leads to a time-scale separation between the fast intra-group interaction due to high intra-group connectivity, and the slow inter-group oscillation, due to the weak inter-group connection. We develop a novel frequency-domain network coherence analysis that captures both the coherent behavior within each group, and the dynamical interaction between groups, leading to a structure-preserving model-reduction methodology for large-scale dynamic networks with multiple clusters under general node dynamics assumptions

    Graduate Catalog of Studies, 2023-2024

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    Undergraduate Catalog of Studies, 2023-2024

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    Graduate Catalog of Studies, 2023-2024

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    Decentralised State Feedback Tracking Control for Large-Scale Interconnected Systems Using Sliding Mode Techniques

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    A class of large-scale interconnected systems with matched and unmatched uncertainties is studied in this thesis, with the objective of proposing a controller based on diffeomorphisms and some techniques to deal with the tracking problem of the system. The main research developed in this thesis includes: 1. Large-scale interconnected system is a complex system consisting of several semi-independent subsystems, which are typically located in distinct geographic or logical locations. In this situation, decentralised control which only collects the local information is the practical method to deal with large-scale interconnected systems. The decentralised methodology is utilised throughout this thesis, guaranteeing that systems exhibit essential robustness against uncertainty. 2. Sliding mode technique is involved in the process of controller design. By introducing a nonsingular local diffeomorphism, the large-scale system can be transformed into a system with a specific regular form, where the matched uncertainty is completely absent from the subspace spanned by the sliding mode dynamics. The sliding mode based controller is proposed in this thesis to successfully achieve high robustness of the closed-loop interconnected systems with some particular uncertainties. 3. The considered large-scale interconnected systems can always track the smooth desired signals in a finite time. Each subsystem can track its own ideal signal or all subsystems can track the same ideal signal. Both situations are discussed in this thesis and the results are mathematically proven by introducing the Lyapunov theory, even when operating under the presence of disturbances. At the end of each chapter, some simulation examples, like a coupled inverted pendulums system, a river pollution system and a high-speed train system, are presented to verify the correctness of the proposed theory. At the conclusion of this thesis, a brief summary of the research findings has been provided, along with a mention of potential future research directions in tracking control of large-scale systems, like more general boundedness of interconnections, possibilities of distributed control, collaboration with intelligent control and so on. Some mathematical theories involved and simulation code are included in the appendix section

    Undergraduate Catalog of Studies, 2022-2023

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    System Identification on the Families of Auto-Regressive with Least-Square-Batch Algorithm

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    The theories of system identification have been highly elaborated so as to achieve the true system. This paper much discuses regarding the stochastic processes along with the divergent of whether or not the system has zero-mean under scenario of either white and coloured noise. The mathematical foundations, including mean, variance, covariance, optimal parameters, along with some modified scenarios among them, are presented in detail from various system along with some basic idea behind them. The families of auto-regressive (1) and (2) are compared both mathematical and simulation in order to obtain the best design approaching the true system, Moreover, the least-square algorithm is used to examine the effectiveness of some number of iteration along with "batch" theoremComment: 8 pages; 10 figure
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