<p dir="ltr">The problem of system identification is to learn the system dynamics from data. While classical system identification theories focused primarily on achieving asymptotic consistency, recent efforts have sought to characterize the number of samples needed to achieve a desired level of accuracy in the learned model. This thesis focuses on finite sample analysis for identifying/learning dynamical systems.</p><p dir="ltr">In the first part of this thesis, we provide novel results on finite sample analysis for learning different linear systems. We first consider the system identification problem of a fully observed system (i.e., all states of the system can be perfectly measured), leveraging data generated from an auxiliary system that shares ``similar" dynamics. We provide insights on the benefits of using the auxiliary data, and guidelines on selecting the weight parameter during the model training process. Subsequently, we consider the system identification problem for a partially observed autonomous linear system, where only a subset of states and multiple short trajectories of the system can be observed. We present a finite sample error bound and characterize the learning rate. </p><p dir="ltr">In the second part of this thesis, we explore the practical usage of finite sample analysis under several different scenarios. We first consider a parameter learning problem in a distributed setting, where a group of agents wishes to collaboratively learn the underlying model. We propose a distributed parameter estimation algorithm and provide finite time bounds on the estimation error. We show that our analysis allows us to determine a time at which the communication can be stopped (due to the costs associated with communications), while meeting a desired estimation accuracy. Subsequently, we consider the problem of online change point detection for a linear system, where the user observes data in an online manner, and the goal is to determine when the underlying system dynamics change. We provide an online change point detection algorithm, and a data-dependent threshold that allows one to achieve a pre-specified upper bound on the probability of making a false alarm. We further provide a finite-sample-based lower bound for the probability of detecting a change point with a certain delay.</p><p dir="ltr">Finally, we extend the results to linear model identification from non-linear systems. We provide a data acquisition algorithm followed by a regularized least squares algorithm, along with an associated finite sample error bound on the learned linearized dynamics. Our error bound demonstrates a trade-off between the error due to nonlinearity and the error due to noise, and shows that one can learn the linearized dynamics with arbitrarily small error given sufficiently many samples.</p>

Laser interferometry, commonly used in high-precision motion control systems, is rarely adopted in experimental vibration analysis because its installation and mounting is invasive to dynamical systems. However, metrology systems that already utilize laser interferometry, such as profilometry in semiconductor manufacturing, may benefit from interferometer feedback for signal processing. This study investigates the use of laser interferometry for system identification through a piezoelectrically actuated cantilevered beam. The model of the beam including piezo actuators and optical measurement components are established through the Euler-Bernoulli beam theory. From the method of separation of variables, the continuous system is transformed into a discrete system represented in a state-space form. By performing the Laplace transformation of the state-space form, we obtain the analytical transfer function of interferometer displacement versus actuator input, which is then validated numerically and experimentally. Adaptive filters based on FIR and IIR are designed to identify the transfer function. Because of the slow convergence of such filters, a recursive LMS algorithm is designed to accelerate computation. It is experimentally demonstrated that the precision measurement of interferometer can lead to highly accurate results of system identification.

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