Identifying a high-fidelity model of nonlinear dynamic systems is a prerequisite for achieving desired specifications in any model-based control design technique. This is because, most control design methods rely on the availability of an accurate model of the system dynamics and coarse dynamics models without generalization guarantees typically induce controllers that are either overly conservative with poor performance or violate spatiotemporal constraints imposed on the system when applied to the true system.This dissertation investigates the finite-time identification of deterministic and stochastic systems. First in Chapter 2, a novel finite-time distributed identification method is introduced for nonlinear interconnected systems. A distributed concurrent learning (CL) based discontinuous gradient descent (GD) update law is presented to learn uncertain interconnected subsystems' dynamics by minimizing the identification error for a batch of previously recorded data collected from each subsystem as well as its neighboring subsystems. The state information of neighboring interconnected subsystems is acquired through direct communication. Finite-time Lyapunov stability analysis is performed and easy-to-check rank conditions on the distributed memories data of subsystems are obtained, under which finite-time stability of the distributed identifier is guaranteed. These rank conditions replace the restrictive persistence of excitation (PE) conditions which are hard and even impossible to achieve and verify.Next, Chapter 3 presents a fixed-time system identifier for continuous-time nonlinear systems. A novel adaptive update law with discontinuous gradient flows of the identification errors is presented that leverages CL to guarantee the learning of uncertain dynamics in a fixed time. The CL approach retrieves a batch of samples stored in a memory and the update law simultaneously minimizes the identification error for current stream of samples as well as past memory samples. Fixed-time Lyapunov stability analysis certifies fixed-time convergence to the stable equilibria of the GD flow of the system identification error under easy-to-verify rank conditions.Chapter 4, an online data-regularized CL-based stochastic GD is presented for function approximation with noisy data. A fixed-size memory of past experiences is repeatedly used in the update law along with the current streaming data to provide probabilistic convergence guarantees with much-improved convergence rates (i.e, linear instead of sublinear) and less restrictive data richness requirements. This approach allows us to leverage the Lyapunov theory to provide probabilistic guarantees that assure convergence of the parameters to a probabilistic ultimate bound exponentially fast, provided that a rank condition on the stored data is satisfied. This analysis shows how the quality of the memory data affects the ultimate bound and can reduce the effects of the noise variance on the error bounds.In Chapter 5, deterministic and stochastic fixed-time stability of autonomous nonlinear discrete time (DT) systems are studied. Lyapunov conditions are first presented under which the fixed-time stability of deterministic DT systems is certified. Extensions to systems under deterministic perturbations as well as stochastic noise are then considered. For the former, the sensitivity to perturbations for fixed-time stable DT systems is analyzed, and it is shown that fixed-time attractiveness is resulted from the presented Lyapunov conditions. For the latter, sufficient Lyapunov conditions for fixed-time stability in probability of nonlinear stochastic DT systems are presented. The fixed upper bound of the settling-time function is derived for both fixed-time stable and fixed time attractive systems, and the stochastic settling-time function fixed upper bound is derived for stochastic DT systems.Finally, using the results of Chapter 5, in Chapter 6, a fixed-time identifier for modeling unknown DT nonlinear systems without requiring the PE condition is developed. A data-driven update law based on a modified GD update law is presented to learn the system parameters, which relies on CL. Fixed-time convergence guarantees are provided for the modified GD update law under a rank condition on the recorded data. To guarantee fixed-time convergence, fixed-time Lyapunov analysis is leveraged.
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