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Thanin Quartz | Applied Mathematics, University of Waterloo
Data-Driven Methods for System Identification and Lyapunov Stability
This thesis focuses on data-driven methods applied to system identification and stability analysis of dynamical systems. In the first major contribution of the theorem we propose a learning framework to simultaneously stabilize an unknown nonlinear system with a neural controller and learn a neural Lyapunov function to certify a region of attraction (ROA) for the closed-loop system. The algorithmic structure consists of two neural networks and a satisfiability modulo theories (SMT) solver. The first neural network is responsible for learning the unknown dynamics. The second neural network aims to identify a valid Lyapunov function and a provably stabilizing nonlinear controller. The SMT solver then verifies that the candidate Lyapunov function indeed satisfies the Lyapunov conditions. We provide theoretical guarantees of the proposed learning framework in terms of the closed-loop stability for the unknown nonlinear system. We illustrate the effectiveness of the approach with a set of numerical experiments. We then examine another popular data driven method for system identification involving the Koopman operator. Methods based on the Koopman operator aim to approximate advancements of the state under the flow operator by a high-dimensional linear operator. This is accomplished by the extended mode decomposition (eDMD) algorithm which takes non-linear measurements of the state. Under the suitable conditions we have a result on the weak convergence of the eigenvalues and eigenfunctions of the eDMD operator that can serve as components of Lyapunov functions. Finally, we review methods for finding the region of attraction of an asymptotically stable fixed point and compare this method to the two methods mentioned above.