PhD Comprehensive Exam | Ruikun Zhou, Learning-based Lyapunov Control of Unknown Nonlinear Systems with Stability Guarantees

MS Teams (please email amgrad@uwaterloo.ca for the meeting link)

Candidate

Ruikun Zhou | Applied Mathematics, University of Waterloo

Title

Learning-based Lyapunov Control of Unknown Nonlinear Systems with Stability Guarantees

 Abstract

Learning for control of dynamical systems with formal guarantees remains a challenging task. Thanks to state-of-the-art machine learning techniques and classical control theories, this issue is being addressed by data-driven control approaches. In this talk, I will present how to learn the unknown dynamics and obtain stability guarantees for nonlinear dynamical systems with two learning-based approaches: neural networks and the Koopman operator.

Frist, I will illustrate 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, where the two different neural networks are responsible for learning the unknown dynamics and identifying a valid Lyapunov function with a provably stabilizing nonlinear controller respectively, while the SMT solver then verifies that the candidate Lyapunov function indeed satisfies the Lyapunov conditions. Next, the proposal on estimating the maximal ROA of nonlinear systems with the Koopman operator and Lyapunov functions will be discussed. The Koopman operator provides a global linear representation of nonlinear systems by mapping it into some infinite-dimensional operator space, where the Koopman eigenfunctions parametrize a pool of Lyapunov function candidates. After being verified by the SMT solver, the sub-level sets of the remaining valid Lyapunov functions depict the estimated ROA. The effectiveness of both approaches is illustrated with numerical experiments on typical nonlinear control problems.