From: 2024-06-05 13:14 to 14:30
Place: Seminar Room M 3170-73 at Dept. of Automatic Control, LTH
Contact: richard [dot] pates [at] control [dot] lth [dot] se

Date & Time: June 5, 13:15-14:30
Location: Seminar Room M 3170-73 at Dept. of Automatic Control, LTH
Speaker: Enrique Mallada, John Hopkins University
Title:  Model-Free Analysis of Dynamical Systems Using Recurrent Sets

Abstract:
In this talk, we develop model-free methods for analyzing dynamical systems using trajectory data. Our critical insight is to replace the notion of invariance, a core concept in Lyapunov Theory, with the more relaxed notion of recurrence. Specifically, a set is τ-recurrent (resp. k-recurrent) if every trajectory that starts within the set returns to it after at most τ seconds (resp. k steps). We leverage this notion of recurrence to develop several analysis tools and algorithms to study dynamical systems. Firstly, we consider the problem of learning an inner approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point using trajectory data. We show that a τ-recurrent set containing a stable equilibrium must be a subset of its ROA under mild assumptions. We then develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling finite-length trajectories. Secondly, we generalize Lyapunov's Direct Method to allow for non-monotonic evolution of the function values by only requiring sub-level sets to be τ-recurrent (instead of invariant). We provide conditions for stability, asymptotic stability, and exponential stability of an equilibrium using τ-decreasing functions (functions whose value along trajectories decrease after at most τ seconds) and develop a verification algorithm that leverages GPU parallel processing to verify such conditions using trajectories. We finalize by discussing future research directions and possible extensions for control.