ESE Fall Seminar – “Non-Parametric Analysis of Dynamical Systems: From Recurrent Sets to Generalized Lyapunov and Barrier Conditions”

This talk presents novel non-parametric methods for analyzing dynamical systems using solely trajectory data. Our critical insight is to replace the notion of invariance, a core concept in Lyapunov Theory, with the more relaxed condition of recurrence. Specifically, a set is τ-recurrent if every trajectory that starts within the set returns to it after at most τ seconds. 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 and develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence obtained by sampling finite-length trajectories. Secondly, we generalize Lyapunov and Barrier Function Methods 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 and safety using τ-monotonic functions (functions whose value along trajectories monotonically increases or decreases after at most τ seconds) and develop a verification algorithm that leverages GPU parallel processing power to verify stability and safety using only trajectory information. We finalize by discussing future research directions and possible extensions for control.