Koopman-Based Learning of Regions of Attraction for Unknown Systems

Estimating the region of attraction (ROA) of an asymptotically stable attractor is crucial in the analysis of nonlinear systems. We propose a lifting approach to map observable data into an infinite-dimensional function space, which generates a flow governed by the proposed Koopman-based operators. We can indirectly approximate the solution (value function) to Zubov’s Equation. The approximator's non-trivial sub-level sets, with values ranging from (0, 1], form the exact ROA. This approximation is achieved by learning the operator over a fixed time interval. We demonstrate that a transformation of such an approximator can be readily utilized as a near-maximal Lyapunov function. We show that this approach reduces the amount of data and can yield desirable estimation results.