Data-Driven Domain of Attraction Estimation via Convergent Koopman-Zubov Approximation

Abstract

The computation of a domain of attraction (DOA) around an equilibrium point is a key issue in nonlinear stability analysis, which boils down to the difficult problem of searching for a Zubov function. With an operator-theoretical viewpoint of nonlinear systems, the concept of Zubov--Koopman operator has been introduced. However, due to the lack of convergence guarantee on the infinite-times action of Zubov--Koopman operator, the Zubov function estimate is unamenable to a theoretical bound under data-based learning errors. In this paper, considering a reproducing kernel Hilbert space (RKHS) with a linear--radial product kernel, the operator is proved to have a spectrum inside the unit circle. Hence, by augmenting this RKHS with constant-valued functions, the Zubov function that characterizes the DOA is obtained as the unique invariant element under the operator's action. This new RKHS formulation allows an efficient kernel-based estimation, which has an at most sectorially bounded error that scales down with the sample size. The proposed approach is tested with numerical examples, showing high accuracy of on-DOA/off-DOA classification of states, with two order-of-magnitude faster computation than neural networks.