Closed-loop stabilization of nonlinear systems using Koopman Lyapunov-based model predictive control

Abstract

This work considers the problem of stabilizing feedback control design for nonlinear systems. To achieve this, we integrate Koopman operator theory with Lyapunov-based model predictive control (LMPC). A bilinear representation of the nonlinear dynamics is determined using Koopman eigenfunctions. Then, a predictive controller is formulated in the space of Koopman eigenfunctions using an auxiliary Control Lyapunov Function (CLF) based bounded controller as a constraint which enables the characterization of stability of the Koopman bilinear system. Unlike previous studies, we show via an inverse mapping - realized by continuously differentiable functions - that the designed controller translates the stability of the Koopman bilinear system to the original closed-loop system. Remarkably, the feedback control design proposed in this work remains completely data-driven and does not require any explicit knowledge of the original system. Moreover, in contrast to standard LMPC, seeking a CLF for the bilinear system is computationally favorable compared to the original nonlinear system. The application of the proposed method is illustrated on a numerical example.