Data-driven Stabilization of Discrete-time Control-affine Nonlinear Systems: A Koopman Operator Approach

TL;DR

This paper introduces a data-driven method using Koopman operators to stabilize discrete-time control-affine nonlinear systems by lifting them to a higher-dimensional bilinear space and designing feedback control laws.

Contribution

It presents a novel Koopman-based approach for stabilization of nonlinear systems directly from data, connecting controllability in the lifted space to the original system.

Findings

Successfully stabilized Van der Pol oscillator from data.

Effectively stabilized chaotic Henon map.

Demonstrated controllability transfer in lifted bilinear systems.

Abstract

In recent years data-driven analysis of dynamical systems has attracted a lot of attention and transfer operator techniques, namely, Perron-Frobenius and Koopman operators are being used almost ubiquitously. Since data is always obtained in discrete-time, in this paper, we propose a purely data-driven approach for the design of a stabilizing feedback control law for a general class of discrete-time control-affine non-linear systems. In particular, we use the Koopman operator to lift a control-affine system to a higher-dimensional space, where the control system's evolution is bilinear. We analyze the controllability of the lifted bilinear system and relate it to the controllability of the underlying non-linear system. We then leverage the concept of Control Lyapunov Function (CLF) to design a state feedback law that stabilizes the origin. Furthermore, we demonstrate the efficacy of the proposed method to stabilize the origin of the Van der Pol oscillator and the chaotic Henon map from the time-series data.