Koopman-based feedback design with stability guarantees

TL;DR

This paper introduces a Koopman-based data-driven feedback control method that guarantees exponential stability for nonlinear systems, extending to multi-input cases with gain-scheduling and heuristic improvements, validated through numerical examples.

Contribution

It presents a novel Koopman-operator approach for stabilizing nonlinear systems directly from data, including stability guarantees and extensions to multi-input and gain-scheduling controllers.

Findings

Provides a semidefinite programming framework for controller design

Achieves larger regions of attraction with heuristic algorithms

Validated effectiveness through numerical simulations

Abstract

We present a method to design a state-feedback controller ensuring exponential stability for nonlinear systems using only measurement data. Our approach relies on Koopman-operator theory and uses robust control to explicitly account for approximation errors due to finitely many data samples. To simplify practical usage across various applications, we provide a tutorial-style exposition of the feedback design and its stability guarantees for single-input systems. Moreover, we extend this controller design to multi-input systems and more flexible nonlinear state-feedback controllers using gain-scheduling techniques to increase the guaranteed region of attraction. As the proposed controller design is framed as a semidefinite program, it allows for an efficient solution. Further, we enhance the geometry of the region of attraction through a heuristic algorithm that establishes a connection between the employed Koopman lifting and the dynamics of the system. Finally, we validate the proposed feedback design procedure by means of numerical examples.