Closed-Loop Koopman Operator Approximation
This addresses a practical limitation in system identification for control engineers, but it is incremental as it extends existing Koopman methods to closed-loop scenarios.
The paper tackles the problem of identifying Koopman models for feedback-controlled systems, which is impractical for unstable systems in open-loop, by proposing a method that simultaneously identifies closed-loop and plant systems using controller knowledge, demonstrated with simulations and experiments on a Duffing oscillator and rotary inverted pendulum.
This paper proposes a method to identify a Koopman model of a feedback-controlled system given a known controller. The Koopman operator allows a nonlinear system to be rewritten as an infinite-dimensional linear system by viewing it in terms of an infinite set of lifting functions. A finite-dimensional approximation of the Koopman operator can be identified from data by choosing a finite subset of lifting functions and solving a regression problem in the lifted space. Existing methods are designed to identify open-loop systems. However, it is impractical or impossible to run experiments on some systems, such as unstable systems, in an open-loop fashion. The proposed method leverages the linearity of the Koopman operator, along with knowledge of the controller and the structure of the closed-loop system, to simultaneously identify the closed-loop and plant systems. The advantages of the proposed closed-loop Koopman operator approximation method are demonstrated in simulation using a Duffing oscillator and experimentally using a rotary inverted pendulum system. An open-source software implementation of the proposed method is publicly available, along with the experimental dataset generated for this paper.