Closed-Loop Koopman Operator Approximation

TL;DR

This work addresses identifying Koopman representations for systems under feedback control by jointly inferring the closed-loop and plant models from closed-loop data with a known controller. The authors formulate a constrained EDMD framework that enforces the closed-loop structure in the lifted space and solve it as a semidefinite program to obtain $\mathbf{U}^{\mathrm{f}}$ and $\mathbf{U}^{\mathrm{p}}$, incorporating regularization to ensure stability. They demonstrate enhanced accuracy and correct stability properties on a simulated Duffing oscillator and an experimental Quanser QUBE-Servo pendulum, and provide open-source software and data. The approach reduces bias inherent in open-loop identification under feedback, enables stable closed-loop models across regularization, and offers a practical path for data-driven control-system identification in challenging settings.

Abstract

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.

Figures (10)

• Figure 1: Overview of the proposed closed-loop Koopman operator identification method. To simplify the presentation, no feedforward signal is used. (a) First, the controller reference, controller state, and plant state are recorded during a series of experiments. The controller state space matrices are known, and $\mbf{C}^\mathrm{p}$ and $\mbf{D}^\mathrm{p}$ are fixed. If the controller state is not directly available, it can be computed from its input and output. Only the Koopman matrix of the plant $\mbf{U}^\mathrm{p}$ is unknown. (b) The plant state is lifted and augmented with the controller state and reference. (c) The Koopman matrices of the closed-loop (CL) system and the plant system are approximated simultaneously by incorporating the known structure of the closed-loop system as a constraint on the Extended DMD (EDMD) problem.
• Figure 2: Series interconnection of the controller and plant.
• Figure 3: Feedback interconnection of the controller and plant.
• Figure 4: Duffing oscillator system with position $x(t)$, force $u(t)$, mass $m$, viscous damping $b$, linear stiffness $k_1$, and nonlinear stiffness $k_2$.
• Figure 5: Prediction errors the closed-loop and plant systems identified using ARX and EDMD methods.