Derivative-Based Koopman Operators for Real-Time Control of Robotic Systems

TL;DR

This paper introduces a derivative-based Koopman operator methodology for real-time control of nonlinear robotic systems, providing robust, data-driven linear models that improve control performance and facilitate efficient control design.

Contribution

The paper develops a novel, error-bounded Koopman modeling approach using higher-order derivatives, enabling real-time, data-driven control of nonlinear systems with robustness to noise.

Findings

The method achieves better control performance than nonlinear modeling techniques like SINDy and NARX.

It enables the use of linear control tools such as LQR for nonlinear systems.

Experimental results demonstrate superior performance over PID controllers and benefits of online model updating.

Abstract

This paper presents a generalizable methodology for data-driven identification of nonlinear dynamics that bounds the model error in terms of the prediction horizon and the magnitude of the derivatives of the system states. Using higher-order derivatives of general nonlinear dynamics that need not be known, we construct a Koopman operator-based linear representation and utilize Taylor series accuracy analysis to derive an error bound. The resulting error formula is used to choose the order of derivatives in the basis functions and obtain a data-driven Koopman model using a closed-form expression that can be computed in real time. Using the inverted pendulum system, we illustrate the robustness of the error bounds given noisy measurements of unknown dynamics, where the derivatives are estimated numerically. When combined with control, the Koopman representation of the nonlinear system has marginally better performance than competing nonlinear modeling methods, such as SINDy and NARX. In addition, as a linear model, the Koopman approach lends itself readily to efficient control design tools, such as LQR, whereas the other modeling approaches require nonlinear control methods. The efficacy of the approach is further demonstrated with simulation and experimental results on the control of a tail-actuated robotic fish. Experimental results show that the proposed data-driven control approach outperforms a tuned PID (Proportional Integral Derivative) controller and that updating the data-driven model online significantly improves performance in the presence of unmodeled fluid disturbance. This paper is complemented with a video: https://youtu.be/9_wx0tdDta0.