Safe Control Design for Unknown Nonlinear Systems with Koopman-based Fixed-Time Identification

TL;DR

This paper introduces a Koopman-based fixed-time identification method for safe control of unknown nonlinear systems, enabling finite-time residual dynamics estimation and safe trajectory tracking.

Contribution

It develops a novel fixed-time identification scheme for Koopman generators and integrates it with control barrier functions for safety guarantees in nonlinear control.

Findings

Successfully estimates residual nonlinear dynamics in fixed time.

Achieves safe trajectory tracking in a quadrotor-inspired case study.

Provides explicit error bounds for residual estimation over time.

Abstract

We consider the problem of safe control design for a class of nonlinear, control-affine systems subject to an unknown, additive, nonlinear disturbance. Leveraging recent advancements in the application of Koopman operator theory to the field of system identification and control, we introduce a novel fixed-time identification scheme for the infinitesimal generator of the infinite-dimensional, but notably linear, Koopman dynamical system analogous to the nonlinear system of interest. That is, we derive a parameter adaptation law that allows us to recover the unknown, residual nonlinear dynamics in the system within a finite-time independent of an initial estimate. We then use properties of fixed-time stability to derive an error bound on the residual vector field estimation error as an explicit function of time, which allows us to synthesize a provably safe controller using control barrier function based methods. We conduct a quadrotor-inspired case study in support of our proposed method, in which we show that safe trajectory tracking is achieved despite unknown, nonlinear dynamics.