A Data-Driven Prescribed-Time Control Framework via Koopman Operator and Adaptive Backstepping

TL;DR

This paper introduces a control framework that combines data-driven Koopman modeling with adaptive backstepping to achieve rapid, guaranteed stabilization of nonlinear systems within a prescribed time.

Contribution

It develops a novel Prescribed-Time Adaptive Backstepping controller based on Koopman linear models with uncertainty quantification, enhancing stability guarantees for complex nonlinear systems.

Findings

Successfully stabilizes Van der Pol oscillator within user-defined time.

Ensures boundedness of all closed-loop signals.

Demonstrates robustness to initial conditions and model uncertainties.

Abstract

Achieving rapid and time-deterministic stabilization for complex systems characterized by strong nonlinearities and parametric uncertainties presents a significant challenge. Traditional model-based control relies on precise system models, whereas purely data-driven methods often lack formal stability guarantees, limiting their applicability in safety-critical systems. This paper proposes a novel control framework that synergistically integrates data-driven modeling with model-based control. The framework first employs the Extended Dynamic Mode Decomposition with Control (EDMDc) to identify a high-dimensional Koopman linear model and quantify its bounded uncertainty from data. Subsequently, a novel Prescribed-Time Adaptive Backstepping (PTAB) controller is synthesized based on this data-driven model. The design leverages the structural advantages of Koopman linearization to systematically handle model errors and circumvent the "explosion of complexity" issue inherent in traditional backstepping. The proposed controller is validated through simulations on the classic Van der Pol oscillator. The results demonstrate that the controller can precisely stabilize the system states to a small neighborhood of the origin within a user-prescribed time, regardless of the initial conditions, while ensuring the boundedness of all closed-loop signals. This research successfully combines the flexibility of data-driven approaches with the rigor of Lyapunov-based analysis. It provides a high-performance control strategy with quantifiable performance and pre-assignable settling time for nonlinear systems, showcasing its great potential for controlling complex dynamics.