Stability Guarantees for Data-Driven Predictive Control of Nonlinear Systems via Approximate Koopman Embeddings

TL;DR

This paper provides stability guarantees for data-driven predictive control of nonlinear systems by using approximate Koopman embeddings, enabling the application of linear stability theories directly to nonlinear data.

Contribution

It introduces a method to certify stability of nonlinear systems using approximate Koopman embeddings without requiring explicit knowledge of the embedding functions.

Findings

Koopman embeddings certify practical exponential stability for nonlinear systems.

The approach allows control directly on raw nonlinear data, bypassing the need for explicit lifting functions.

Demonstrated on a synchronous generator with explicit error bounds.

Abstract

Data-driven model predictive control based on Willems' fundamental lemma has proven effective for linear systems, but extending stability guarantees to nonlinear systems remains an open challenge. In this paper, we establish conditions under which data-driven MPC, applied directly to input-output data from a nonlinear system, yields practical exponential stability. The key insight is that the existence of an approximate Koopman linear embedding certifies that the nonlinear data can be interpreted as noisy data from a linear time-invariant system, enabling the application of existing robust stability theories. Crucially, the Koopman embedding serves only as a theoretical certificate; the controller itself operates on raw nonlinear data without knowledge of the lifting functions. We further show that the proportional structure of the embedding residual can be exploited to obtain an ultimate bound that depends only on the irreducible offset, rather than the worst-case embedding error. The framework is demonstrated on a synchronous generator connected to an infinite bus, for which we construct an explicit physics-informed embedding with error bounds.