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

Amin Taghieh, SangWoo Park

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.

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

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.
Paper Structure (15 sections, 7 theorems, 57 equations)

This paper contains 15 sections, 7 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Motivation and Background
  3. Notation
  4. Synchronous Generator Model
  5. Continuous-Time Dynamics
  6. Discrete-Time Formulation
  7. Analysis of Approximate Koopman Embedding for the Generator Model
  8. Koopman Linear Embedding: Definition and Existence
  9. Koopman Embedding for the Synchronous Generator
  10. Stability of Data-Driven MPC Under Approximate Koopman Embeddings
  11. Data-Driven Representation
  12. From Exact to Approximate: Impact of Koopman Error on Data-Driven Control
  13. Tighter Bounds via Proportional Error Structure
  14. Conclusion
  15. Proof of Theorem \ref{['thm:gen_koopman']}

Key Result

Theorem 1

Consider the discretized generator eq:disc1--eq:disc3 under Assumption ass:operating. Define the lifting: Then $z_k = \Phi(x_k)$ satisfies an approximate Koopman linear embedding centered at the equilibrium $(x_s, u_s)$ with where $c_6, c_7$ depend on $\Delta t/T'_{d0}$, $\mu$, $E'_{q,\max}$, and $E_{fd}$ (see Appendix app:koopman_proof for details). The bound is proportional with offset: The p

Theorems & Definitions (21)

  • Definition 1: Koopman Linear Embedding shang2024
  • Definition 2: Approximate Koopman Embedding
  • Remark 1: Interpretation of Approximate Embedding
  • Theorem 1
  • proof
  • Remark 2: Non-Controllability and Non-Observability of the Embedding
  • Definition 3: Lifted Excitation shang2024
  • Theorem 2: Data-Driven Representation for the Generator (Exact Case)
  • proof
  • Lemma 3: Koopman Error as Bounded Output Noise
  • ...and 11 more