Data-driven Model Predictive Control: Asymptotic Stability despite Approximation Errors exemplified in the Koopman framework

TL;DR

This paper proves that data-driven model predictive control can achieve asymptotic stability despite approximation errors, using Koopman-based surrogates and error bounds, validated through numerical simulations.

Contribution

It establishes conditions under which stability is maintained in data-driven MPC with approximation errors, specifically using Koopman operator theory and kernel methods.

Findings

Asymptotic stability is achievable without stabilizing terminal conditions.

Proportional bounds on approximation errors preserve cost controllability.

Numerical simulations confirm theoretical results.

Abstract

In this paper, we analyze stability of nonlinear model predictive control (MPC) using data-driven surrogate models in the optimization step. First, we establish asymptotic stability of the origin, a controlled steady state, w.r.t. the MPC closed loop without stabilizing terminal conditions for sufficiently long prediction horizons. To this end, we prove that cost controllability of the original system is preserved if sufficiently accurate proportional bounds on the approximation error hold. Here, proportional refers to state and control. The proportionality of the error bounds is a key element to derive asymptotic stability in presence of modeling errors and not only practical asymptotic stability. Second, we exemplarily verify the imposed assumptions for data-driven surrogates generated with kernel extended dynamic mode decomposition based on Koopman operator theory. Hereby, we do not impose invariance assumptions on finite dictionaries, but rather derive all conditions under non-restrictive conditions. Finally, we demonstrate our findings with numerical simulations.