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

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.

💡 Research Summary

This paper addresses the stability of nonlinear model predictive control (MPC) when the prediction model is a data‑driven surrogate rather than the true plant dynamics. Classical MPC stability proofs rely on terminal costs, terminal constraints, or the exactness of the model used in the optimization. In the presence of modeling errors, only practical asymptotic stability—convergence to a neighbourhood of the desired equilibrium—can usually be guaranteed. The authors propose a new framework that yields true asymptotic stability of the origin without any terminal ingredients, provided that the prediction horizon is sufficiently long and that the surrogate model satisfies specific proportional error bounds.

The central technical assumption is a proportional error bound (P‑bound): the discrepancy between the true dynamics f(x,u) and the surrogate f_ε(x,u) is zero at the equilibrium and grows at most linearly with the magnitudes of the state and input, i.e.,
‖f(x,u) – f_ε(x,u)‖ ≤ c_ε^x‖x‖ + c_ε^u‖u‖.
A complementary uniform bound (U‑bound) provides a global constant η_ε such that the error never exceeds η_ε on a prescribed compact set. Both constants shrink to zero as the approximation accuracy parameter ε→0. Additionally, the surrogate model is required to be uniformly Lipschitz continuous with respect to the state, uniformly in ε.

Under these assumptions the authors prove that cost controllability—a property guaranteeing that the optimal finite‑horizon cost can be bounded by a multiple of the minimal stage cost—holds for the surrogate model whenever it holds for the true system. Cost controllability is the key ingredient that replaces terminal constraints: if the horizon N is long enough, a relaxed Lyapunov inequality of the form

V_N(f(x, μ_N^ε(x))) ≤ V_N(x) – α_N ℓ(x, μ_N^ε(x)), α_N∈(0,1]

holds for the value function V_N associated with the surrogate‑based optimal control problem. This inequality, together with the relaxed dynamic programming principle, yields asymptotic stability of the closed‑loop system driven by the MPC feedback μ_N^ε.

The second major contribution is the verification of the proportional and uniform error bounds for a concrete class of data‑driven models: kernel‑based extended dynamic mode decomposition (kEDMD) derived from Koopman operator theory. Koopman theory lifts a nonlinear system into a (potentially infinite‑dimensional) linear space of observables. kEDMD approximates the Koopman operator by projecting onto a reproducing kernel Hilbert space (RKHS) spanned by kernel functions centered at the data points. By exploiting recent finite‑sample error analyses for RKHS regression, the authors derive explicit expressions for c_ε^x, c_ε^u, and η_ε that satisfy the proportionality requirement and vanish as the number of samples grows. Importantly, they avoid restrictive invariance assumptions on a fixed dictionary; instead, the kernel automatically generates a flexible basis, and a carefully designed sampling scheme reduces the data burden. They also prove that the Lipschitz constant of the kEDMD surrogate remains uniformly bounded with respect to ε.

Numerical experiments on two benchmark nonlinear systems (a Van‑der‑Pol oscillator and a two‑degree‑of‑freedom robotic arm) illustrate the theory. The MPC controller uses only the surrogate model in the optimization, applies no terminal cost or constraint, and employs prediction horizons ranging from 15 to 25 steps. The closed‑loop trajectories converge to the origin, and the observed errors stay within the analytically predicted bounds. Comparisons with a nominal MPC that uses the true model confirm that the performance loss due to approximation is modest and fully explained by the derived error terms.

In summary, the paper establishes that:

1. Proportional error bounds are sufficient to preserve cost controllability and thus guarantee asymptotic stability of MPC without terminal ingredients.
2. Kernel‑based Koopman approximations can be constructed to satisfy these bounds under realistic data‑quantity assumptions.
3. The framework is model‑agnostic and can be extended to other learning‑based surrogates such as neural networks or Gaussian processes.

Future work suggested includes adaptive schemes that estimate and update the proportional error online, extensions to constrained systems, and scalability improvements for high‑dimensional applications.