Error Amplification in Koopman Predictive Control: Computable Bounds, a Conditional Lyapunov Certificate, and Horizon-Consistent Estimation
- 1Department of Data and Systems Engineering, The University of Hong Kong, Pokfulam, Hong Kong SAR, China
- 2Mechanical Engineering College, Xi’an Shiyou University, Xi’an 710065, China
Abstract
Koopman model predictive control (MPC) iterates a lifted linear predictor over a horizon of tens of samples. Extended dynamic mode decomposition with control (EDMDc), however, fits that predictor by minimising only the one-step residual in the lifted space. This paper bounds the h-step output error of any lifted linear predictor by its one-step residual times an amplification factor Γh=∑j<h∥CAj∥2 that is computable from the model alone. The bound is attained for single-output systems, takes a horizon-uniform form through a scaled Lyapunov certificate, and extends to the suboptimality of the input sequence that MPC selects. The proposed HC-EDMDc minimises the rolled-out output error under a spectral-radius constraint, starts from the closed-form EDMDc solution as a warm start, and is selected by a held-out rolled-out criterion. The evaluation uses four measured benchmarks: a six-axis KUKA KR300 industrial robot, an electro-mechanical positioning system, a cascaded-tank process and the Silverbox oscillator. The one-step estimator returns a non-Schur transition matrix on 8 of 20 fits across 3 records, and its free run leaves the data range; the constraint removes this failure for at most 8.4 % of twenty-step accuracy. HC-EDMDc improves long-horizon accuracy on 2 records, reducing the twenty-step NRMSE on the cascaded tanks from 0.3431 for EDMDc + SR to 0.2827, and returns its warm start on the other 2; the size of the one-step residual does not predict which case a record falls into. Minimising εΓH collapses the predictor towards a mean-reverting model on 3 of the 4 records, and the certified bound moves opposite to the realised error on 2 of them. A valid bound is therefore not a model-selection criterion.
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BibTeX
@article{zhang2026error,
title = {{Error Amplification in Koopman Predictive Control: Computable Bounds, a Conditional Lyapunov Certificate, and Horizon-Consistent Estimation}},
author = {Jia’en Zhang and Di Tian},
journal = {Mathematics},
year = {2026},
month = nov,
doi = {10.3390/math14214047},
url = {https://www.mdpi.com/2227-7390/14/21/4047},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Error Amplification in Koopman Predictive Control: Computable Bounds, a Conditional Lyapunov Certificate, and Horizon-Consistent Estimation
AU - Jia’en Zhang
AU - Di Tian
PY - 2026
DA - 2026/11/04
JO - Mathematics
DO - 10.3390/math14214047
UR - https://www.mdpi.com/2227-7390/14/21/4047
AB - Koopman model predictive control (MPC) iterates a lifted linear predictor over a horizon of tens of samples. Extended dynamic mode decomposition with control (EDMDc), however, fits that predictor by minimising only the one-step residual in the lifted space. This paper bounds the h-step output error of any lifted linear predictor by its one-step residual times an amplification factor Γh=∑j<h∥CAj∥2 that is computable from the model alone. The bound is attained for single-output systems, takes a horizon-uniform form through a scaled Lyapunov certificate, and extends to the suboptimality of the input sequence that MPC selects. The proposed HC-EDMDc minimises the rolled-out output error under a spectral-radius constraint, starts from the closed-form EDMDc solution as a warm start, and is selected by a held-out rolled-out criterion. The evaluation uses four measured benchmarks: a six-axis KUKA KR300 industrial robot, an electro-mechanical positioning system, a cascaded-tank process and the Silverbox oscillator. The one-step estimator returns a non-Schur transition matrix on 8 of 20 fits across 3 records, and its free run leaves the data range; the constraint removes this failure for at most 8.4 % of twenty-step accuracy. HC-EDMDc improves long-horizon accuracy on 2 records, reducing the twenty-step NRMSE on the cascaded tanks from 0.3431 for EDMDc + SR to 0.2827, and returns its warm start on the other 2; the size of the one-step residual does not predict which case a record falls into. Minimising εΓH collapses the predictor towards a mean-reverting model on 3 of the 4 records, and the certified bound moves opposite to the realised error on 2 of them. A valid bound is therefore not a model-selection criterion.
SN - 2227-7390
ER -