Adaptive Control for General Nonlinear Stochastic Systems
Abstract
This paper extends the least-squares and certainty-equivalence minimum-variance framework of classical linear self-tuning regulators to a broad class of nonlinearly parameterized systems with compact parameter spaces, including commonly used deep neural network models. First, historical nonlinear least-squares fitting guarantees are converted into online one-step prediction bounds under a general modulus of continuity, with explicit regret rates under parameter Lipschitz continuity and finite-moment noise. Second, approximate NLS estimation combined with certainty-equivalence control achieves closed-loop stability and asymptotically optimal tracking under mild conditions without requiring excitation or parameter identifiability. Explicit closed-loop regret and tracking error rates are established in the parameter-Lipschitz case, and closed-loop examples show that the regret rates cannot, in general, be improved for this regulator. Finally, robustness to small unmodeled dynamics is established by showing that closed-loop stability is preserved and the tracking error bounds comprise a vanishing term and a residual term determined by the model discrepancy.
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BibTeX
@misc{li2026adaptive,
title = {{Adaptive Control for General Nonlinear Stochastic Systems}},
author = {Haizheng Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35182},
eprint = {2610.35182},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35182},
}
RIS
TY - UNPB
TI - Adaptive Control for General Nonlinear Stochastic Systems
AU - Haizheng Li
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35182
UR - https://arxiv.org/abs/2610.35182
AB - This paper extends the least-squares and certainty-equivalence minimum-variance framework of classical linear self-tuning regulators to a broad class of nonlinearly parameterized systems with compact parameter spaces, including commonly used deep neural network models. First, historical nonlinear least-squares fitting guarantees are converted into online one-step prediction bounds under a general modulus of continuity, with explicit regret rates under parameter Lipschitz continuity and finite-moment noise. Second, approximate NLS estimation combined with certainty-equivalence control achieves closed-loop stability and asymptotically optimal tracking under mild conditions without requiring excitation or parameter identifiability. Explicit closed-loop regret and tracking error rates are established in the parameter-Lipschitz case, and closed-loop examples show that the regret rates cannot, in general, be improved for this regulator. Finally, robustness to small unmodeled dynamics is established by showing that closed-loop stability is preserved and the tracking error bounds comprise a vanishing term and a residual term determined by the model discrepancy.
ER -