Neural Closed-Loop Identification of Nonlinear Systems with Stability and Generalization Guarantees

Neural networks are expressive models of nonlinear dynamics, but identifying such systems from closed-loop data raises two difficulties: the feedback loop correlates inputs with measurement noise, biasing standard open-loop estimators, and the identified model must be stabilized by the data-collection controller, so that its closed-loop behavior matches the true plant's. We derive a complete parameterization of all nonlinear systems stabilized by a given controller, generalizing the classical dual-Youla parameterization to the nonlinear setting. Building on this characterization, we propose Internal Controller Identification (ICI), a computationally tractable framework for identifying potentially unstable systems while ensuring closed-loop stability by construction; in the linear case, our indirect method reduces to an open-loop identification problem, for which standard consistency results apply. The framework accommodates contextual information, letting a single certified model identify a family of systems, and supports a PAC-Bayesian training objective that, when data are scarce and prior knowledge is available, bounds the identified operator's expected prediction loss. Experiments show that ICI identifies open-loop unstable plants where uncertified baselines diverge, outperforms direct methods under weak excitation and feedback-correlated noise, generalizes zero-shot to environments unseen during training, and, with limited data, avoids the overfitting of empirical risk minimization while providing a non-vacuous generalization bound.