Neural Identification of Feedback-Stabilized Nonlinear Systems

TL;DR

This work addresses the challenge of identifying nonlinear, potentially unstable dynamical systems from closed-loop data by extending a dual Youla-inspired parameterization to nonlinear settings. It parameterizes all systems stabilized by a given controller as an interconnection of the controller and a trainable strictly causal operator in $\mathcal{L}_p^{SC}$, implemented via Recurrent Equilibrium Networks, and demonstrates asymptotic consistency in the linear case. The proposed Internal Controller Identification (ICI) framework enables three learning strategies, with the indirect-ICI approach via output linearization achieving stable, open-loop–equivalent identification that yields superior closed-loop predictive performance in both unstable scalar and planar-robot experiments. The method offers a scalable, stability-guaranteed path to fit expressive neural models to closed-loop data, with potential impact on adaptive control and safety-critical identification tasks.

Abstract

Neural networks have demonstrated remarkable success in modeling nonlinear dynamical systems. However, identifying these systems from closed-loop experimental data remains a challenge due to the correlations induced by the feedback loop. Traditional nonlinear closed-loop system identification methods struggle with reliance on precise noise models, robustness to data variations, or computational feasibility. Additionally, it is essential to ensure that the identified model is stabilized by the same controller used during data collection, ensuring alignment with the true system's closed-loop behavior. The dual Youla parameterization provides a promising solution for linear systems, offering statistical guarantees and closed-loop stability. However, extending this approach to nonlinear systems presents additional complexities. In this work, we propose a computationally tractable framework for identifying complex, potentially unstable systems while ensuring closed-loop stability using a complete parameterization of systems stabilized by a given controller. We establish asymptotic consistency in the linear case and validate our method through numerical comparisons, demonstrating superior accuracy over direct identification baselines and compatibility with the true system in stability properties.

Figures (4)

• Figure 1: Left: The closed-loop of system $\mathbf{G}$ and controller $\mathbf{K}$, with input $\mathbf{u}$, noisy output $\mathbf{y}$, excitation signal $\mathbf{r}$, and noise $\mathbf{v}$. Right: In the ICI framework, the model $\hat{\mathbf{G}}$ is an interconnection of a copy of the controller $\mathbf{K}$ and a trainable operator $\hat{\mathbf{S}} \in \mathcal{L}_p^{SC}$. The model input and output are denoted by $\hat{\mathbf{u}}$ and $\hat{\mathbf{y}}^\circ$, respectively.
• Figure 2: Top: Closed-loop system of $\hat{\mathbf{G}}$ and $\mathbf{K}$ (left) and its linearized approximation (right). Bottom: In the linear case, noise injection can be shifted to enable loop cancellation, yielding the exact simplified form on the right, consistent with \ref{['eq:linearized_out']}.
• Figure 3: Comparison of system and model outputs in the scalar unstable experiment. Top: true system output; Bottom: output of the model learned by \ref{['strat3']}; Left: open-loop; Right: closed-loop. Solid lines show the average output across multiple realizations of noise and excitation signals and shaded regions indicate $95\%$ confidence bounds. The open-loop confidence bounds are not visible due to their much smaller scale relative to the output. The learned model mirrors the true system’s stability properties: it is unstable in open-loop but stable in closed-loop. Closed-loop responses closely match, with a low MSE of $0.0034 \pm 0.0006$.
• Figure 4: Training trajectories in the robot experiment. Left: excitation standard deviation $\sigma = 10$; Right:$\sigma = 50$. The robot has an initial velocity of $[10, 0]^\top$ and operates in a loop with a controller that drives it to the origin. As expected, the robot deviates less from its intended trajectory under lower excitation, and more significantly when the excitation is higher. Colors indicate time progress.

Theorems & Definitions (9)

• Example 1: From state-space to input-output models
• Definition 1: Closed-loop $\mathcal{L}_p$-stability PerfBoost
• Remark 1: Recursive implementation of $\hat{\mathbf{S}}$
• Theorem 1
• Remark 2: Validity of linear approximations
• Remark 3: Open-loop equivalence
• Definition 2: Consistency Vandenhof98CLissues
• Proposition 1: Corollary 6 in Forssell99Revisited
• Corollary 1