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

Laura Meroi; Mahrokh G. Boroujeni; Leonardo Massai; Giancarlo Ferrari-Trecate

eess.SYOct 20, 2026

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

Authors: Laura Meroi, Mahrokh G. Boroujeni, Leonardo Massai, Giancarlo Ferrari-Trecate

Organizations: Institute of Mechanical Engineering, EPFL, Switzerland.

Abstract

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.

Figures & tables

Panel of Figure 1 — Figure 1: Block diagram of the closed-loop system formed by system $\mathbf{G}$ and controller $\mathbf{K}$ , with system input $\mathbf{u}$ , noisy output $\mathbf{y}$ , excitation signal $\mathbf{r}$ , and noise $\mathbf{v}$ .

Panel of Figure 2 — Figure 2: In the ICI framework, the model $\hat{\mathbf{G}}_{\text{ICI}}$ consists of the negative feedback 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 $\hat{\mathbf{y}}$ , respectively.

Panel of Figure 3 — Figure 3: Top: Closed-loop system connecting the true plant $\mathbf{G}$ (represented within the ICI framework by the operator $\mathbf{S}$ ) and the controller $\mathbf{K}$ (left), alongside 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.

Section	System	Observation	ICI operator $\hat{\mathbf{S}}$	Baseline class	$N$	$T$	$\sigma_{r}$	$\sigma_{v}$	$N_{test}$	$T_{test}$
6.2	1	full state	deep SSM	ReLU RNN	$40$	$100$	$0.5$	$0.1$	$100$	$200$
6.3	2	positions	deep SSM	LSTM	$200$	$100$	$\{0.5,\dots,20\}$	$\{0,\dots,2\}$ , AR(1)	$100$	$100$
6.4	2	full state	deep SSM	deep SSM, no context	$60$	$50$	$20$	$0.1$	$100$ / $20$	$50$
6.5	2	positions	REN	—	$1$ – $64$	$50$	$20$	$0.5$	$100$	$50$

Table 1: Model classes and data of the four experiments.

N

T

are the number and length of the training trajectories,

\sigma_{r}

\sigma_{v}

the excitation and measurement-noise levels, and

N_{test}

T_{test}

the number and length of the test trajectories. Initial conditions are

\bar{x}\sim\mathcal{U}(-2,2)

(training) and

\bar{x}\sim\mathcal{U}(-4,4)

(redeployment test) in Section 6.2 , and

p_{0}\sim\mathcal{U}([-10,10]^{2})

with zero initial velocity elsewhere. The redeployment test of Section 6.2 is repeated at

\sigma_{v}\in\{0,\,0.3,\,0.6\}

for each of the

S=40

training runs per strategy.

Strategy	Identifiability of unstable G	Certified CL stability	Unstable CL runs
Indirect ICI	✓	✓	$0\%$
Direct ICI	—	✓	$0\%$
CLOE	✓	—	$2.5\%$
Direct OE	—	—	$25\%$

Table 2: Unstable scalar plant ( 29 ): identifiability of the unstable plant, certified closed-loop stability with the known controller, and fraction of the

S=40

independent training runs whose model the data-collection controller fails to stabilize (

\sigma_{v}=0.3

Figure 4: Unstable scalar plant. Open-loop trajectories (log scale,

8

steps) from

\bar{x}=0.1

of the models returned by

S=40

independent training runs (solid

=

median over runs, shaded

=

10

–

90

percentile band); the indirect strategies reproduce the diverging behavior, while both direct strategies return an inert surrogate.

	$\sigma_{v}{=}0.1$		$\sigma_{v}{=}0.5$		$\sigma_{v}{=}1$		$\sigma_{v}{=}2$
Strategy	OL	CL	OL	CL	OL	CL	OL	CL
Direct OE (LSTM)	$72.3\pm 13.3$	$83.0\pm 7.2$	$66.3\pm 17.2$	$80.7\pm 7.9$	$45.9\pm 22.5$	$72.4\pm 11.9$	$-36.2\pm 55.0$	$43.5\pm 30.7$
CLOE (LSTM)	$78.9\pm 10.6$	$87.0\pm 5.4$	$70.2\pm 16.5$	$82.8\pm 8.4$	$39.1\pm 34.1$	$77.1\pm 14.5$	$-86.1\pm 123.0$	$30.3\pm 54.8$
Direct ICI (SSM)	$82.9\pm 6.5$	$92.1\pm 3.1$	$79.0\pm 6.8$	$91.7\pm 3.0$	$62.0\pm 9.7$	$90.2\pm 3.0$	$-7.0\pm 33.6$	$87.8\pm 4.3$
Indirect ICI (SSM)	$86.7\pm 5.1$	$92.0\pm 4.4$	$84.3\pm 6.0$	$91.7\pm 3.7$	$74.0\pm 10.0$	$89.6\pm 3.9$	$60.9\pm 15.7$	$82.7\pm 6.1$
	$\sigma_{r}{=}1$		$\sigma_{r}{=}2$		$\sigma_{r}{=}5$		$\sigma_{r}{=}10$

Table 3: Point-mass robot under partial observation (Section 6.3 ): open-loop and closed-loop FIT [%], mean

\pm

std over the

100

test trajectories. Top: nominal excitation

\sigma_{r}{=}10

and AR(1) measurement noise (

\rho{=}0.9

) of increasing level

\sigma_{v}

. Bottom: noise-free training (

\sigma_{v}{=}0

) and increasing excitation level

\sigma_{r}

; at

\sigma_{r}{=}0.5

the LSTM baselines fall below

-400\%

open-loop FIT. Colored cells indicate the best result per column (ties are highlighted in both entries). Both sweeps are shown in Figure 6 ; the complete grids, including

\sigma_{r}\in\{0.5,20\}

, are in the repository.

Figure 5: Out-of-distribution excitation waveforms (square, multisine) at the training amplitude; training cell

\sigma_{r}=20

\sigma_{v}=1

, AR(1); channel

p_{y}

(the analogous

p_{x}

figure and all other training cells are in the repository). Indirect ICI transfers to the unseen shapes more accurately.

Figure 6: Point-mass robot: open-loop (top) / closed-loop (bottom) FIT, mean

\pm

std over

100

test trajectories. Left: noise-free training vs. excitation

\sigma_{r}

(observation (a)); right: AR(1) noise sweep at

\sigma_{r}=10

(observation (b)).

Figure 7: Distribution of the open-loop per-trajectory FIT for the two models (box

=

median and interquartile range, whiskers

=

Tukey,

\diamond=

mean; all seeds and test trajectories pooled). Columns vary the training regime (Case A, a single field / Case B, a family of fields) and rows the test set (seen fields, top / unseen fields, bottom); the

y

-axis is FIT in percent. The contextual model dominates the uncontextualized one in every panel, and under family training its seen- and unseen-field distributions nearly coincide.

Figure 8: Case B, open-loop response on a seen (left) and unseen (right) field:

p_{x}

(top),

v_{x}

(bottom); shaded

=

strong friction (

s_{e}>2.5

). The no-context prediction departs from the truth inside the patches, while the contextual model tracks it; the analogous

p_{y}

v_{y}

figures are in the public repository.

Figure 9: Case B: true open-loop test trajectories (black) and SSM-context rollouts (dashed) over the friction field $s_{e}(p)$ (darker $=$ stickier), on a seen (left) and unseen (right) field, under the causal predicted-position context. Dots mark the initial positions.

Figure 10: Certificate and overfitting diagnostics vs. sample size

N

. Left: the ( 27 ) bound for the transfer prior (blue) and the transfer-free data-split prior (light blue); the shaded band marks the vacuous region (

C=1

). Right: train (dark) vs. test (light) empirical risk ( 16 ) built on the untransformed loss ( 21 ) for the ERM (orange) and PAC-Bayes (blue) models, against noisy outputs (log scale). The left panel bounds the true risk in ( 15 ) for a draw from

\mathcal{Q}^{*}

; the right is an empirical diagnostic. The MSE noise floor is

\sigma_{v}^{2}=0.25

(dotted), and training below it indicates noise memorization.

Figure 11: Deployment FIT vs.

N

(mean over seeds): left, open loop; right, closed loop; against the noise-free plant. Key comparisons: 1) ERM methods (orange/gold) overfit to noise under data scarcity, degrading performance below the baseline prior. 2) The SVGD estimator (blue), anchored to the transfer prior , resists overfitting and preserves the prior’s quality at scarce

N

. ERM overtakes it only at larger sample sizes. 3) Relying solely on target data, the data-split prior approach (green) adapts well and maintains a valid certificate. 4) Using a wide prior (pink) yields high FIT but results in a vacuous certificate.

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