Scalable Incremental Robustness Analysis of Neural Network Feedback Systems

Zichen Wang; Peter Seiler; Geir Dullerud; Bin Hu

math.OCSep 18, 2026

Scalable Incremental Robustness Analysis of Neural Network Feedback Systems

Authors: Zichen Wang, Peter Seiler, Geir Dullerud, Bin Hu

Organizations: Coordinated Science Laboratory and the Department of Electrical and Computer Engineering, University of Illinois Urbana–Champaign · Department of Electrical Engineering and Computer Science, at the University of Michigan · Electrical and Computer Engineering Department at the University of Minnesota

Abstract

Semidefinite programming (SDP) certificates for feedback systems containing deep neural networks (NNs) typically scale with the total number of neurons, whereas small-gain tests are scalable but can be highly conservative. This paper develops a unified and scalable framework for incremental robust stability and performance analysis of feedback interconnections involving high-dimensional NNs and unmodeled dynamics. By combining a structured decomposition of the full-order SDP condition with scalable Lipschitz constant estimation algorithms, we derive reduced verification conditions that certify incremental convergence and incremental $\ell_2$ -gain bounds. The dimensions of the resulting control-analysis linear matrix inequalities (LMIs) depend only on the widths of the last two network layers and are independent of network depth. The framework preserves the coupling between the plant and the NN, with the incremental small-gain condition recovered as a special case. To further reduce conservatism, we develop a multi-round alternating update scheme that iteratively refines the coupling variables while preserving scalability. Numerical experiments show that the proposed framework achieves state-of-the-art incremental $\ell_2$ -gain bounds for large-scale NN feedback systems.

Figures & tables

Panel of Figure 1 — Fig. 1 : Block diagram of the feedback interconnection $F(G,\Delta_{\mathrm{NN}},\Delta_{\text{U}})$ with exogenous input $d$ and performance output $e$ .

Method	Main control-analysis LMI dimension	Depth dependence	Coupling retained
Full-order SDP (Theorem 1 )	$m+\sum_{l}n_{l}$	Yes	Full layerwise structure
Chordal-DeepSDP [ 17 ]	Small cliques; number grows with $N$	Yes	Full structure via sparse decomposition
Small gain (Proposition 1 )	$m+n_{w}$	Via Lipschitz estimation only	Global scalar Lipschitz bound
Reduced condition (Theorem 2 )	$m+n_{N-1}+n_{w}$	Via Lipschitz estimation only	General plant–NN coupling via $Y_{1},Y_{2},Y_{3}$

TABLE I : Comparison of the dimensions and structural properties of representative verification conditions.

Baselines
Method	20	40	60	80	100	5000
Theorem 1	6.48	6.47	6.41	$\sim$	$\sim$	$\sim$
Chordal-DeepSDP	6.48	6.47	6.41	$\sim$	$\sim$	$\sim$
Small-Gain + ECLipsE	7.86	7.89	7.39	7.56	7.67	$>$ 1h
Small-Gain + ECLipsE-SN	10.75	10.23	9.16	10.40	14.74	10.61
Small-Gain + ECLipsE-GC	9.28	8.80	8.11	8.46	9.47	7.61

TABLE II : Upper bounds on the incremental

\ell_{2}

gain versus network depth (Instance 1)

Baselines
Method	20	40	60	80	100	5000
Theorem 1	7.76	7.80	7.68	$\sim$	$\sim$	$\sim$
Chordal-DeepSDP	7.76	7.80	7.68	$\sim$	$\sim$	$\sim$
Small-Gain + ECLipsE	10.64	10.68	10.58	10.59	10.81	$>$ 1h
Small-Gain + ECLipsE-SN	11.60	11.65	11.57	11.53	12.45	11.48
Small-Gain + ECLipsE-GC	10.74	10.84	10.73	10.73	11.40	10.29

TABLE III : Upper bounds on the incremental

\ell_{2}

gain versus network depth (Instance 2)

Panel of Figure 5 — Fig. 2 : Computational time versus network depth.

Baselines
Method	$\ell_{2}$ Gain Bound
Theorem 1	1.15
Chordal-DeepSDP	1.15
Small-Gain + ECLipsE	1.96
Small-Gain + ECLipsE-SN	2.10
Small-Gain + ECLipsE-GC	2.08

TABLE IV : Upper bound of the incremental

\ell_{2}

gain for the spacecraft docking example.

Neural Network Robustness

Neural network robustness studies how models withstand adversarial inputs, data shifts, noise, and hardware faults while maintaining reliable predictions.

See all 227 papers

Learning Provable Neural Network Observer for Uncertain Dynamical Systems
Sep 25, 2026Zhangyi Wang, Jiaxu Liu, Song Chen +2Neural Network Stability Lyapunov Stability
College of Control Science and Engineering, Zhejiang University, China · School of Science, Huzhou Normal University, China · Department of Mathematics, National University of Singapore, Singapore
$(\text{DNN})^2$ : Doubly Non-Negative Relaxations for Deep Neural Networks
Aug 25, 2026Hanna Jiamei Zhang, Alan Papalia, Michael Everett +1Neural Network Verification ReLU Neural Networks
Departments of Computer Science, Electrical and Computer Engineering, and Mathematics Northeastern University, Boston, MA. · Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, MI.
Closing the Loop: Branch-and-Bound for Scalable Verification of Nonlinear Neural Feedback Systems
Sep 14, 2026I. Samuel Akinwande, Mykel J. Kochenderfer, Clark BarrettNeural Network Verification Branch and Bound
Department of Aeronautics and Astronautics, Stanford University Stanford, CA, 94305, USA · Department of Computer Science, Stanford University Stanford, CA, 94305, USA

Sep 25, 2026J/K move · Enter open · S save

Learning Provable Neural Network Observer for Uncertain Dynamical Systems

Zhangyi Wang, Jiaxu Liu, Song Chen, Chao Xu+1

College of Control Science and Engineering, Zhejiang University, China · School of Science, Huzhou Normal University, China · Department of Mathematics, National University of Singapore, Singapore

Neural Network Stability Lyapunov Stability Dynamical Systems Neural Network Verification

Open paper AskPDF arXiv

In many safety-critical applications, control of uncertain dynamical systems relies on observers that estimate states and external disturbances. Neural network observers can improve estimation accuracy, but certifying their Lyapunov stability via Linear Matrix Inequality (LMI) constraints leads to large-scale semidefinite programs (SDPs) that are difficult to solve for large networks. To overcome this scalability bottleneck, we propose a novel two-stage training framework for provably stable neural network observers. Our approach decouples the optimization into a point-guided Lyapunov pre-training phase, which rapidly achieves high estimation accuracy and local stability over sampled states, followed by an LMI fine-tuning phase that efficiently satisfies a strict global Lyapunov stability certificate. We provide formal theoretical guarantees for local stability radii and probabilistic coverage over a prescribed compact error-state domain under specified regularity and sampling assumptions. Experiments on nonlinear control benchmarks and X-29 aircraft ablations show that our LMI-certified neural network observers train significantly faster than direct LMI-based methods and generalize robustly across diverse systems, achieving improved tracking accuracy over a range of observer baselines. The code is available at https://github.com/Berry-Myon/LearningNeuralNetworkObserver.

Abstract

Neural Network Robustness

Explore similar work

Learning Provable Neural Network Observer for Uncertain Dynamical Systems

(DNN)2(\text{DNN})^2(DNN)2: Doubly Non-Negative Relaxations for Deep Neural Networks

Closing the Loop: Branch-and-Bound for Scalable Verification of Nonlinear Neural Feedback Systems

$(\text{DNN})^2$ : Doubly Non-Negative Relaxations for Deep Neural Networks