Learning Provable Neural Network Observer for Uncertain Dynamical Systems

Zhangyi Wang; Jiaxu Liu; Song Chen; Chao Xu; Shengze Cai

cs.LGSep 25, 2026

Learning Provable Neural Network Observer for Uncertain Dynamical Systems

Authors: Zhangyi Wang, Jiaxu Liu, Song Chen, Chao Xu, Shengze Cai

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

Abstract

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.

Figures & tables

Figure 1: Two-stage training framework for neural network observers. Stage I uses sampled error states for point-guided Lyapunov pre-training, and Stage II applies lightweight LMI fine-tuning to obtain an observer satisfying the global LMI-based Lyapunov stability certificate.

Architecture	Total Neurons	Solving/Training Method	Time (s)
Linear	0	Pole Placement	<1
Tiny NN	9	Direct LMI Solver	<1
Small NN	128	Direct LMI Solver	Infeasible
Large NN	576	Direct LMI Solver	Failed
Large NN	576	LMI Gradient Descent (Only)	2210.39
Large NN	576	Point & LMI Tuning (Ours)	895.31

Table 1: Computational efficiency and scalability of the two-stage training framework. Comparison of solving or training time across observer capacities and optimization methods.

Training Method	Time (s)	RMSE (m)
Point-Guided (Only)	99.53	1.0317
LMI Gradient Descent (Only)	2210.39	0.2659
Point & LMI Tuning (Ours)	895.31	0.2234

Table 2: Ablation study on training strategy for the Large NN observer. Comparison of training time and root mean squared estimation error (RMSE) for point-guided pre-training, pure LMI optimization, and the proposed two-stage Point & LMI Tuning method.

Figure 2: Ablation on training strategy for the Large NN observer. RMSE under Point-Only , LMI-Only , and the proposed two-stage scheme.

Figure 3: Generalization performance on a Quad-UAV under ground effect. Vertical trajectory tracking for landing (a) and take-off (b). Shaded regions denote the mean $\pm$ standard deviation (SD).

Method	Data Required	Training Time	Landing (m)	Take-off (m)
Basic PID	$\times$	—	$2.2006\pm 0.1456$	$0.1255\pm 0.0266$
PID + Neural Lander ( Shi et al. (2019) )	✓	88.2 s (per task)	$0.5732\pm 0.1188$	$0.2444\pm 0.0297$
PID + Neural Network Observer (Ours)	$\times$	233.8 s (once)	$\mathbf{0.1471\pm 0.0602}$	$\mathbf{0.0002\pm 0.0000}$

Table 3: Comparative analysis of tracking errors and data requirements for Quad-UAV control. Here, “per task” means that once the environment or disturbance changes, new data must be collected and the model must be retrained. The label “once” means that the same trained parameters can be reused across changes in the environment as long as the underlying system is unchanged.

Figure 4: AUV navigation in a simulated von Kármán vortex street. (a) The WaterLily fluid environment showing pressure fields and the AUV’s path. (b) Trajectory tracking results for different controllers. Shaded regions denote the mean $\pm$ SD.

Method	Tracking Error (m)
Basic NMPC	$1.4932\pm 0.0199$
NMPC + ESO ( Guo and Zhao (2011) )	$0.0959\pm 0.0088$
NMPC + Neural Network Observer (Ours)	$\mathbf{0.0499\pm 0.0019}$

Table 4: Disturbance rejection performance in dynamic fluid vortices. Tracking error of different control configurations in the vortex environment.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Residual network parameterization for the neural observer. The network uses hidden layers with the activation function $\sigma$ and a linear shortcut $W^{[L+2]}$ from the input to the output.

Architecture	Tracking Error (m)	Success Rate	Architecture	Tracking Error (m)	Success Rate
Linear	$0.2052\pm 0.0402$	0.49	Small NN	$0.0558\pm 0.0312$	0.52
Tiny NN	$0.2056\pm 0.0402$	0.49	Large NN	$\mathbf{0.0118\pm 0.0028}$	0.56

Appendix

Table 5: Quantitative performance comparison under extreme perturbation (

\mathbf{\sigma=0.5}

). Results are reported as mean

\pm

standard error of the mean (SEM) across 100 independent trials.

Figure 6: Robustness of neural network observers under structural perturbations in the X-29 aircraft benchmark. Tracking error (top) and success rate (bottom) across different network capacities. Error bars denote the mean $\pm$ SEM across 100 independent trials.

Scenario	Method	Steady-state RMSE
Landing	Neural Lander	$1.7295^{\dagger}$
Landing	Neural observer	$0.02788\pm 0.01144$
Take-off	Neural Lander	$0.1815\pm 0.0257$
Take-off	Neural observer	$(2.588\pm 0.6381)\times 10^{-7}$

Appendix

Table 6: Quad-UAV steady-state ground-effect acceleration RMSE. Mean

\pm

sample SD over ten seeds per scenario, in

\mathrm{m/s^{2}}

. The marked entry reports the mean.

Channel	ESO	Neural observer
$v$	$0.003759\pm 0.000318$	$(3.540\pm 1.055)\times 10^{-4}$
$\omega$	$(6.039\pm 0.136)\times 10^{-5}$	$(1.691\pm 0.390)\times 10^{-5}$
$d_{v}$	$0.12520\pm 0.01238$	$0.04993\pm 0.01238$
$d_{\omega}$	$(1.743\pm 0.132)\times 10^{-6}$	$(2.363\pm 0.621)\times 10^{-3}$

Appendix

Table 7: WaterLily steady-state observer-error RMSE. Mean

\pm

sample SD over five seeds for each observer, in native simulation units.

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LMI-Net: Linear Matrix Inequality--Constrained Neural Networks via Differentiable Projection Layers

Sunbochen Tang, Andrea Goertzen, Navid Azizan

Laboratory for Information and Decision Systems (LIDS), Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Constrained Optimization Differentiable Optimization Semidefinite Programming

Open paper AskPDF arXiv

Linear matrix inequalities (LMIs) have played a central role in certifying stability, robustness, and forward invariance of dynamical systems. Despite rapid development in learning-based methods for control design and certificate synthesis, existing approaches often fail to preserve the hard matrix inequality constraints required for formal guarantees. We propose LMI-Net, an efficient and modular differentiable projection layer that enforces LMI constraints by construction. Our approach lifts the set defined by LMI constraints into the intersection of an affine equality constraint and the positive semidefinite cone, performs the forward pass via Douglas-Rachford splitting, and supports efficient backward propagation through implicit differentiation. We establish theoretical guarantees that the projection layer converges to a feasible point, certifying that LMI-Net transforms a generic neural network into a reliable model satisfying LMI constraints. Evaluated on experiments including invariant ellipsoid synthesis and joint controller-and-certificate design for a family of disturbed linear systems, LMI-Net substantially improves feasibility over soft-constrained models under distribution shift while retaining fast inference speed, bridging semidefinite-program-based certification and modern learning techniques.

Abstract

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