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
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 ± standard deviation (SD).
Method
Data Required
Training Time
Landing (m)
Take-off (m)
Basic PID
×
—
2.2006±0.1456
0.1255±0.0266
PID + Neural Lander ( Shi et al. (2019) )
✓
88.2 s (per task)
0.5732±0.1188
0.2444±0.0297
PID + Neural Network Observer (Ours)
×
233.8 s (once)
0.1471±0.0602
0.0002±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 ± SD.
Method
Tracking Error (m)
Basic NMPC
1.4932±0.0199
NMPC + ESO ( Guo and Zhao (2011) )
0.0959±0.0088
NMPC + Neural Network Observer (Ours)
0.0499±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 σ 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±0.0402
0.49
Small NN
0.0558±0.0312
0.52
Tiny NN
0.2056±0.0402
0.49
Large NN
0.0118±0.0028
0.56
Appendix
Table 5: Quantitative performance comparison under extreme perturbation ( σ=0.5 ). Results are reported as mean ± 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 ± SEM across 100 independent trials.
Scenario
Method
Steady-state RMSE
Landing
Neural Lander
1.7295†
Landing
Neural observer
0.02788±0.01144
Take-off
Neural Lander
0.1815±0.0257
Take-off
Neural observer
(2.588±0.6381)×10−7
Appendix
Table 6: Quad-UAV steady-state ground-effect acceleration RMSE. Mean ± sample SD over ten seeds per scenario, in m/s2 . The marked entry reports the mean.
Channel
ESO
Neural observer
v
0.003759±0.000318
(3.540±1.055)×10−4
ω
(6.039±0.136)×10−5
(1.691±0.390)×10−5
dv
0.12520±0.01238
0.04993±0.01238
dω
(1.743±0.132)×10−6
(2.363±0.621)×10−3
Appendix
Table 7: WaterLily steady-state observer-error RMSE. Mean ± sample SD over five seeds for each observer, in native simulation units.
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.
Sunbochen Tang, Andrea Goertzen, Navid Azizan
Laboratory for Information and Decision Systems (LIDS), Massachusetts Institute of Technology, Cambridge, MA 02139, USA
We propose a framework for training neural network controllers with certified robust forward invariant polytopes. First, we parameterize a family of lifted control systems in a higher dimensional space, where the original neural controlled system evolves on an invariant subspace of each lifted system. We use interval analysis and neural network verifiers to further construct a family of lifted embedding systems, carefully capturing the knowledge of this invariant subspace. If the vector field of any lifted embedding system satisfies a sign constraint at a single point, then a certain convex polytope of the original system is robustly forward invariant. Treating the neural network controller and the lifted system parameters as variables, we propose an algorithm to train controllers with certified forward invariant polytopes in the closed-loop control system. Through two examples, we demonstrate how the simplicity of the sign constraint allows our approach to scale with system dimension to over 50 states, and outperform state-of-the-art Lyapunov-based sampling approaches in runtime.
Akash Harapanahalli, Samuel Coogan
School of Electrical and Computer Engineering, Georgia Institute of Technology
A learning-enabled disturbance-rejection framework based on a Neural Extended State Observer (Neural-ESO) is presented in this letter. Unlike existing learning-based control methods that largely rely on the learned model once deployed, Neural-ESO adopts a dual-pathway architecture: a predictive pathway uses a neural network to provide a feedforward disturbance estimate that accelerates convergence, while a corrective pathway employs a conventional ESO to compensate prediction errors and prevent over-reliance on the neural component. Using Lyapunov theory and a small-gain analysis, we show that enforcing a Lipschitz bound on the learning component guarantees uniform ultimate boundedness of the closed-loop error dynamics. The proposed framework is validated on a quadrotor landing task subject to strong ground-effect disturbances across normal and out-of-distribution scenarios, demonstrating accuracy-robustness trade-off and greater operational reliability during training, deployment, and transfer compared with state-of-the-art baselines.
Fan Zhang, Richie Suganda, Jinfeng Chen +4
Department of Engineering Technology, University of Houston, USA · Department of Electrical and Computer Engineering, University of Houston, USA · Department of Electrical Engineering, Rice University, USA