Safe and Robust Domains of Attraction for Discrete-Time Systems: A Set-Based Characterization and Certifiable Neural Network Estimation

2026-03-03

AI Summary

This work addresses the theoretical and computational challenges in accurately characterizing safe robust regions of attraction for discrete-time nonlinear systems subject to uncertainties and state constraints. The authors propose a unified framework based on set-valued functions in the space of compact sets, leveraging Bellman- or Zubov-type functional equations to characterize robustly invariant sets with uniform ℓp stability. To enhance generalization, they integrate physics-informed neural networks that embed the structural properties of these equations. Furthermore, formal verification tools are employed to rigorously certify the learned estimates, yielding provably safe and robust regions of attraction. Experimental evaluations on four representative uncertain nonlinear systems demonstrate that the proposed approach achieves superior accuracy and reduced conservativeness compared to existing methods.

Technology Category

• Intelligent Robots: State Estimation
• Natural Language Processing: Safety and Robustness
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Analyzing nonlinear systems with attracting robust invariant sets (RISs) requires estimating their domains of attraction (DOAs). Despite extensive research, accurately characterizing DOAs for general nonlinear systems remains challenging due to both theoretical and computational limitations, particularly in the presence of uncertainties and state constraints. In this paper, we propose a novel framework for the accurate estimation of safe (state-constrained) and robust DOAs for discrete-time nonlinear uncertain systems with continuous dynamics, open safe sets, compact disturbance sets, and uniformly locally $\ell_p$-stable compact RISs. The notion of uniform $\ell_p$ stability is quite general and encompasses, as special cases, uniform exponential and polynomial stability. The DOAs are characterized via newly introduced value functions defined on metric spaces of compact sets. We establish their fundamental mathematical properties and derive the associated Bellman-type (Zubov-type) functional equations. Building on this characterization, we develop a physics-informed neural network (NN) framework to learn the corresponding value functions by embedding the derived Bellman-type equations directly into the training process. To obtain certifiable estimates of the safe robust DOAs from the learned neural approximations, we further introduce a verification procedure that leverages existing formal verification tools. The effectiveness and applicability of the proposed methodology are demonstrated through four numerical examples involving nonlinear uncertain systems subject to state constraints, and its performance is compared with existing methods from the literature.

Problem

• domain of attraction
• robust invariant set
• state constraints
• uncertain systems
• discrete-time nonlinear systems

Innovation

• Domain of Attraction
• Robust Invariant Set
• Physics-Informed Neural Network
• Bellman-Type Equation
• Formal Verification