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

TL;DR

This paper introduces a set-based framework for accurately estimating safe and robust domains of attraction in discrete-time nonlinear uncertain systems, combining mathematical characterization with neural network learning and formal verification.

Contribution

It proposes a novel set-based characterization of DOAs using value functions and Bellman-type equations, and develops a neural network method with verification for certifiable estimates.

Findings

Effective estimation of safe and robust DOAs demonstrated on numerical examples.

The neural network approach accurately approximates value functions.

The verification procedure ensures certifiable safety guarantees.

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.