Develops a framework for estimating safe, robust domains of attraction in uncertain, constrained nonlinear discrete-time systems.
The paper "Safe and Robust Domains of Attraction for Discrete-Time Systems: A Set-Based Characterization and Certifiable Neural Network Estimation" introduces a novel framework for estimating safe and robust domains of attraction (DOAs) in discrete-time nonlinear uncertain systems subject to state constraints and disturbances . The approach centers on characterizing DOAs through newly defined value functions on metric spaces of compact sets, which are associated with robust invariant sets (RISs) that exhibit uniform $$\ell_p$$ stability—a generalization encompassing uniform exponential and polynomial stability .
These value functions satisfy Bellman-type (Zubov-type) functional equations, whose solutions enable a precise characterization of the safe robust DOA . To overcome computational challenges in solving these equations directly, the authors develop a physics-informed neural network (NN) framework that learns the value functions by embedding the derived Bellman equations into the training process, ensuring the learned function adheres to the system's physical dynamics .
To ensure reliability, the method includes a verification step that uses formal verification tools to generate certifiable estimates of the DOAs from the neural network approximations, thereby providing formal safety and robustness guarantees . This integration of learning and verification is particularly relevant to AI research in safety-critical control systems, where ensuring stability and constraint satisfaction under uncertainty is paramount .
The framework is demonstrated on four numerical examples involving nonlinear uncertain systems with state constraints, showing improved performance over existing methods . Unlike prior approaches that rely on fixed Lyapunov function templates or restrictive assumptions like polynomial dynamics, this method requires only continuity of the system dynamics, openness of the safe set, and uniform $$\ell_p$$ stability of the RIS, significantly broadening its applicability
This research addresses the critical challenge of estimating the Domain of Attraction (DOA) for discrete-time nonlinear systems that operate under both state constraints and model uncertainties. The authors formulate the concept of a "Safe and Robust Domain of Attraction" (SR-DOA), defined as the set of initial conditions from which a system is guaranteed to converge to a desired equilibrium while strictly adhering to safety constraints despite bounded disturbances. Unlike traditional approaches that often focus solely on stability or rely on restrictive linearization, this work provides a comprehensive set-based characterization that rigorously accounts for the interplay between robust stability and safety in a discrete-time framework, which is essential for digital control implementations.
The key contribution of the paper is a novel computational framework that leverages neural networks to approximate the complex SR-DOA sets that are typically intractable to compute exactly for high-dimensional nonlinear systems. Crucially, the authors move beyond standard black-box learning by introducing a "certifiable" estimation method. By encoding the conditions for the SR-DOA into the training process and utilizing formal verification techniques—likely involving linear matrix inequalities (LMIs) or sum-of-squares (SOS) programming—the framework ensures that the neural network's output provides a mathematically rigorous under-approximation of the true safe region. This allows for the efficient computation of large, provably safe regions of attraction that scale better than purely analytical methods.
This work matters significantly for the deployment of control systems in safety-critical applications such as autonomous driving, robotics, and aerospace, where system failures due to model uncertainty or constraint violations are unacceptable. By bridging the gap between the high representational capacity of deep learning and the mathematical rigor of formal verification, the authors provide a practical tool for engineers to design controllers that are both robust and safe. This approach helps to overcome the scalability limitations of traditional control theory, enabling the analysis of complex, uncertain nonlinear systems that were previously beyond the reach of rigorous certifiable analysis.
Safe and Robust Domains of Attraction for Discrete-Time Systems: A Set-Based Characterization and Certifiable Neural Network Estimation
This paper presents a novel framework for estimating safe and robust domains of attraction (DoA) in uncertain, constrained nonlinear discrete-time systems. The authors leverage set-theoretic methods to characterize these domains, ensuring both stability and safety under bounded uncertainties and constraints. A key contribution is the use of certifiable neural networks to overapproximate the DoA, providing computationally efficient and verifiable estimates. The approach addresses the challenge of balancing computational tractability with theoretical guarantees, making it suitable for real-time applications in control and robotics.
The work is significant because it bridges the gap between rigorous theoretical analysis and practical implementation. By formalizing the DoA estimation problem using set-based methods and neural network overapproximation, the authors enable safe control in uncertain environments without requiring exact system models. This is particularly valuable for systems with complex dynamics, where traditional methods may fail due to conservatism or computational intractability. The paper's insights could advance the deployment of neural networks in safety-critical control applications, where robustness and certifiability are paramount.