Problem
Research questions and friction points this paper is trying to address.
domains of stabilization
stabilizable controlled invariant sets
input-constrained systems
discrete-time nonlinear systems
value function characterization
Set-Based Value Function Characterization and Neural Approximation of Stabilization Domains for Input-Constrained Discrete-Time Systems
🤖 AI Summary
This work addresses the challenging problem of jointly estimating the domain of stability (DOS) and a stabilizing controller for discrete-time nonlinear systems subject to input constraints. The authors propose a novel set-valued function defined in a metric space of sets, which enables the first-ever characterization of the DOS via a Zubov-type Bellman functional equation. This equation is seamlessly embedded into a physics-informed neural network architecture to facilitate end-to-end learning. By innovatively integrating set-valued functions with physics-informed mechanisms, the framework simultaneously estimates both the domain of stability and a stabilizing control policy. Numerical experiments on two benchmark examples demonstrate that the proposed approach accurately reconstructs the true domain of stability and successfully synthesizes effective stabilizing controllers.
📝 Abstract
Analyzing nonlinear systems with stabilizable controlled invariant sets (CISs) requires accurate estimation of their domains of stabilization (DOS) together with associated stabilizing controllers. Despite extensive research, estimating DOSs for general nonlinear systems remains challenging due to fundamental theoretical and computational limitations. In this paper, we propose a novel framework for estimating DOSs for controlled input-constrained discrete-time systems. The DOS is characterized via newly introduced value functions defined on metric spaces of compact sets. We establish the fundamental properties of these value functions and derive the associated Bellman-type (Zubov-type) functional equations. Building on this characterization, we develop a physics-informed neural network (NN) framework that learns the value functions by embedding the derived functional equations directly into the training process. The proposed methodology is demonstrated through two numerical examples, illustrating its ability to accurately estimate DOSs and synthesize stabilizing controllers from the learned value functions.
Innovation
Methods, ideas, or system contributions that make the work stand out.
value function on sets
domain of stabilization
controlled invariant set
physics-informed neural networks
Zubov-type equation
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