A New Neural Framework for Certifying Safe Domains of Attraction in Uncertain Nonlinear Systems
Accurately determining the domain of attraction (DOA) for complex nonlinear systems remains a formidable challenge in control theory, especially when systems are subject to uncertainties and must operate within strict state constraints. A new research paper introduces a novel, certifiable framework that leverages physics-informed neural networks (PINNs) to estimate safe and robust DOAs for discrete-time nonlinear uncertain systems, offering a significant advance over existing methods.
The work specifically addresses systems with continuous dynamics, open safe sets, compact disturbance sets, and uniformly locally ℓp-stable compact robust invariant sets (RIS). This stability notion is highly general, subsuming special cases like uniform exponential and polynomial stability. The core innovation lies in characterizing DOAs through newly defined value functions on metric spaces of compact sets, for which the researchers establish fundamental mathematical properties and derive associated Bellman-type (Zubov-type) functional equations.
Embedding Physics into Neural Network Training
The theoretical characterization is made computationally tractable through a dedicated physics-informed neural network (NN) framework. Instead of relying solely on data, this approach directly embeds the derived Bellman-type equations into the neural network's training process. This physics-informed learning guides the NN to approximate the complex value functions that define the DOA boundaries more accurately and efficiently.
To transform these neural approximations into reliable, certifiable estimates of the safe robust DOA, the authors introduce a subsequent verification procedure. This step leverages existing formal verification tools to provide mathematical guarantees on the neural network's output, ensuring the estimated safe regions are rigorously validated.
Demonstrated Effectiveness and Comparative Performance
The proposed methodology's effectiveness was demonstrated through four distinct numerical examples involving nonlinear uncertain systems under state constraints. In these tests, the framework's performance was compared against existing methods from the literature, showing its applicability and advantages in providing accurate, verifiable DOA estimates for challenging real-world system models.
Why This Matters: Key Takeaways
- Bridges Theory and Computation: This research provides a concrete pathway from theoretical DOA characterization (via value functions and Bellman equations) to practical, computable solutions using modern machine learning.
- Enhances Safety Guarantees: By integrating a formal verification step, the framework moves beyond heuristic neural network predictions to deliver certifiable estimates of safe operating regions, which is critical for safety-critical applications in robotics, aerospace, and power systems.
- Addresses Real-World Complexity: The direct incorporation of state constraints and compact disturbance sets makes the approach particularly relevant for engineering systems that must operate reliably despite uncertainties and within predefined safe limits.