[Paper Review] Learning Stability Certificates from Data

Nicholas M. Boffi, Stephen Tu+3 more|arXiv (Cornell University)|Aug 13, 2020

Model Reduction and Neural Networks20 citations

TL;DR

This paper proposes a data-driven method to learn stability certificates for nonlinear dynamical systems directly from trajectory data, bypassing the need for explicit analytical models. By deriving generalization error bounds from data, the approach provides global stability guarantees and enables applications like adaptive control in complex robotic systems.

ABSTRACT

Many existing tools in nonlinear control theory for establishing stability or safety of a dynamical system can be distilled to the construction of a certificate function that guarantees a desired property. However, algorithms for synthesizing certificate functions typically require a closed-form analytical expression of the underlying dynamics, which rules out their use on many modern robotic platforms. To circumvent this issue, we develop algorithms for learning certificate functions only from trajectory data. We establish bounds on the generalization error - the probability that a certificate will not certify a new, unseen trajectory - when learning from trajectories, and we convert such generalization error bounds into global stability guarantees. We demonstrate empirically that certificates for complex dynamics can be efficiently learned, and that the learned certificates can be used for downstream tasks such as adaptive control.

Motivation & Objective

• To address the limitation of existing stability certificate synthesis methods that require closed-form analytical models of system dynamics.
• To develop algorithms that learn certificate functions solely from observed trajectory data, enabling application to modern robotic platforms with unknown or complex dynamics.
• To establish theoretical bounds on generalization error for learned certificates, ensuring reliability on unseen trajectories.
• To convert these generalization error bounds into global stability guarantees for the system.
• To demonstrate the practical utility of learned certificates in downstream control tasks such as adaptive control.

Proposed method

• The method formulates the certificate learning problem as a supervised learning task using trajectory data as training signals.
• It introduces a generalization error bound framework to quantify the risk that a learned certificate fails on new, unseen trajectories.
• The framework leverages statistical learning theory to derive probabilistic guarantees on certificate performance across unseen system trajectories.
• The approach uses a neural network or function approximator to represent the certificate function, trained to satisfy Lyapunov-like conditions on observed trajectories.
• It transforms the generalization error bound into a global stability certificate by ensuring the learned function satisfies a negative definite derivative condition in expectation.
• The method is validated through empirical evaluation on systems with complex, high-dimensional dynamics.

Experimental results

Research questions

• RQ1 Can stability certificates be reliably learned from trajectory data without requiring explicit analytical models of the system dynamics?
• RQ2 What is the generalization error of a learned certificate, and how can it be bounded to ensure reliability on unseen trajectories?
• RQ3 How can generalization error bounds be converted into formal global stability guarantees for the system?
• RQ4 Can learned certificates be effectively used in practical control applications such as adaptive control?
• RQ5 What is the empirical performance of the method on systems with complex or unknown dynamics?

Key findings

• The proposed method successfully learns stability certificates from trajectory data alone, without requiring closed-form system models.
• Generalization error bounds are derived and shown to provide reliable probabilistic guarantees on the performance of learned certificates.
• These bounds are formally converted into global stability guarantees, ensuring the system remains stable under the learned certificate.
• Empirical results demonstrate that the learned certificates can be effectively used in adaptive control tasks, improving system performance.
• The method is scalable and effective for complex, high-dimensional dynamical systems where analytical modeling is infeasible.
• The framework provides a practical pathway to apply formal stability verification in modern robotic systems with black-box or partially known dynamics.