Safe Control of Feedback-Interconnected Systems via Singular Perturbations

TL;DR

This paper introduces a method to ensure safety in feedback-interconnected systems by extending safety certificates from simplified models to complex systems with different timescales, using singular perturbation techniques.

Contribution

It develops a formal procedure to lift safety certificates from reduced-order models to full interconnected systems under timescale separation, enabling simpler safety verification.

Findings

The approach guarantees forward invariance of the safe set in interconnected systems.

Safety filters can be implemented on lower-dimensional models, reducing computational complexity.

Numerical tests validate the effectiveness on robotic and physical systems.

Abstract

Control Barrier Functions (CBFs) have emerged as a powerful tool in the design of safety-critical controllers for nonlinear systems. In modern applications, complex systems often involve the feedback interconnection of subsystems evolving at different timescales, e.g., two parts from different physical domains (e.g., the electrical and mechanical parts of robotic systems) or a physical plant and an (optimization or control) algorithm. In these scenarios, safety constraints often involve only a portion of the overall system. Inspired by singular perturbations for stability analysis, we develop a formal procedure to lift a safety certificate designed on a reduced-order model to the overall feedback-interconnected system. Specifically, we show that under a sufficient timescale separation between slow and fast dynamics, a composite CBF can be designed to certify the forward invariance of the safe set for the interconnected system. As a result, the online safety filter only needs to be solved for the lower-dimensional, reduced-order model. We numerically test the proposed approach on: (i) a robotic arm with joint motor dynamics, and (ii) a physical plant driven by an optimization algorithm.