Overview
This research presents a formal procedure for ensuring the safe control of feedback-interconnected systems, particularly those exhibiting different timescales in their dynamics. The approach leverages singular perturbations to extend safety certificates designed for reduced-order models to the full-order interconnected system. A key outcome is the design of a composite Control Barrier Function (CBF) that certifies the forward invariance of the safe set for the interconnected system, provided there is sufficient timescale separation between fast and slow dynamics.
Research Context
Control Barrier Functions (CBFs) are recognized tools for designing safety-critical controllers for nonlinear systems. Contemporary applications frequently involve complex systems structured as feedback interconnections of subsystems that operate at distinct timescales. Examples include robotic systems, where electrical and mechanical components may have different dynamic rates, or a physical plant integrated with an optimization or control algorithm. In such complex scenarios, safety constraints are often relevant to only a subset of the overall system's components.
Approach
Inspired by methodologies from singular perturbations used in stability analysis, the researchers developed a formal process to transfer a safety certificate formulated on a reduced-order model to the complete feedback-interconnected system. This procedure focuses on scenarios where there is a sufficient difference in the timescales of the slow and fast dynamics within the interconnected system. The method involves constructing a composite CBF. This composite CBF is designed to certify the forward invariance of the safe set for the overall interconnected system. A practical implication of this approach is that the online safety filter, which is responsible for maintaining safety during operation, only needs to be computed for the lower-dimensional, reduced-order model.
Findings
The research demonstrates that a composite CBF can be engineered to certify the forward invariance of the safe set for a feedback-interconnected system. This is achievable under the condition of sufficient timescale separation between the fast and slow dynamics of the system. A direct consequence of this finding is that the computational burden for the online safety filter can be significantly reduced, as it operates on the lower-dimensional, reduced-order model rather than the full-order system. Numerical tests were conducted to evaluate the proposed approach. These tests included two distinct applications: (i) a robotic arm that incorporates joint motor dynamics, and (ii) a physical plant that is regulated by an optimization algorithm.
Why This Matters
This approach simplifies the design and implementation of safety-critical controllers for complex, multi-timescale systems by reducing the computational load of online safety filters. It provides a formal method to ensure safety for interconnected systems where dynamics operate at different rates, a common characteristic in modern robotic and control applications.