Safe Control of Feedback-Interconnected Systems via Singular Perturbations

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.

Figures (5)

• Figure C1: Block diagram representation of \ref{['eq:interconnected_system_generic']}.
• Figure C2: Block diagram representation of \ref{['eq:bl']} in original coordinates $(x,z)$.
• Figure D1: Graphical representation of the safety guarantees provided by Theorem \ref{['th:SP']} for a toy 2D example. (a) Level sets of the composite CBF. (b) Time evolution of the slow state $x$. (c) Space-time evolution of the feedback-interconnected system. Safe trajectories are depicted in solid green ($\epsilon < \bar{\epsilon}$), whereas unsafe trajectories are represented by dash-dotted red lines ($\epsilon \geq \bar{\epsilon}$).
• Figure F1: Results for the 2-DoF robotic arm with joint motors. Time evolution of $h(q, \omega)$ (left) and of $q = [\theta_1, \theta_2]^\top$ (middle). System trajectories in the joint space (right), where red areas indicate unsafe regions. Blue-to-cyan solid lines indicate safe trajectories (corresponding to increasing $\epsilon < \bar{\epsilon}$), whereas the dash-dotted red lines denote safety violations ($\epsilon \geq \bar{\epsilon}$).
• Figure F2: Time evolution of the CBF $h(x)$ for varying values of $\epsilon$. The green-to-red color gradient indicates increasing values of $\epsilon$, with the ideal CBF-QP solution represented by the dashed black line. Unsafe trajectories are dash-dotted.

Theorems & Definitions (4)

• Definition 1
• Definition 2
• Lemma 1
• Theorem 1