Control Barrier Functions With Unmodeled Input Dynamics Using Integral Quadratic Constraints

Abstract

This letter considers the design of controllers that achieve safety for systems with unmodeled dynamics at the plant input. Simplified, low-order models are often used in the design of such controllers. Unmodeled dynamics (e.g., actuator dynamics, time delays, etc.) can lead to safety violations. This letter proposes a method to achieve safety in the presence of these unmodeled dynamics. The approach combines control barrier functions (CBFs) and integral quadratic constraints (IQCs). First, the input-output behavior of the unmodeled dynamics is bounded in the time domain using an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> -IQC. Next, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> -IQC is incorporated into the CBF constraint to ensure safety. A safe controller is implemented by solving, in real-time, a convex optimization subject to the CBF constraint with the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> -IQC. The approach is demonstrated with a simple example.