Safety Embedded Adaptive Control Using Barrier States

TL;DR

The paper tackles safe control for nonlinear systems under parametric uncertainty by embedding safety directly into augmented dynamics via Barrier States (BaS). It introduces Barrier States Embedded Adaptive Control Lyapunov Functions (BaS-aCLFs) to jointly achieve safety and stabilization despite unknown parameters, using a single adaptation law. A composite Lyapunov analysis guarantees adaptive asymptotic stability of the safety-embedded system, thereby ensuring forward safety for the original system. Numerical experiments on a planar quadrotor, an inverted pendulum, and adaptive cruise control demonstrate that BaS-aCLFs handle parameter uncertainty effectively, outperforming vanilla BaS and raCBFs in several scenarios. This framework advances safe nonlinear adaptive control by unifying safety embedding with adaptive design, reducing conservatism and easing design compared to traditional barrier-function approaches.

Abstract

In this work, we explore the application of barrier states (BaS) in the realm of safe nonlinear adaptive control. Our proposed framework derives barrier states for systems with parametric uncertainty, which are augmented into the uncertain dynamical model. We employ an adaptive nonlinear control strategy based on a control Lyapunov functions approach to design a stabilizing controller for the augmented system. The developed theory shows that the controller ensures safe control actions for the original system while meeting specified performance objectives. We validate the effectiveness of our approach through simulations on diverse systems, including a planar quadrotor subject to unknown drag forces and an adaptive cruise control system, for which we provide comparisons with existing methodologies.

Figures (4)

• Figure 1: An illustration of our framework where the blue quadrotor can navigate safely even with unmodeled external forces affecting it. Plain safe control without adaptation (red quadrotor) cannot handle the scenario because it does not consider the presence of the unmodeled external forces.
• Figure 2: Our method successfully stabilizes the quadrotor at $x=[per-mode = symbol]{0}{\meter}, y=[per-mode = symbol]{1}{\meter}, \psi = [per-mode = symbol]{0}{\rad}$ while BaS without adaptation cannot handle the task because of the uncertain $d_x$ and $d_y$.
• Figure 3: The proposed approach safely stabilizes the pendulum at $q=[per-mode = symbol]{0}{\rad}$. Vanilla BaS cannot handle the situation due to the uncertain $g$ and $b$.
• Figure 4: Our method increases the speed until the safe set's boundary set is approached and then it decreases the speed. Notice that it can approach the boundary with a small gain ($\Gamma = I$), while raCBF is more conservative even with a very high gain ($\Gamma = 200I$).

Theorems & Definitions (11)

• Definition 1
• Theorem 1: almubarak2023barrier
• Definition 2: Safe Adaptive Control
• Lemma 1
• proof
• Definition 3
• Lemma 2
• proof
• Remark 1
• Theorem 2