Safety-critical Control with Control Barrier Functions: A Hierarchical Optimization Framework

TL;DR

This paper proposes a hierarchical optimization framework that decomposes the multi-objective optimization problem into nested optimization sub-problems in a safety-first approach and addresses potential infeasibility on the premise of ensuring safety and performance as much as possible.

Abstract

The control barrier function (CBF) has become a fundamental tool in safety-critical systems design since its invention. Typically, the quadratic optimization framework is employed to accommodate CBFs, control Lyapunov functions (CLFs), other constraints and nominal control design. However, the constrained optimization framework involves hyper-parameters to tradeoff different objectives and constraints, which, if not well-tuned beforehand, impact system performance and even lead to infeasibility. In this paper, we propose a hierarchical optimization framework that decomposes the multi-objective optimization problem into nested optimization sub-problems in a safety-first approach. The new framework addresses potential infeasibility on the premise of ensuring safety and performance as much as possible and applies easily in multi-certificate cases. With vivid visualization aids, we systematically analyze the advantages of our proposed method over existing QP-based ones in terms of safety, feasibility and convergence rates. Moreover, two numerical examples are provided that verify our analysis and show the superiority of our proposed method.

Figures (12)

• Figure 1: A motivating example for sub-safe. With an unsafe initial distance between two vehicles, the existing method CLF-CBF QP QP_TAC is infeasible while Optimal-decay CLF-CBF QP optimal-decay_ACC drives the vehicle to speed up and collide. To consider such unsafe situations, we develop the new concept of sub-safe.
• Figure 2: A priority list shows different priority of all the metrics considered. A higher level means that the metric has a higher priority, such as $\mathscr{C}_{21}$ is prior to $\mathscr{C}_{11}$. The weight $c_{ij}$ denotes the relative importance among metrics within the same priority level.
• Figure 3: Illustrations of feasibility and infeasiblity of SU CLF-CBF QP.
• Figure 4: The bound determined by $\gamma$ when $h(x)<0$ at $t=0$.
• Figure 5: Convergence degradation caused by slack variables $\delta_1,\delta_2$.

Theorems & Definitions (10)

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