Prescribed-Time Safety Design for a Chain of Integrators

TL;DR

This paper introduces a prescribed-time safety control method for chains of integrators that ensures safety only within a specified finite time, using a novel combination of control barrier functions, backstepping, and quadratic programming.

Contribution

It presents a new prescribed-time safety design that explicitly specifies gains and operates in the full safe set without initial restrictions, unlike traditional methods.

Findings

Successfully enforces safety within a finite time for a double-integrator system.

Operates in the entire original safe set without initial condition restrictions.

Uses explicit gains instead of class K functions for high-relative degree constraints.

Abstract

Safety in dynamical systems is commonly pursued using control barrier functions (CBFs) which enforce safety-constraints over the entire duration of a system's evolution. We propose a prescribed-time safety (PTSf) design which enforces safety only for a finite time of interest to the user. While traditional CBF designs would keep the system away from the barrier longer than necessary, our PTSf design lets the system reach the barrier by the prescribed time and obey the operator's intent thereafter. To emphasize the capability of our design for safety constraints with high relative degrees, we focus our exposition on a chain of integrators where the safety condition is defined for the state furthest from the control input. In contrast to existing CBF-based methods for high-relative degree constraints, our approach involves choosing explicitly specified gains (instead of class $\mathcal{K}$ functions), and, with the aid of backstepping, operates in the entirety of the original safe set with no additional restriction on the initial conditions. With Quadratic Programming (QP) being employed in the design, in addition to backstepping and CBFs with a PTSf property, we refer to our design as a QP-backstepping PT-CBF design. For illustration, we include a simulation for the double-integrator system.